7.7 homework handout geometry answer key

Right Triangles and Trigonometry (Geometry - Unit 7) | All Things Algebra®

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This Right Triangles and Trigonometry Unit Bundle contains guided notes, homework assignments, three quizzes, a study guide and a unit test that cover the following topics:

• Pythagorean Theorem and Applications

• Pythagorean Theorem Converse and Classifying Triangles

• Special Right Triangles: 45-45-90 and 30-60-90

• Similar Right Triangles

• Geometric Mean

• Trigonometric Ratios: Sine, Cosine, and Tangent

• Finding Missing Sides using Trigonometry

• Finding Missing Angles using Trigonometry

• Trigonometry Applications: Angle of Elevation and Angle of Depression

• Law of Sines

• Law of Cosines

• Solving Triangles

• Law of Sines and Law of Cosines Applications

ADDITIONAL COMPONENTS INCLUDED:

(1) Links to Instructional Videos: Links to videos of each lesson in the unit are included. Videos were created by fellow teachers for their students using the guided notes and shared in March 2020 when schools closed with no notice.  Please watch through first before sharing with your students. Many teachers still use these in emergency substitute situations. (2) Editable Assessments: Editable versions of each quiz and the unit test are included. PowerPoint is required to edit these files. Individual problems can be changed to create multiple versions of the assessment. The layout of the assessment itself is not editable. If your Equation Editor is incompatible with mine (I use MathType), simply delete my equation and insert your own.

(3) Google Slides Version of the PDF: The second page of the Video links document contains a link to a Google Slides version of the PDF. Each page is set to the background in Google Slides. There are no text boxes;  this is the PDF in Google Slides.  I am unable to do text boxes at this time but hope this saves you a step if you wish to use it in Slides instead! 

This resource is included in the following bundle(s):

Geometry Second Semester Notes Bundle

Geometry Curriculum

Geometry Curriculum (with Activities)

More Geometry Units:

Unit 1 – Geometry Basics

Unit 2 – Logic and Proof

Unit 3 – Parallel and Perpendicular Lines

Unit 4 – Congruent Triangles

Unit 5 – Relationships in Triangles

Unit 6 – Similar Triangles

Unit 8 – Polygons and Quadrilaterals

Unit 9 – Transformations

Unit 10 – Circles

Unit 11 – Volume and Surface Area Unit 12 – Probability

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COPYRIGHT TERMS: This resource may not be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by students.

© All Things Algebra (Gina Wilson), 2012-present

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Geometry Worksheets(pdf)

Free worksheets with answer keys.

Enjoy these free sheets. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

• Graphic Organizer on All Formulas
• Interior Angles of Polygons
• Exterior Angles of Polygons
• Similar Polygons
• Area of Triangle
• Interior Angles of Triangle

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

Popular pages @ mathwarehouse.com.

7.1 Solving Trigonometric Equations with Identities

sin 2 θ − 1 tan θ sin θ − tan θ = ( sin θ + 1 ) ( sin θ − 1 ) tan θ ( sin θ − 1 ) = sin θ + 1 tan θ sin 2 θ − 1 tan θ sin θ − tan θ = ( sin θ + 1 ) ( sin θ − 1 ) tan θ ( sin θ − 1 ) = sin θ + 1 tan θ

This is a difference of squares formula: 25 − 9 sin 2 θ = ( 5 − 3 sin θ ) ( 5 + 3 sin θ ) . 25 − 9 sin 2 θ = ( 5 − 3 sin θ ) ( 5 + 3 sin θ ) .

7.2 Sum and Difference Identities

2 + 6 4 2 + 6 4

2 − 6 4 2 − 6 4

1 − 3 1 + 3 1 − 3 1 + 3

cos ( 5 π 14 ) cos ( 5 π 14 )

7.3 Double-Angle, Half-Angle, and Reduction Formulas

cos ( 2 α ) = 7 32 cos ( 2 α ) = 7 32

cos 4 θ − sin 4 θ = ( cos 2 θ + sin 2 θ ) ( cos 2 θ − sin 2 θ ) = cos ( 2 θ ) cos 4 θ − sin 4 θ = ( cos 2 θ + sin 2 θ ) ( cos 2 θ − sin 2 θ ) = cos ( 2 θ )

cos ( 2 θ ) cos θ = ( cos 2 θ − sin 2 θ ) cos θ = cos 3 θ − cos θ sin 2 θ cos ( 2 θ ) cos θ = ( cos 2 θ − sin 2 θ ) cos θ = cos 3 θ − cos θ sin 2 θ

10 cos 4 x = 10 cos 4 x = 10 ( cos 2 x ) 2              = 10 [ 1 + cos ( 2 x ) 2 ] 2 Substitute reduction formula for cos 2 x .              = 10 4 [ 1 + 2 cos ( 2 x ) + cos 2 ( 2 x ) ]              = 10 4 + 10 2 cos ( 2 x ) + 10 4 ( 1 + cos 2 ( 2 x ) 2 ) Substitute reduction formula for cos 2 x .              = 10 4 + 10 2 cos ( 2 x ) + 10 8 + 10 8 cos ( 4 x )              = 30 8 + 5 cos ( 2 x ) + 10 8 cos ( 4 x )              = 15 4 + 5 cos ( 2 x ) + 5 4 cos ( 4 x ) 10 cos 4 x = 10 cos 4 x = 10 ( cos 2 x ) 2              = 10 [ 1 + cos ( 2 x ) 2 ] 2 Substitute reduction formula for cos 2 x .              = 10 4 [ 1 + 2 cos ( 2 x ) + cos 2 ( 2 x ) ]              = 10 4 + 10 2 cos ( 2 x ) + 10 4 ( 1 + cos 2 ( 2 x ) 2 ) Substitute reduction formula for cos 2 x .              = 10 4 + 10 2 cos ( 2 x ) + 10 8 + 10 8 cos ( 4 x )              = 30 8 + 5 cos ( 2 x ) + 10 8 cos ( 4 x )              = 15 4 + 5 cos ( 2 x ) + 5 4 cos ( 4 x )

− 2 5 − 2 5

7.4 Sum-to-Product and Product-to-Sum Formulas

1 2 ( cos 6 θ + cos 2 θ ) 1 2 ( cos 6 θ + cos 2 θ )

1 2 ( sin 2 x + sin 2 y ) 1 2 ( sin 2 x + sin 2 y )

− 2 − 3 4 − 2 − 3 4

2 sin ( 2 θ ) cos ( θ ) 2 sin ( 2 θ ) cos ( θ )

tan θ cot θ − cos 2 θ = ( sin θ cos θ ) ( cos θ sin θ ) − cos 2 θ = 1 − cos 2 θ = sin 2 θ tan θ cot θ − cos 2 θ = ( sin θ cos θ ) ( cos θ sin θ ) − cos 2 θ = 1 − cos 2 θ = sin 2 θ

7.5 Solving Trigonometric Equations

x = 7 π 6 , 11 π 6 x = 7 π 6 , 11 π 6

π 3 ± π k π 3 ± π k

θ ≈ 1.7722 ± 2 π k θ ≈ 1.7722 ± 2 π k and θ ≈ 4.5110 ± 2 π k θ ≈ 4.5110 ± 2 π k

cos θ = − 1 , θ = π cos θ = − 1 , θ = π

π 2 , 2 π 3 , 4 π 3 , 3 π 2 π 2 , 2 π 3 , 4 π 3 , 3 π 2

7.6 Modeling with Trigonometric Functions

The amplitude is 3 , 3 , and the period is 2 3 . 2 3 .

y = 8 sin ( π 12 t ) + 32 y = 8 sin ( π 12 t ) + 32 The temperature reaches freezing at noon and at midnight.

initial displacement =6, damping constant = -6, frequency = 2 π 2 π

y = 10 e − 0.5 t cos ( π t ) y = 10 e − 0.5 t cos ( π t )

y = 5 cos ( 6 π t ) y = 5 cos ( 6 π t )

7.1 Section Exercises

All three functions, F F , G G , and H , H , are even.

This is because F ( − x ) = sin ( − x ) sin ( − x ) = ( − sin x ) ( − sin x ) = sin 2 x = F ( x ) , G ( − x ) = cos ( − x ) cos ( − x ) = cos x cos x = cos 2 x = G ( x ) F ( − x ) = sin ( − x ) sin ( − x ) = ( − sin x ) ( − sin x ) = sin 2 x = F ( x ) , G ( − x ) = cos ( − x ) cos ( − x ) = cos x cos x = cos 2 x = G ( x ) and H ( − x ) = tan ( − x ) tan ( − x ) = ( − tan x ) ( − tan x ) = tan 2 x = H ( x ) . H ( − x ) = tan ( − x ) tan ( − x ) = ( − tan x ) ( − tan x ) = tan 2 x = H ( x ) .

When cos t = 0 , cos t = 0 , then sec t = 1 0 , sec t = 1 0 , which is undefined.

sin x sin x

sec x sec x

csc t csc t

sec 2 x sec 2 x

sin 2 x + 1 sin 2 x + 1

1 sin x 1 sin x

1 cot x 1 cot x

tan x tan x

− 4 sec x tan x − 4 sec x tan x

± 1 cot 2 x + 1 ± 1 cot 2 x + 1

± 1 − sin 2 x sin x ± 1 − sin 2 x sin x

Answers will vary. Sample proof:

cos x − cos 3 x = cos x ( 1 − cos 2 x ) cos x − cos 3 x = cos x ( 1 − cos 2 x ) = cos x sin 2 x = cos x sin 2 x

Answers will vary. Sample proof: 1 + sin 2 x cos 2 x = 1 cos 2 x + sin 2 x cos 2 x = sec 2 x + tan 2 x = tan 2 x + 1 + tan 2 x = 1 + 2 tan 2 x 1 + sin 2 x cos 2 x = 1 cos 2 x + sin 2 x cos 2 x = sec 2 x + tan 2 x = tan 2 x + 1 + tan 2 x = 1 + 2 tan 2 x

Answers will vary. Sample proof: cos 2 x − tan 2 x = 1 − sin 2 x − ( sec 2 x − 1 ) = 1 − sin 2 x − sec 2 x + 1 = 2 − sin 2 x − sec 2 x cos 2 x − tan 2 x = 1 − sin 2 x − ( sec 2 x − 1 ) = 1 − sin 2 x − sec 2 x + 1 = 2 − sin 2 x − sec 2 x

Proved with negative and Pythagorean Identities

True 3 sin 2 θ + 4 cos 2 θ = 3 sin 2 θ + 3 cos 2 θ + cos 2 θ = 3 ( sin 2 θ + cos 2 θ ) + cos 2 θ = 3 + cos 2 θ 3 sin 2 θ + 4 cos 2 θ = 3 sin 2 θ + 3 cos 2 θ + cos 2 θ = 3 ( sin 2 θ + cos 2 θ ) + cos 2 θ = 3 + cos 2 θ

7.2 Section Exercises

The cofunction identities apply to complementary angles. Viewing the two acute angles of a right triangle, if one of those angles measures x , x , the second angle measures π 2 − x . π 2 − x . Then sin x = cos ( π 2 − x ) . sin x = cos ( π 2 − x ) . The same holds for the other cofunction identities. The key is that the angles are complementary.

sin ( − x ) = − sin x , sin ( − x ) = − sin x , so sin x sin x is odd. cos ( − x ) = cos ( 0 − x ) = cos x , cos ( − x ) = cos ( 0 − x ) = cos x , so cos x cos x is even.

6 − 2 4 6 − 2 4

− 2 − 3 − 2 − 3

− 2 2 sin x − 2 2 cos x − 2 2 sin x − 2 2 cos x

− 1 2 cos x − 3 2 sin x − 1 2 cos x − 3 2 sin x

csc θ csc θ

cot x cot x

tan ( x 10 ) tan ( x 10 )

sin ( a − b ) = ( 4 5 ) ( 1 3 ) − ( 3 5 ) ( 2 2 3 ) = 4 − 6 2 15 sin ( a − b ) = ( 4 5 ) ( 1 3 ) − ( 3 5 ) ( 2 2 3 ) = 4 − 6 2 15 cos ( a + b ) = ( 3 5 ) ( 1 3 ) − ( 4 5 ) ( 2 2 3 ) = 3 − 8 2 15 cos ( a + b ) = ( 3 5 ) ( 1 3 ) − ( 4 5 ) ( 2 2 3 ) = 3 − 8 2 15

cot ( π 6 − x ) cot ( π 6 − x )

cot ( π 4 + x ) cot ( π 4 + x )

sin x 2 + cos x 2 sin x 2 + cos x 2

They are the same.

They are the different, try g ( x ) = sin ( 9 x ) − cos ( 3 x ) sin ( 6 x ) . g ( x ) = sin ( 9 x ) − cos ( 3 x ) sin ( 6 x ) .

They are the different, try g ( θ ) = 2 tan θ 1 − tan 2 θ . g ( θ ) = 2 tan θ 1 − tan 2 θ .

They are different, try g ( x ) = tan x − tan ( 2 x ) 1 + tan x tan ( 2 x ) . g ( x ) = tan x − tan ( 2 x ) 1 + tan x tan ( 2 x ) .

− 3 − 1 2 2 ,  or  − 0.2588 − 3 − 1 2 2 ,  or  − 0.2588

1 + 3 2 2 , 1 + 3 2 2 , or 0.9659

tan ( x + π 4 ) = tan x + tan ( π 4 ) 1 − tan x tan ( π 4 ) = tan x + 1 1 − tan x ( 1 ) = tan x + 1 1 − tan x tan ( x + π 4 ) = tan x + tan ( π 4 ) 1 − tan x tan ( π 4 ) = tan x + 1 1 − tan x ( 1 ) = tan x + 1 1 − tan x

cos ( a + b ) cos a cos b = cos a cos b cos a cos b − sin a sin b cos a cos b = 1 − tan a tan b cos ( a + b ) cos a cos b = cos a cos b cos a cos b − sin a sin b cos a cos b = 1 − tan a tan b

cos ( x + h ) − cos x h = cos x cosh − sin x sinh − cos x h = cos x ( cosh − 1 ) − sin x sinh h = cos x cos h − 1 h − sin x sin h h cos ( x + h ) − cos x h = cos x cosh − sin x sinh − cos x h = cos x ( cosh − 1 ) − sin x sinh h = cos x cos h − 1 h − sin x sin h h

True. Note that sin ( α + β ) = sin ( π − γ ) sin ( α + β ) = sin ( π − γ ) and expand the right hand side.

7.3 Section Exercises

Use the Pythagorean identities and isolate the squared term.

1 − cos x sin x , sin x 1 + cos x , 1 − cos x sin x , sin x 1 + cos x , multiplying the top and bottom by 1 − cos x 1 − cos x and 1 + cos x , 1 + cos x , respectively.

a) 3 7 32 3 7 32 b) 31 32 31 32 c) 3 7 31 3 7 31

a) 3 2 3 2 b) − 1 2 − 1 2 c) − 3 − 3

cos θ = − 2 5 5 , sin θ = 5 5 , tan θ = − 1 2 , csc θ = 5 , sec θ = − 5 2 , cot θ = − 2 cos θ = − 2 5 5 , sin θ = 5 5 , tan θ = − 1 2 , csc θ = 5 , sec θ = − 5 2 , cot θ = − 2

2 sin ( π 2 ) 2 sin ( π 2 )

2 − 2 2 2 − 2 2

2 − 3 2 2 − 3 2

2 + 3 2 + 3

− 1 − 2 − 1 − 2

a) 3 13 13 3 13 13 b) − 2 13 13 − 2 13 13 c) − 3 2 − 3 2

a) 10 4 10 4 b) 6 4 6 4 c) 15 3 15 3

120 169 , – 119 169 , – 120 119 120 169 , – 119 169 , – 120 119

2 13 13 , 3 13 13 , 2 3 2 13 13 , 3 13 13 , 2 3

cos ( 74 ∘ ) cos ( 74 ∘ )

cos ( 18 x ) cos ( 18 x )

3 sin ( 10 x ) 3 sin ( 10 x )

− 2 sin ( − x ) cos ( − x ) = − 2 ( − sin ( x ) cos ( x ) ) = sin ( 2 x ) − 2 sin ( − x ) cos ( − x ) = − 2 ( − sin ( x ) cos ( x ) ) = sin ( 2 x )

sin ( 2 θ ) 1 + cos ( 2 θ ) tan 2 θ = 2 sin ( θ ) cos ( θ ) 1 + cos 2 θ − sin 2 θ tan 2 θ = 2 sin ( θ ) cos ( θ ) 2 cos 2 θ tan 2 θ = sin ( θ ) cos θ tan 2 θ = tan θ tan 2 θ = tan 3 θ sin ( 2 θ ) 1 + cos ( 2 θ ) tan 2 θ = 2 sin ( θ ) cos ( θ ) 1 + cos 2 θ − sin 2 θ tan 2 θ = 2 sin ( θ ) cos ( θ ) 2 cos 2 θ tan 2 θ = sin ( θ ) cos θ tan 2 θ = tan θ tan 2 θ = tan 3 θ

1 + cos ( 12 x ) 2 1 + cos ( 12 x ) 2

3 + cos ( 12 x ) − 4 cos ( 6 x ) 8 3 + cos ( 12 x ) − 4 cos ( 6 x ) 8

2 + cos ( 2 x ) − 2 cos ( 4 x ) − cos ( 6 x ) 32 2 + cos ( 2 x ) − 2 cos ( 4 x ) − cos ( 6 x ) 32

3 + cos ( 4 x ) − 4 cos ( 2 x ) 3 + cos ( 4 x ) + 4 cos ( 2 x ) 3 + cos ( 4 x ) − 4 cos ( 2 x ) 3 + cos ( 4 x ) + 4 cos ( 2 x )

1 − cos ( 4 x ) 8 1 − cos ( 4 x ) 8

3 + cos ( 4 x ) − 4 cos ( 2 x ) 4 ( cos ( 2 x ) + 1 ) 3 + cos ( 4 x ) − 4 cos ( 2 x ) 4 ( cos ( 2 x ) + 1 )

( 1 + cos ( 4 x ) ) sin x 2 ( 1 + cos ( 4 x ) ) sin x 2

4 sin x cos x ( cos 2 x − sin 2 x ) 4 sin x cos x ( cos 2 x − sin 2 x )

2 tan x 1 + tan 2 x = 2 sin x cos x 1 + sin 2 x cos 2 x = 2 sin x cos x cos 2 x + sin 2 x cos 2 x = 2 tan x 1 + tan 2 x = 2 sin x cos x 1 + sin 2 x cos 2 x = 2 sin x cos x cos 2 x + sin 2 x cos 2 x = 2 sin x cos x . cos 2 x 1 = 2 sin x cos x = sin ( 2 x ) 2 sin x cos x . cos 2 x 1 = 2 sin x cos x = sin ( 2 x )

2 sin x cos x 2 cos 2 x − 1 = sin ( 2 x ) cos ( 2 x ) = tan ( 2 x ) 2 sin x cos x 2 cos 2 x − 1 = sin ( 2 x ) cos ( 2 x ) = tan ( 2 x )

sin ( x + 2 x ) = sin x cos ( 2 x ) + sin ( 2 x ) cos x = sin x ( cos 2 x − sin 2 x ) + 2 sin x cos x cos x = sin x cos 2 x − sin 3 x + 2 sin x cos 2 x = 3 sin x cos 2 x − sin 3 x sin ( x + 2 x ) = sin x cos ( 2 x ) + sin ( 2 x ) cos x = sin x ( cos 2 x − sin 2 x ) + 2 sin x cos x cos x = sin x cos 2 x − sin 3 x + 2 sin x cos 2 x = 3 sin x cos 2 x − sin 3 x

1 + cos ( 2 t ) sin ( 2 t ) − cos t = 1 + 2 cos 2 t − 1 2 sin t cos t − cos t = 2 cos 2 t cos t ( 2 sin t − 1 ) = 2 cos t 2 sin t − 1 1 + cos ( 2 t ) sin ( 2 t ) − cos t = 1 + 2 cos 2 t − 1 2 sin t cos t − cos t = 2 cos 2 t cos t ( 2 sin t − 1 ) = 2 cos t 2 sin t − 1

( cos 2 ( 4 x ) − sin 2 ( 4 x ) − sin ( 8 x ) ) ( cos 2 ( 4 x ) − sin 2 ( 4 x ) + sin ( 8 x ) ) = = ( cos ( 8 x ) − sin ( 8 x ) ) ( cos ( 8 x ) + sin ( 8 x ) ) = cos 2 ( 8 x ) − sin 2 ( 8 x ) = cos ( 16 x ) ( cos 2 ( 4 x ) − sin 2 ( 4 x ) − sin ( 8 x ) ) ( cos 2 ( 4 x ) − sin 2 ( 4 x ) + sin ( 8 x ) ) = = ( cos ( 8 x ) − sin ( 8 x ) ) ( cos ( 8 x ) + sin ( 8 x ) ) = cos 2 ( 8 x ) − sin 2 ( 8 x ) = cos ( 16 x )

7.4 Section Exercises

Substitute α α into cosine and β β into sine and evaluate.

Answers will vary. There are some equations that involve a sum of two trig expressions where when converted to a product are easier to solve. For example: sin ( 3 x ) + sin x cos x = 1. sin ( 3 x ) + sin x cos x = 1. When converting the numerator to a product the equation becomes: 2 sin ( 2 x ) cos x cos x = 1 2 sin ( 2 x ) cos x cos x = 1

8 ( cos ( 5 x ) − cos ( 27 x ) ) 8 ( cos ( 5 x ) − cos ( 27 x ) )

sin ( 2 x ) + sin ( 8 x ) sin ( 2 x ) + sin ( 8 x )

1 2 ( cos ( 6 x ) − cos ( 4 x ) ) 1 2 ( cos ( 6 x ) − cos ( 4 x ) )

2 cos ( 5 t ) cos t 2 cos ( 5 t ) cos t

2 cos ( 7 x ) 2 cos ( 7 x )

2 cos ( 6 x ) cos ( 3 x ) 2 cos ( 6 x ) cos ( 3 x )

1 4 ( 1 + 3 ) 1 4 ( 1 + 3 )

1 4 ( 3 − 2 ) 1 4 ( 3 − 2 )

1 4 ( 3 − 1 ) 1 4 ( 3 − 1 )

cos ( 80° ) − cos ( 120° ) cos ( 80° ) − cos ( 120° )

1 2 ( sin ( 221° ) + sin ( 205° ) ) 1 2 ( sin ( 221° ) + sin ( 205° ) )

2 cos ( 31° ) 2 cos ( 31° )

2 cos ( 66.5 ° ) sin ( 34.5 ° ) 2 cos ( 66.5 ° ) sin ( 34.5 ° )

2 sin ( −1.5° ) cos ( 0.5° ) 2 sin ( −1.5° ) cos ( 0.5° )

2 sin ( 7 x ) − 2 sin x = 2 sin ( 4 x + 3 x ) − 2 sin ( 4 x − 3 x ) = 2 ( sin ( 4 x ) cos ( 3 x ) + sin ( 3 x ) cos ( 4 x ) ) − 2 ( sin ( 4 x ) cos ( 3 x ) − sin ( 3 x ) cos ( 4 x ) ) = 2 sin ( 4 x ) cos ( 3 x ) + 2 sin ( 3 x ) cos ( 4 x ) ) − 2 sin ( 4 x ) cos ( 3 x ) + 2 sin ( 3 x ) cos ( 4 x ) ) = 4 sin ( 3 x ) cos ( 4 x ) 2 sin ( 7 x ) − 2 sin x = 2 sin ( 4 x + 3 x ) − 2 sin ( 4 x − 3 x ) = 2 ( sin ( 4 x ) cos ( 3 x ) + sin ( 3 x ) cos ( 4 x ) ) − 2 ( sin ( 4 x ) cos ( 3 x ) − sin ( 3 x ) cos ( 4 x ) ) = 2 sin ( 4 x ) cos ( 3 x ) + 2 sin ( 3 x ) cos ( 4 x ) ) − 2 sin ( 4 x ) cos ( 3 x ) + 2 sin ( 3 x ) cos ( 4 x ) ) = 4 sin ( 3 x ) cos ( 4 x )

sin x + sin ( 3 x ) = 2 sin ( 4 x 2 ) cos ( − 2 x 2 ) = sin x + sin ( 3 x ) = 2 sin ( 4 x 2 ) cos ( − 2 x 2 ) = 2 sin ( 2 x ) cos x = 2 ( 2 sin x cos x ) cos x = 2 sin ( 2 x ) cos x = 2 ( 2 sin x cos x ) cos x = 4 sin x cos 2 x 4 sin x cos 2 x

2 tan x cos ( 3 x ) = 2 sin x cos ( 3 x ) cos x = 2 ( .5 ( sin ( 4 x ) − sin ( 2 x ) ) ) cos x 2 tan x cos ( 3 x ) = 2 sin x cos ( 3 x ) cos x = 2 ( .5 ( sin ( 4 x ) − sin ( 2 x ) ) ) cos x = 1 cos x ( sin ( 4 x ) − sin ( 2 x ) ) = sec x ( sin ( 4 x ) − sin ( 2 x ) ) = 1 cos x ( sin ( 4 x ) − sin ( 2 x ) ) = sec x ( sin ( 4 x ) − sin ( 2 x ) )

2 cos ( 35 ∘ ) cos ( 23 ∘ ) ,  1 .5081 2 cos ( 35 ∘ ) cos ( 23 ∘ ) ,  1 .5081

− 2 sin ( 33 ∘ ) sin ( 11 ∘ ) , − 0.2078 − 2 sin ( 33 ∘ ) sin ( 11 ∘ ) , − 0.2078

1 2 ( cos ( 99 ∘ ) − cos ( 71 ∘ ) ) , − 0.2410 1 2 ( cos ( 99 ∘ ) − cos ( 71 ∘ ) ) , − 0.2410

It is and identity.

It is not an identity, but 2 cos 3 x 2 cos 3 x is.

tan ( 3 t ) tan ( 3 t )

2 cos ( 2 x ) 2 cos ( 2 x )

− sin ( 14 x ) − sin ( 14 x )

Start with cos x + cos y . cos x + cos y . Make a substitution and let x = α + β x = α + β and let y = α − β , y = α − β , so cos x + cos y cos x + cos y becomes cos ( α + β ) + cos ( α − β ) = cos α cos β − sin α sin β + cos α cos β + sin α sin β = 2 cos α cos β cos ( α + β ) + cos ( α − β ) = cos α cos β − sin α sin β + cos α cos β + sin α sin β = 2 cos α cos β

Since x = α + β x = α + β and y = α − β , y = α − β , we can solve for α α and β β in terms of x and y and substitute in for 2 cos α cos β 2 cos α cos β and get 2 cos ( x + y 2 ) cos ( x − y 2 ) . 2 cos ( x + y 2 ) cos ( x − y 2 ) .

cos ( 3 x ) + cos x cos ( 3 x ) − cos x = 2 cos ( 2 x ) cos x − 2 sin ( 2 x ) sin x = − cot ( 2 x ) cot x cos ( 3 x ) + cos x cos ( 3 x ) − cos x = 2 cos ( 2 x ) cos x − 2 sin ( 2 x ) sin x = − cot ( 2 x ) cot x

cos ( 2 y ) − cos ( 4 y ) sin ( 2 y ) + sin ( 4 y ) = − 2 sin ( 3 y ) sin ( − y ) 2 sin ( 3 y ) cos y = 2 sin ( 3 y ) sin ( y ) 2 sin ( 3 y ) cos y = tan y cos ( 2 y ) − cos ( 4 y ) sin ( 2 y ) + sin ( 4 y ) = − 2 sin ( 3 y ) sin ( − y ) 2 sin ( 3 y ) cos y = 2 sin ( 3 y ) sin ( y ) 2 sin ( 3 y ) cos y = tan y

cos x − cos ( 3 x ) = − 2 sin ( 2 x ) sin ( − x ) = 2 ( 2 sin x cos x ) sin x = 4 sin 2 x cos x cos x − cos ( 3 x ) = − 2 sin ( 2 x ) sin ( − x ) = 2 ( 2 sin x cos x ) sin x = 4 sin 2 x cos x

tan ( π 4 − t ) = tan ( π 4 ) − tan t 1 + tan ( π 4 ) tan ( t ) = 1 − tan t 1 + tan t tan ( π 4 − t ) = tan ( π 4 ) − tan t 1 + tan ( π 4 ) tan ( t ) = 1 − tan t 1 + tan t

7.5 Section Exercises

There will not always be solutions to trigonometric function equations. For a basic example, cos ( x ) = −5. cos ( x ) = −5.

If the sine or cosine function has a coefficient of one, isolate the term on one side of the equals sign. If the number it is set equal to has an absolute value less than or equal to one, the equation has solutions, otherwise it does not. If the sine or cosine does not have a coefficient equal to one, still isolate the term but then divide both sides of the equation by the leading coefficient. Then, if the number it is set equal to has an absolute value greater than one, the equation has no solution.

π 3 , 2 π 3 π 3 , 2 π 3

3 π 4 , 5 π 4 3 π 4 , 5 π 4

π 4 , 5 π 4 π 4 , 5 π 4

π 4 , 3 π 4 , 5 π 4 , 7 π 4 π 4 , 3 π 4 , 5 π 4 , 7 π 4

π 4 , 7 π 4 π 4 , 7 π 4

7 π 6 , 11 π 6 7 π 6 , 11 π 6

π 18 π 18 , 5 π 18 5 π 18 , 13 π 18 13 π 18 , 17 π 18 17 π 18 , 25 π 18 25 π 18 , 29 π 18 29 π 18

3 π 12 , 5 π 12 , 11 π 12 , 13 π 12 , 19 π 12 , 21 π 12 3 π 12 , 5 π 12 , 11 π 12 , 13 π 12 , 19 π 12 , 21 π 12

1 6 , 5 6 , 13 6 , 17 6 , 25 6 , 29 6 , 37 6 1 6 , 5 6 , 13 6 , 17 6 , 25 6 , 29 6 , 37 6

0 , π 3 , π , 5 π 3 0 , π 3 , π , 5 π 3

π 3 , π , 5 π 3 π 3 , π , 5 π 3

π 3 , 3 π 2 , 5 π 3 π 3 , 3 π 2 , 5 π 3

0 , π 0 , π

π − sin − 1 ( − 1 4 ) , 7 π 6 , 11 π 6 , 2 π + sin − 1 ( − 1 4 ) π − sin − 1 ( − 1 4 ) , 7 π 6 , 11 π 6 , 2 π + sin − 1 ( − 1 4 )

1 3 ( sin − 1 ( 9 10 ) ) , π 3 − 1 3 ( sin − 1 ( 9 10 ) ) , 2 π 3 + 1 3 ( sin − 1 ( 9 10 ) ) , π − 1 3 ( sin − 1 ( 9 10 ) ) , 4 π 3 + 1 3 ( sin − 1 ( 9 10 ) ) , 5 π 3 − 1 3 ( sin − 1 ( 9 10 ) ) 1 3 ( sin − 1 ( 9 10 ) ) , π 3 − 1 3 ( sin − 1 ( 9 10 ) ) , 2 π 3 + 1 3 ( sin − 1 ( 9 10 ) ) , π − 1 3 ( sin − 1 ( 9 10 ) ) , 4 π 3 + 1 3 ( sin − 1 ( 9 10 ) ) , 5 π 3 − 1 3 ( sin − 1 ( 9 10 ) )

θ = sin - 1 2 3 , π - sin - 1 2 3 , π + sin - 1 2 3 , 2 π - sin - 1 2 3 θ = sin - 1 2 3 , π - sin - 1 2 3 , π + sin - 1 2 3 , 2 π - sin - 1 2 3

3 π 2 , π 6 , 5 π 6 3 π 2 , π 6 , 5 π 6

0 , π 3 , π , 4 π 3 0 , π 3 , π , 4 π 3

There are no solutions.

cos − 1 ( 1 3 ( 1 − 7 ) ) , 2 π − cos − 1 ( 1 3 ( 1 − 7 ) ) cos − 1 ( 1 3 ( 1 − 7 ) ) , 2 π − cos − 1 ( 1 3 ( 1 − 7 ) )

tan − 1 ( 1 2 ( 29 − 5 ) ) , π + tan − 1 ( 1 2 ( − 29 − 5 ) ) , π + tan − 1 ( 1 2 ( 29 − 5 ) ) , 2 π + tan − 1 ( 1 2 ( − 29 − 5 ) ) tan − 1 ( 1 2 ( 29 − 5 ) ) , π + tan − 1 ( 1 2 ( − 29 − 5 ) ) , π + tan − 1 ( 1 2 ( 29 − 5 ) ) , 2 π + tan − 1 ( 1 2 ( − 29 − 5 ) )

0 , 2 π 3 , 4 π 3 0 , 2 π 3 , 4 π 3

sin − 1 ( 3 5 ) , π 2 , π − sin − 1 ( 3 5 ) , 3 π 2 sin − 1 ( 3 5 ) , π 2 , π − sin − 1 ( 3 5 ) , 3 π 2

cos − 1 ( − 1 4 ) cos − 1 ( − 1 4 ) , 2 π − cos − 1 ( − 1 4 ) 2 π − cos − 1 ( − 1 4 )

π 3 , cos − 1 ( − 3 4 ) , 2 π − cos − 1 ( − 3 4 ) , 5 π 3 π 3 , cos − 1 ( − 3 4 ) , 2 π − cos − 1 ( − 3 4 ) , 5 π 3

cos − 1 ( 3 4 ) , cos − 1 ( − 2 3 ) , 2 π − cos − 1 ( − 2 3 ) cos − 1 ( 3 4 ) , cos − 1 ( − 2 3 ) , 2 π − cos − 1 ( − 2 3 ) , 2 π − cos − 1 ( 3 4 ) 2 π − cos − 1 ( 3 4 )

0 , π 2 , π , 3 π 2 0 , π 2 , π , 3 π 2

π 3 , cos −1 ( − 1 4 ) , 2 π − cos −1 ( − 1 4 ) , 5 π 3 π 3 , cos −1 ( − 1 4 ) , 2 π − cos −1 ( − 1 4 ) , 5 π 3

π + tan −1 ( −2 ) π + tan −1 ( −2 ) , π + tan −1 ( − 3 2 ) , 2 π + tan −1 ( −2 ) , 2 π + tan −1 ( − 3 2 ) π + tan −1 ( − 3 2 ) , 2 π + tan −1 ( −2 ) , 2 π + tan −1 ( − 3 2 )

2 π k + 0.2734 , 2 π k + 2.8682 2 π k + 0.2734 , 2 π k + 2.8682

π k − 0.3277 π k − 0.3277

0.6694 , 1.8287 , 3.8110 , 4.9703 0.6694 , 1.8287 , 3.8110 , 4.9703

1.0472 , 3.1416 , 5.2360 1.0472 , 3.1416 , 5.2360

0.5326 , 1.7648 , 3.6742 , 4.9064 0.5326 , 1.7648 , 3.6742 , 4.9064

sin − 1 ( 1 4 ) , π − sin − 1 ( 1 4 ) , 3 π 2 sin − 1 ( 1 4 ) , π − sin − 1 ( 1 4 ) , 3 π 2

π 2 , 3 π 2 π 2 , 3 π 2

7.2 ∘ 7.2 ∘

5.7 ∘ 5.7 ∘

82.4 ∘ 82.4 ∘

31.0 ∘ 31.0 ∘

88.7 ∘ 88.7 ∘

59.0 ∘ 59.0 ∘

36.9 ∘ 36.9 ∘

7.6 Section Exercises

Physical behavior should be periodic, or cyclical.

Since cumulative rainfall is always increasing, a sinusoidal function would not be ideal here.

y = − 3 cos ( π 6 x ) − 1 y = − 3 cos ( π 6 x ) − 1

5 sin ( 2 x ) + 2 5 sin ( 2 x ) + 2

y = 4 - 6 cos ( x π 2 ) y = 4 - 6 cos ( x π 2 )

y = tan ( x π 8 ) y = tan ( x π 8 )

tan ( x π 12 ) tan ( x π 12 )

From June 15 through November 16

From day 31 through day 58

Floods: April 16 to July 15. Drought: October 16 to January 15.

Amplitude: 8, period: 1 3 , 1 3 , frequency: 3 Hz

Amplitude: 4, period: 4 , 4 , frequency: 1 4 1 4 Hz

P ( t ) = − 19 cos ( π 6 t ) + 800 + 160 12 t P ( t ) = − 19 cos ( π 6 t ) + 800 + 40 3 t P ( t ) = − 19 cos ( π 6 t ) + 800 + 160 12 t P ( t ) = − 19 cos ( π 6 t ) + 800 + 40 3 t

P ( t ) = − 33 cos ( π 6 t ) + 900 + ( 1.07 ) t P ( t ) = − 33 cos ( π 6 t ) + 900 + ( 1.07 ) t

D ( t ) = 10 ( 0.85 ) t cos ( 36 π t ) D ( t ) = 10 ( 0.85 ) t cos ( 36 π t )

D ( t ) = 17 ( 0.9145 ) t cos ( 28 π t ) D ( t ) = 17 ( 0.9145 ) t cos ( 28 π t )

15.4 seconds

Spring 2 comes to rest first after 7.3 seconds.

234.3 miles, at 72.2°

y = 6 ( 4 ) x + 5 sin ( π 2 x ) y = 6 ( 4 ) x + 5 sin ( π 2 x )

y = 4 ( – 2 ) x + 8 sin ( π 2 x ) y = 4 ( – 2 ) x + 8 sin ( π 2 x )

y = 3 ( 2 ) x cos ( π 2 x ) + 1 y = 3 ( 2 ) x cos ( π 2 x ) + 1

Review Exercises

sin − 1 ( 3 3 ) , π − sin − 1 ( 3 3 ) , π + sin − 1 ( 3 3 ) , 2 π − sin − 1 ( 3 3 ) sin − 1 ( 3 3 ) , π − sin − 1 ( 3 3 ) , π + sin − 1 ( 3 3 ) , 2 π − sin − 1 ( 3 3 )

sin − 1 ( 1 4 ) , π − sin − 1 ( 1 4 ) sin − 1 ( 1 4 ) , π − sin − 1 ( 1 4 )

cos ( 4 x ) − cos ( 3 x ) cos x = cos ( 2 x + 2 x ) − cos ( x + 2 x ) cos x                                    = cos ( 2 x ) cos ( 2 x ) − sin ( 2 x ) sin ( 2 x ) − cos x cos ( 2 x ) cos x + sin x sin ( 2 x ) cos x                                    = ( cos 2 x − sin 2 x ) 2 − 4 cos 2 x sin 2 x − cos 2 x ( cos 2 x − sin 2 x ) + sin x ( 2 ) sin x cos x cos x                                    = ( cos 2 x − sin 2 x ) 2 − 4 cos 2 x sin 2 x − cos 2 x ( cos 2 x − sin 2 x ) + 2 sin 2 x cos 2 x                                    = cos 4 x − 2 cos 2 x sin 2 x + sin 4 x − 4 cos 2 x sin 2 x − cos 4 x + cos 2 x sin 2 x + 2 sin 2 x cos 2 x                                    = sin 4 x − 4 cos 2 x sin 2 x + cos 2 x sin 2 x                                    = sin 2 x ( sin 2 x + cos 2 x ) − 4 cos 2 x sin 2 x                                    = sin 2 x − 4 cos 2 x sin 2 x cos ( 4 x ) − cos ( 3 x ) cos x = cos ( 2 x + 2 x ) − cos ( x + 2 x ) cos x                                    = cos ( 2 x ) cos ( 2 x ) − sin ( 2 x ) sin ( 2 x ) − cos x cos ( 2 x ) cos x + sin x sin ( 2 x ) cos x                                    = ( cos 2 x − sin 2 x ) 2 − 4 cos 2 x sin 2 x − cos 2 x ( cos 2 x − sin 2 x ) + sin x ( 2 ) sin x cos x cos x                                    = ( cos 2 x − sin 2 x ) 2 − 4 cos 2 x sin 2 x − cos 2 x ( cos 2 x − sin 2 x ) + 2 sin 2 x cos 2 x                                    = cos 4 x − 2 cos 2 x sin 2 x + sin 4 x − 4 cos 2 x sin 2 x − cos 4 x + cos 2 x sin 2 x + 2 sin 2 x cos 2 x                                    = sin 4 x − 4 cos 2 x sin 2 x + cos 2 x sin 2 x                                    = sin 2 x ( sin 2 x + cos 2 x ) − 4 cos 2 x sin 2 x                                    = sin 2 x − 4 cos 2 x sin 2 x

tan ( 5 8 x ) tan ( 5 8 x )

− 24 25 , − 7 25 , 24 7 − 24 25 , − 7 25 , 24 7

2 ( 2 + 2 ) 2 ( 2 + 2 )

2 10 , 7 2 10 , 1 7 , 3 5 , 4 5 , 3 4 2 10 , 7 2 10 , 1 7 , 3 5 , 4 5 , 3 4

cot x cos ( 2 x ) = cot x ( 1 − 2 sin 2 x )                      = cot x − cos x sin x ( 2 ) sin 2 x                      = − 2 sin x cos x + cot x                      = − sin ( 2 x ) + cot x cot x cos ( 2 x ) = cot x ( 1 − 2 sin 2 x )                      = cot x − cos x sin x ( 2 ) sin 2 x                      = − 2 sin x cos x + cot x                      = − sin ( 2 x ) + cot x

10 sin x − 5 sin ( 3 x ) + sin ( 5 x ) 8 ( cos ( 2 x ) + 1 ) 10 sin x − 5 sin ( 3 x ) + sin ( 5 x ) 8 ( cos ( 2 x ) + 1 )

− 2 2 − 2 2

1 2 ( sin ( 6 x ) + sin ( 12 x ) ) 1 2 ( sin ( 6 x ) + sin ( 12 x ) )

2 sin ( 13 2 x ) cos ( 9 2 x ) 2 sin ( 13 2 x ) cos ( 9 2 x )

3 π 4 , 7 π 4 3 π 4 , 7 π 4

0 , π 6 , 5 π 6 , π 0 , π 6 , 5 π 6 , π

3 π 2 3 π 2

No solution

0.2527 , 2.8889 , 4.7124 0.2527 , 2.8889 , 4.7124

1.3694 1.3694 , 1.9106 1.9106 , 4.3726 4.3726 , 4.9137 4.9137

3 sin ( x π 2 ) − 2 3 sin ( x π 2 ) − 2

71.6 ∘ 71.6 ∘

P ( t ) = 950 − 450 sin ( π 6 t ) P ( t ) = 950 − 450 sin ( π 6 t )

Amplitude: 3, period: 2, frequency: 1 2 1 2 Hz

C ( t ) = 20 sin ( 2 π t ) + 100 ( 1.4427 ) t C ( t ) = 20 sin ( 2 π t ) + 100 ( 1.4427 ) t

Practice Test

π 2 , 3π 2 π 2 , 3π 2

2 cos ( 3 x ) cos ( 5 x ) 2 cos ( 3 x ) cos ( 5 x )

x = cos –1 ( 1 5 ) x = cos –1 ( 1 5 )

3 5 , − 4 5 , − 3 4 3 5 , − 4 5 , − 3 4

tan 3 x – tan x sec 2 x = tan x ( tan 2 x – sec 2 x ) = tan x ( tan 2 x – ( 1 + tan 2 x ) ) = tan x ( tan 2 x – 1 – tan 2 x ) = – tan x = tan ( – x ) = tan – x ) tan 3 x – tan x sec 2 x = tan x ( tan 2 x – sec 2 x ) = tan x ( tan 2 x – ( 1 + tan 2 x ) ) = tan x ( tan 2 x – 1 – tan 2 x ) = – tan x = tan ( – x ) = tan – x )

sin ( 2 x ) sin x – cos ( 2 x ) cos x = 2 sin x cos x sin x – 2 cos 2 x – 1 cos x = 2 cos x – 2 cos x + 1 cos x = 1 cos x = sec x = sec x sin ( 2 x ) sin x – cos ( 2 x ) cos x = 2 sin x cos x sin x – 2 cos 2 x – 1 cos x = 2 cos x – 2 cos x + 1 cos x = 1 cos x = sec x = sec x

Amplitude: 1 4 1 4 , period 1 60 1 60 , frequency: 60 Hz

Amplitude: 8 8 , fast period: 1 500 1 500 , fast frequency: 500 Hz, slow period: 1 10 1 10 , slow frequency: 10 Hz

D ( t ) = 20 ( 0.9086 ) t cos ( 4 π t ) D ( t ) = 20 ( 0.9086 ) t cos ( 4 π t ) , 31 seconds

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• Book title: Precalculus
• Publication date: Oct 23, 2014
• Location: Houston, Texas
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• Section URL: https://openstax.org/books/precalculus/pages/chapter-7

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Answer Key 7.4

• [latex]-21\times 2=-42[/latex] [latex]-21+2=-19[/latex] [latex]7x^2-21x+2x-6[/latex] [latex]7x(x-3)+2(x-3)[/latex] [latex](x-3)(7x+2)[/latex]
• [latex]-6\times 4=-24[/latex] [latex]-6+4=-2[/latex] [latex]3n^2-6n+4n-8[/latex] [latex]3n(n-2)+4(n-2)[/latex] [latex](n-2)(3n+4)[/latex]
• [latex]14\times 1=14[/latex] [latex]14+1=15[/latex] [latex]7b^2+14b+b+2[/latex] [latex]7b(b+2)+1(b+2)[/latex] [latex](b+2)(7b+1)[/latex]
• [latex]-14\times 3=-42[/latex] [latex]-14+3=-11[/latex] [latex]21v^2-14v+3v-2[/latex] [latex]7v(3v-2)+1(3v-2)[/latex] [latex](3v-2)(7v+1)[/latex]
• [latex]15\times -2=-30[/latex] [latex]15+-2=13[/latex] [latex]5a^2+15a-2a-6[/latex] [latex]5a(a+3)-2(a+3)[/latex] [latex](a+3)(5a-2)[/latex]
• [latex]-20\times 2=-40[/latex] [latex]-20+2=-18[/latex] [latex]5n^2-20n+2n-8[/latex] [latex]5n(n-4)+2(n-4)[/latex] [latex](n-4)(5n+2)[/latex]
• [latex]-1\times -4=4[/latex] [latex]-1+-4=-5[/latex] [latex]2x^2-x-4x+2[/latex] [latex]x(2x-1)-2(2x-1)[/latex] [latex](2x-1)(x-2)[/latex]
• [latex]-6\times 2=-12[/latex] [latex]-6+2=-4[/latex] [latex]3r^2-6r+2r-4[/latex] [latex]3r(r-2)+2(r-2)[/latex] [latex](r-2)(3r+2)[/latex]
• [latex]14\times 5=70[/latex] [latex]14+5=19[/latex] [latex]2x^2+14x+5x+35[/latex] [latex]2x(x+7)+5(x+7)[/latex] [latex](x+7)(2x+5)[/latex]
• [latex]9\times -5=-45[/latex] [latex]9+-5=4[/latex] [latex]3x^2+9x-5x-15[/latex] [latex]3x(x+3)-5(x+3)[/latex] [latex](x+3)(3x-5)[/latex]
• [latex]-3\times 2=-6[/latex] [latex]-3+2=-1[/latex] [latex]2b^2-3b+2b-3[/latex] [latex]b(2b-3)+1(2b-3)[/latex] [latex](2b-3)(b+1)[/latex]
• [latex]8\times -3=-24[/latex] [latex]8+-3=5[/latex] [latex]2k^2+8k-3k-12[/latex] [latex]2k(k+4)-3(k+4)[/latex] [latex](k+4)(2k-3)[/latex]
• [latex]15\times 2=30[/latex] [latex]15+2=17[/latex] [latex]3x^2+15xy+2xy+10y^2[/latex] [latex]3x(x+5y)+2y(x+5y)[/latex] [latex](x+5y)(3x+2y)[/latex]
• [latex]-7\times 5=-35[/latex] [latex]-7+5=-2[/latex] [latex]7x^2-7xy+5xy-5y^2[/latex] [latex]7x(x-y)+5y(x-y)[/latex] [latex](x-y)(7x+5y)[/latex]
• [latex]15\times -4=-60[/latex] [latex]15+-4=11[/latex] [latex]3x^2+15xy-4xy-20y^2[/latex] [latex]3x(x+5y)-4y(x+5y)[/latex] [latex](x+5y)(3x-4y)[/latex]
• [latex]18\times -2=-36[/latex] [latex]18+-2=16[/latex] [latex]12u^2+18uv-2uv-3v^2[/latex] [latex]6u(2u+3v)-v(2u+3v)[/latex] [latex](2u+3v)(6u-v)[/latex]
• [latex]-16\times -1=16[/latex] [latex]-16+-1=-17[/latex] [latex]4k^2-16k-k+4[/latex] [latex]4k(k-4)-1(k-4)[/latex] [latex](k-4)(4k-1)[/latex]
• [latex]7\times -4=-28[/latex] [latex]7+-4=3[/latex] [latex]4r^2+7r-4r-7[/latex] [latex]r(4r+7)-1(4r+7)[/latex] [latex](4r+7)(r-1)[/latex]
• [latex]-12\times 3=-36[/latex] [latex]-12+3=-9[/latex] [latex]4m^2-12mn+3mn-9n^2[/latex] [latex]4m(m-3n)+3n(m-3n)[/latex] [latex](m-3n)(4m+3n)[/latex]
• [latex]\text{Cannot be factored.}[/latex]
• [latex]12\times 1=12[/latex] [latex]12+1=13[/latex] [latex]4x^2+12xy+xy+3y^2[/latex] [latex]4x(x+3y)+y(x+3y)[/latex] [latex](x+3y)(4x+y)[/latex]
• [latex]8\times -3=-24[/latex] [latex]8+-3=5[/latex] [latex]6u^2+8uv-3uv-4v^2[/latex] [latex]2u(3u+4v)-v(3u+4v)[/latex] [latex](3u+4v)(2u-v)[/latex]
• [latex]20\times -1=-20[/latex] [latex]20+-1=19[/latex] [latex]10x^2+20xy-xy-2y^2[/latex] [latex]10x(x+2y)-1(x+2y)[/latex] [latex](x-2y)(10x-y)[/latex]
• [latex]-15\times 2=-30[/latex] [latex]-15+2=-13[/latex] [latex]6x^2-15xy+2xy-5y^2[/latex] [latex]3x(2x-5y)+y(2x-5y)[/latex] [latex](2x-5y)(3x+y)[/latex]

Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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CHAPTER 7 KEYS:

Reviews & skills keys:.

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Go Math Grade 4 Answer Key Chapter 7 Add and Subtract Fractions

One of the best study guides for grade 4 students is Go Math Grade 4 Answer Key Chapter 7 Add and Subtract Fractions . Make use of these pdf formatted chapter 7 Go Math HMH 4th Grade Answer Key for free and learn the topics efficiently. Download the Go Math Grade 4 Answer Key Chapter 7 Add and Subtract Fractions pdf from here and get the step-wise answers to all the questions. From this page, you’ll find the different possible models & techniques that students use to find the correct way to solve the fractions.

Approaching the best ways will make you understand the concepts of adding and subtracting fractions. Master in the Go Math Grade 4 Chapter 7 Add and Subtract Fractions by using the clear cut explanation for all the questions with images. Obtain the knowledge to write the fractions as sum and subtractions from Go Math Grade 4 Solution Key of Chapter 7 Add and Subtract Fractions.

Lesson: 1 – Add and Subtract Parts of a Whole

Add and Subtract Parts of a Whole Page No – 389

Add and subtract parts of a whole page no – 390.

Lesson: 2 – Add and Subtract Parts of a Whole

Add and Subtract Parts of a Whole Page No – 393

Add and subtract parts of a whole page no – 394.

Lesson: 3 – Add and Subtract Parts of a Whole

Add and Subtract Parts of a Whole Page No – 395

Add and subtract parts of a whole page no – 396.

Lesson: 4 – Add and Subtract Parts of a Whole

Add and Subtract Parts of a Whole Page No – 399

Add and subtract parts of a whole page no – 400.

Lesson: 5 – Add Fractions Using Models

Add Fractions Using Models – Page No 401

Add fractions using models – lesson check – page no 402, add fractions using models – lesson check – page no 405, add fractions using models – lesson check – page no 406.

Lesson: 6 – Subtract Fractions Using Models

Subtract Fractions Using Models – Page No 407

Subtract fractions using models – page no 408.

Lesson: 7 – Subtract Fractions Using Models

Subtract Fractions Using Models – Page No 411

Sense or nonsense – page no. 412.

Lesson: 8 – Add and Subtract Fractions

Add and Subtract Fractions – Page No. 413

Add and subtract fractions – lesson check – page no. 414.

Lesson: 9 – Add and Subtract Fractions

Add and Subtract Fractions – Page No. 415

Add and subtract fractions – page no. 416.

Lesson: 10 – Add and Subtract Fractions

Add and Subtract Fractions – Page No. 419

Add and subtract fractions – page no. 420.

Lesson: 11 – Rename Fractions and Mixed Numbers

Rename Fractions and Mixed Numbers – Page No. 421

Rename fractions and mixed numbers – lesson check – page no. 422.

Lesson: 12 – Rename Fractions and Mixed Numbers

Rename Fractions and Mixed Numbers – Page No. 425

Rename fractions and mixed numbers – page no. 426.

Lesson: 13 – Add and Subtract Mixed Numbers

Add and Subtract Mixed Numbers – Page No. 427

Add and subtract mixed numbers – lesson check – page no. 428.

Lesson: 14 – Add and Subtract Mixed Numbers

Add and Subtract Mixed Numbers – Page No. 431

Add and subtract mixed numbers – page no. 432.

Lesson: 15 – Record Subtraction with Renaming

Record Subtraction with Renaming – Page No. 433

Record Subtraction with Renaming – Lesson Check – Page No. 434

Lesson: 16 – Record Subtraction with Renaming

Record Subtraction with Renaming – Page No. 437

Record subtraction with renaming – page no. 438.

Lesson: 17 – Fractions and Properties of Addition

Fractions and Properties of Addition – Page No. 439

Fractions and properties of addition – lesson check – page no. 440, fractions and properties of addition – lesson check – page no. 443, fractions and properties of addition – lesson check – page no. 444.

Lesson: 18 – Fractions and Properties of Addition

Fractions and Properties of Addition – Page No. 445

Fractions and properties of addition – lesson check – page no. 446.

Lesson: 19 – Fractions and Properties of Addition

Fractions and Properties of Addition – Page No. 447

Fractions and properties of addition – page no. 448.

Lesson: 20 – Fractions and Properties of Addition

Fractions and Properties of Addition – Page No. 449

Fractions and properties of addition – page no. 450.

Lesson: 21 – Fractions and Properties of Addition

Fractions and Properties of Addition – Page No. 451

Fractions and properties of addition – page no. 452.

Lesson: 22 – Fractions and Properties of Addition

Fractions and Properties of Addition – Page No. 457

Fractions and properties of addition – page no. 458.

Use the model to write an equation.

Answer: 3/8 + 2/8 = 5/8

Explanation: By seeing the above 3 figures we can say that the fraction of the shaded part of the first circle is 3/8, the fraction of the second figure is 2/8 By adding the 2 fractions we get the fraction of the third circle. 3/8 + 2/8 = 5/8

Answer: 4/5 – 3/5 = 1/5

Explanation: The fraction of the shaded part for the above rectangle is 4/5 The fraction of the box is 3/5 The equation for the above figure is 4/5 – 3/5 = 1/5

Answer: 1/4 + 2/4 = 3/4

Explanation: The name of the fraction for the shaded part of first figure is 1/4 The name of the fraction for the shaded part of second figure is 1/4 The name of the fraction for the shaded part of third figure is 3/4 So, The equation for the above figure is 1/4 + 2/4 = 3/4

Answer: \(\frac { 2 }{ 6 } +\frac { 3 }{ 6 } =\frac { 5 }{ 6 } \)

Explanation: The name of the fraction for the shaded part of first figure is 2/6 The name of the fraction for the shaded part of second figure is 3/6 The name of the fraction for the shaded part of third figure is 5/6 So, The equation for the above figure is \(\frac { 2 }{ 6 } +\frac { 3 }{ 6 } =\frac { 5 }{ 6 } \)

Answer: \(\frac { 3 }{ 5 } -\frac { 2 }{ 5 } =\frac { 1 }{ 5 } \)

Explanation: The name of the fraction for the shaded part of figure is 3/5 The name of the fraction for the shaded part of closed box is 2/5 So, The equation for the above figure is \(\frac { 3 }{ 5 } -\frac { 2 }{ 5 } =\frac { 1 }{ 5 } \)

Question 6: Jake ate \(\frac { 4 }{ 8 } \) of a pizza. Millie ate \(\frac { 3}{ 8 } \) of the same pizza. How much of the pizza was eaten by Jake and Millie?

Answer: 7/8 of pizza

Explanation: Given that, Jake ate \(\frac { 4 }{ 8 } \) of a pizza. Millie ate \(\frac { 3}{ 8 } \) of the same pizza. To find how much of the pizza was eaten by Jake and Millie We have to add both the fractions \(\frac { 4 }{ 8 } \) + \(\frac { 3 }{ 8 } \) = \(\frac { 7 }{ 8 } \) Thus the fraction of the pizza eaten by Jake and Millie is \(\frac { 7 }{ 8 } \)

Question 7: Kate ate \(\frac { 1 }{ 4 } \) of her orange. Ben ate \(\frac { 2 }{ 4 } \) of his banana. Did Kate and Ben eat \(\frac { 1 }{ 4 } +\frac { 2}{ 4 } =\frac { 3}{ 4 } \) of their fruit?

Answer: No, one whole refers to orange and the other whole to a banana.

Question 1: A whole pie is cut into 8 equal slices. Three of the slices are served. How much of the pie is left? (a) \(\frac { 1 }{ 8 } \) (b) \(\frac { 3 }{ 8 } \) (c) \(\frac { 5 }{ 8} \) (d)\(\frac { 7 }{ 8 } \)

Answer: \(\frac { 5 }{ 8} \)

Explanation: Given, A whole pie is cut into 8 equal slices. Three of the slices are served. The fraction of 8 slices is 8/8. Out of which 3/8 are served. 8/8 – 3/8 = 5/8 Therefore \(\frac { 5 }{ 8} \) of the pie is left. Thus the correct answer is option c.

Question 2: An orange is divided into 6 equal wedges. Jody eats 1 wedge. Then she eats 3 more wedges. How much of the orange did Jody eat? (a) \(\frac { 1 }{ 6} \) (b) \(\frac { 4}{ 6 } \) (c) \(\frac { 5}{ 6 } \) (d) \(\frac { 6}{ 6} \)

Answer: \(\frac { 4}{ 6 } \)

Explanation: Given, An orange is divided into 6 equal wedges. Jody eats 1 wedge. Then she eats 3 more wedges. The fraction of orange that Jody eat is \(\frac { 4}{ 6 } \). Thus the correct answer is option b.

Question 3: Which list of distances is in order from least to greatest? (a) \(\frac { 1 }{ 8 } \) Mile, \(\frac { 3 }{ 16 } \) Mile, \(\frac { 3 }{ 4 } \) Mile (b) \(\frac { 3 }{ 4 } \) Mile, \(\frac { 1 }{ 8 } \) Mile, \(\frac { 3 }{ 16 } \) Mile (c) \(\frac { 1 }{ 8} \) Mile, \(\frac { 3 }{ 4 } \) Mile, \(\frac { 3 }{ 16 } \) Mile (d)\(\frac { 3 }{ 16 } \) Mile, \(\frac { 1 }{ 8 } \) Mile, \(\frac { 3 }{ 4 } \) Mile

Answer: \(\frac { 1 }{ 8 } \) Mile, \(\frac { 3 }{ 16 } \) Mile, \(\frac { 3 }{ 4 } \) Mile

Explantion: Compare the three fractions 1/8, 3/4 and 3/16 Make the common denominators. 1/8 × 2/2 = 2/16 3/4 × 4/4 = 12/16 The fractions are 2/16, 12/16 and 3/16 The numerator with the highest number will be the greatest. The fractions from least to greatest is \(\frac { 1 }{ 8 } \) Mile, \(\frac { 3 }{ 16 } \) Mile, \(\frac { 3 }{ 4 } \) Mile. Thus the correct answer is option d.

Question 4: Jeremy walked 6/8 of the way to school and ran the rest of the way. What fraction, in simplest form, shows the part of the way that Jeremy walked? (a) \(\frac { 1 }{ 4 } \) (b) \(\frac { 3 }{ 8 } \) (c) \(\frac { 1 }{ 2} \) (d)\(\frac { 3 }{ 4 } \)

Answer: \(\frac { 3 }{ 4 } \)

Explanation: Given, Jeremy walked 6/8 of the way to school and ran the rest of the way. The simplest form of 6/8 is 3/8. The simplest form of part of the way that Jeremy walked is 3/8. Thus the correct answer is option b.

Question 5: An elevator starts on the 100th floor of a building. It descends 4 floors every 10 seconds. At what floor will the elevator be 60 seconds after it starts? (a) 60th floor (b) 66th floor (c) 72nd floor (d) 76th floor

Answer: 76th floor

Explanation: Given, An elevator starts on the 100th floor of a building. It descends 4 floors every 10 seconds. 4 floors – 10 seconds ? – 60 seconds 60 × 4/10 = 240/10 = 24 floors 100 – 24 = 76th floor Thus the correct answer is option d.

Question 6: For a school play, the teacher asked the class to set up chairs in 20 rows with 25 chairs in each row. After setting up all the chairs, they were 5 chairs short. How many chairs did the class set up? (a) 400 (b) 450 (c) 495 (d) 500

Answer: 495

Explanation: Given, For a school play, the teacher asked the class to set up chairs in 20 rows with 25 chairs in each row. After setting up all the chairs, they were 5 chairs short. 20 × 25 = 500 500 – 5 = 495 Therefore the class set up 495 chairs. Thus the correct answer is c.

Answer: The sum of the unit fraction for 3/4 is 1/4 + 1/4 + 1/4

Explanation: A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. The sum of the unit fraction for 3/4 is 1/4 + 1/4 + 1/4.

Answer: The sum of the unit fraction for 5/6 is 1/6 + 1/6 + 1/6 + 1/6 + 1/6

Explanation: A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. The sum of the unit fraction for 5/6 is 1/6 + 1/6 + 1/6 + 1/6 + 1/6

Answer: The sum of the unit fraction for 2/3 is 1/3 + 1/3.

Explanation: A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. The sum of the unit fraction for 2/3 is 1/3 + 1/3.

Question 4: \(\frac { 4 }{ 12 } = \)

Answer: The sum of the unit fraction for 4/12 is 1/12 + 1/12 + 1/12 + 1/12

Explanation: A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. The sum of the unit fraction for 4/12 is 1/12 + 1/12 + 1/12 + 1/12

Question 5: \(\frac { 6 }{ 8 } = \)

Answer: The sum of the unit fraction for 6/8 is 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8

Explanation: A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. The sum of the unit fraction for 6/8 is 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8

Question 6: \(\frac { 8 }{ 10 } = \)

Answer: The sum of the unit fraction for 8/10 is 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10

Explanation: A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. The sum of the unit fraction for 8/10 is 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10

Question 7: \(\frac { 6 }{ 6 } = \)

Answer: The sum of the unit fraction for 6/6 is 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6

Explanation: A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. The sum of the unit fraction for 6/6 is 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6

Question 8: Compare Representations How many different ways can you write a fraction that has a numerator of 2 as a sum of fractions? Explain.

Answer: Let’s say we have the fraction 2/9. We can split this one fraction into two by modifying the numerator, like so: 2/9 = 1/9 + 1/9 This works because since both fractions have a numerator of 9, you can easily add the numerators to give 2, and that will give 2/9 in return. However, you can’t separate the denominators. 2/9 is not equal to 2/6 + 2/3 2/9 = 1/9 + 1/9 2/9 = 0.5/9 + 1.5/9 (which simplifies to 1/18 + 3/18, also giving 2/9) 2/9 = 0.5/9 + 0.5/9 + 0.5/9 + 0.5/9 = 1/18 + 1/18 + 1/18 + 1/18 I basically split it up into more and more fractions that add up to give 2/9. So, in short, there are infinitely many ways to do it.

Answer: We need the information about the equal sections and fence the garden into 3 areas by grouping some equal sections together.

b. How can writing an equation help you solve the problem?

Answer: The equation helps to find what part of the garden could each fenced area be.

Explanation: If you write an equation with 3 addends whose sum is 5/5, you could find the possible sizes of each fenced area. The size of each section is 1/5. Each addend represents the size of a fenced area.

c. How can drawing a model help you write an equation?

Answer: If you draw a model that shows 5 fifth-size parts representing the sections, you can see how to group the parts into 3 areas in different ways.

d. Show how you can solve the problem.

Question 9: Complete the sentence. The garden can be fenced into ______, ______, and ______ parts or ______, ______, and ______ parts.

Answer: 3/5, 1/5 and 1/5 parts or 2/5, 2/5 and 1/5 parts

Explanation: The sum of the unit fractions for 4/5 is 1/5 + 1/5 + 1/5 + 1/5.

Question 2: \(\frac { 3 }{ 8 }= \)

Answer: 1/8 + 1/8 + 1/8

Explanation: The sum of the unit fractions for 3/8 is 1/8 + 1/8 + 1/8

Question 3: \(\frac { 6 }{ 12 }= \)

Answer: 1/12 + 1/12 + 1/12 + 1/12 + 1/12 + 1/12

Explanation: The sum of the unit fractions for 6/12 is 1/12 + 1/12 + 1/12 + 1/12 + 1/12 + 1/12

Question 4: \(\frac { 4 }{ 4 }= \)

Answer: 1/4 + 1/4 + 1/4 + 1/4

Explanation: The sum of the unit fractions for 4/4 is 1/4 + 1/4 + 1/4 + 1/4

Question 5: \(\frac { 7 }{ 10 }= \)

Answer: 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10

Explanation: The sum of the unit fractions for 7/10 is 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10

Question 6: \(\frac { 6 }{ 6 } =\)

Answer: 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6

Explanation: The sum of the unit fractions for 6/6 is 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6

Question 7: Miguel’s teacher asks him to color \(\frac { 4 }{ 8 }\) of his grid. He must use 3 colors: red, blue, and green. There must be more green sections than red sections. How can Miguel color the sections of his grid to follow all the rules?

Answer: 1/8 red, 1/8 blue, and 2/8 green

Explanation: If there are 8 tiles, coloring \(\frac { 4 }{ 8 }\) means coloring 4 tiles. Using those three colors, we could use each 1 time with 1 leftover. Since we must have more green, we would use it twice; this would give us 2 green, 1 red and 1 blue. Since the grid is not necessarily 8 squares, we must account for this by saying 2/8 green, 1/8 red, and 1/8 blue.

Question 8: Petra is asked to color \(\frac { 6 }{ 6 }\) of her grid. She must use 3 colors: blue, red, and pink. There must be more blue sections than red sections or pink sections. What are the different ways Petra can color the sections of her grid and follow all the rules?

Answer: 3/6 blue, 2/6 red, 1/6 pink

Explanation: 1. 3 blues, 2 red, 1 pink. 2. 3 blues, 2 pink, 1 red. 3. 4 blues, 1 red, 1 pink The different ways in which Petra can color the sections of her grid and follow the rules are; 1. 3 blues, 2 red, 1 pink. 2. 3 blues, 2 pink, 1 red. 3. 4 blues, 1 red, 1 pink All these three ways follows the rules that; there must be three colors an also Blue sections are more than red sections or pink sections.

Question 1: Jorge wants to write \(\frac { 4 }{ 5 } \) as a sum of unit fractions. Which of the following should he write? (a) \(\frac { 3 }{ 5 } +\frac { 1 }{ 5 } \) (b) \(\frac { 2 }{ 5 } +\frac { 2 }{ 5 } \) (c) \(\frac { 1 }{ 5 } +\frac { 1 }{ 5 }+\frac { 2 }{ 5 } \) (d) \(\frac { 1 }{ 5 } +\frac { 1 }{ 5 } +\frac { 1 }{ 5 } +\frac { 1 }{ 5 } \)

Answer: \(\frac { 1 }{ 5 } +\frac { 1 }{ 5 } +\frac { 1 }{ 5 } +\frac { 1 }{ 5 } \)

Explanation: Given, Jorge wants to write \(\frac { 4 }{ 5 } \) as a sum of unit fractions. The sum of the unit fraction for \(\frac { 4 }{ 5 } \) is \(\frac { 1 }{ 5 } +\frac { 1 }{ 5 } +\frac { 1 }{ 5 } +\frac { 1 }{ 5 } \) Thus the correct answer is option d.

Question 2: Which expression is equivalent to \(\frac { 7 }{ 8 } \) ? (a) \(\frac { 5 }{ 8 } +\frac { 2 }{ 8}+\frac { 1 }{ 8 } \) (b) \(\frac { 3 }{ 8 } +\frac {3 }{ 8 } +\frac { 1 }{ 8 } +\frac { 1 }{ 8 } \) (c) \(\frac { 4 }{ 8 } +\frac { 2 }{ 8 }+\frac { 1 }{ 8 } \) (d) \(\frac { 4 }{ 8 } +\frac { 2 }{ 8 }+\frac { 2 }{ 8 } \)

Answer: \(\frac { 4 }{ 8 } +\frac { 2 }{ 8 }+\frac { 1 }{ 8 } \)

Explanation: The fraction equivalent to \(\frac { 7 }{ 8 } \) is \(\frac { 4 }{ 8 } +\frac { 2 }{ 8 }+\frac { 1 }{ 8 } \). Thus the correct answer is option c.

Question 3: An apple is cut into 6 equal slices. Nancy eats 2 of the slices. What fraction of the apple is left? (a) \(\frac { 1 }{ 6 } \) (b) \(\frac { 2 }{ 6 } \) (c) \(\frac { 3 }{ 6 } \) (d) \(\frac { 4 }{ 6 } \)

Answer: \(\frac { 4 }{ 6 } \)

Explanation: Given, An apple is cut into 6 equal slices. Nancy eats 2 of the slices. 6 – 2 = 4 \(\frac { 6 }{ 6 } \) – \(\frac { 2 }{ 6 } \) = \(\frac { 4 }{ 6 } \) Thus the correct answer is option d.

Question 4: Which of the following numbers is a prime number? (a) 1 (b) 11 (c) 21 (d) 51

Explanation: A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. 11 is a multiple of 1 and itself. Thus the correct answer is option b.

Question 5: A teacher has a bag of 100 unit cubes. She gives an equal number of cubes to each of the 7 groups in her class. She gives each group as many cubes as she can. How many unit cubes are left over? (a) 1 (b) 2 (c) 3 (d) 6

Explanation: Given, A teacher has a bag of 100 unit cubes. She gives an equal number of cubes to each of the 7 groups in her class. She gives each group as many cubes as she can. 100 divided by 7 is 14 r 2, so there are 2 leftover. Thus the correct answer is option b.

Question 6: Jessie sorted the coins in her bank. She made 7 stacks of 6 dimes and 8 stacks of 5 nickels. She then found 1 dime and 1 nickel. How many dimes and nickels does Jessie have in all? (a) 84 (b) 82 (c) 80 (d) 28

Explanation: Given, Jessie sorted the coins in her bank. She made 7 stacks of 6 dimes and 8 stacks of 5 nickels. She then found 1 dime and 1 nickel. 43 dimes and 41 nickles 43 + 41 = 84 Jessie has 84 dimes and nickels in all. Thus the correct answer is option a.

Answer: 4/5

Explanation: Given, Adrian’s cat ate \(\frac { 3 }{ 5 } \) of a bag of cat treats in September and \(\frac { 1 }{ 5 } \) of the same bag of cat treats in October. From the above figure, we can see that there are 4 fifth size pieces. \(\frac { 3 }{ 5 } \)+\(\frac { 1 }{ 5 } \) = \(\frac { 4 }{ 5 } \).

Answer: 3/4

Explanation: From the above figure, we can see that there are 3 one-fourth shaded parts. So, \(\frac { 1 }{ 4 } +\frac { 2 }{ 4 } =\frac { 3 }{ 4 } \)

Answer: 9/10

Explanation: From the above figure, we can see that there are 9 one-tenth shaded parts. So, \(\frac { 6 }{ 10 } +\frac { 3 }{ 10 } =\frac { 9 }{ 10 } \).

Find the sum. Use models to help. Question 4: \(\frac { 3 }{ 6 } +\frac { 3 }{ 6 } =\frac { }{ } \)

Answer: 6/6 = 1

Explanation: 3/6 and 3/6 has same numerators and same denominators so we have to add both the fractions. \(\frac { 3 }{ 6 } +\frac { 3 }{ 6 } =\frac { 6 }{ 6 } \) 6/6 = 1

Question 5: \(\frac { 1 }{ 3 } +\frac { 1 }{ 3 } =\frac { }{ } \)

Answer: 2/3

Explanation: 1/3 and 1/3 has same numerators and same denominators so we have to add both the fractions. \(\frac { 1 }{ 3 } +\frac { 1 }{ 3 } =\frac { 2 }{ 3 } \)

Question 6: \(\frac { 5 }{ 8 } +\frac { 2 }{ 8 } =\frac { }{ } \)

Answer: 7/8

Explanation: Given the expressions 5/8 and 2/8. The above fractions have the same denominators but the numerators are different. So, \(\frac { 5 }{ 8 } +\frac { 2 }{ 8 } =\frac { 7 }{ 8 } \)

Find the sum. Use models or iTools to help. Question 7: \(\frac { 5 }{ 8 } +\frac { 2 }{ 8 } =\frac { }{ } \) Answer: 7/8

Question 8: \(\frac { 2 }{ 5 } +\frac { 2 }{ 5 } =\frac { }{ } \) Answer: 4/5

Explanation: 2/5 and 2/5 have the same numerators and same denominators so we have to add both the fractions. \(\frac { 2 }{ 5 } +\frac { 2 }{ 5 } =\frac { 4 }{ 5 } \)

Question 9: \(\frac { 4 }{ 6 } +\frac { 1 }{ 6 } =\frac { }{ } \) Answer: 5/6

Explanation: Given the fractions 4/6 and 1/6. The above fractions have the same denominators but the numerators are different. \(\frac { 4 }{ 6 } +\frac { 1 }{ 6 } =\frac { 5 }{ 6 } \)

Question 10: Jason is making a fruit drink. He mixes \(\frac { 2 }{ 8 } \) quart of grape juice with \(\frac { 3 }{ 8 } \) quart of apple juice. Then he adds \(\frac { 1 }{ 8 } \) quart of lemonade. How much fruit drink does Jason make? \(\frac { }{ } \) quart. Answer: \(\frac { 6 }{ 8 } \) quart.

Explanation: Given that, Jason is making a fruit drink. He mixes \(\frac { 2 }{ 8 } \) quart of grape juice with \(\frac { 3 }{ 8 } \) quart of apple juice. Then he adds \(\frac { 1 }{ 8 } \) quart of lemonade Add all the three fractions to how much fruit drink Jason makes. 2/8 + 3/8 + 1/8 = \(\frac { 6 }{ 8 } \) quart.

Question 11: A sum has five addends. Each addend is a unit fraction. The sum is 1. What are the addends?

Answer: 1/5

Explanation: Given that, A sum has five addends. Each addend is a unit fraction. The sum is 1. 1/5 + 1/5 + 1/5 + 1/5 + 1/5 = 5/5 = 1 Thus the addend is 1/5.

Explanation: Given that, In a survey, \(\frac { 4 }{ 12 } \) of the students chose Friday and \(\frac { 5 }{ 12 } \) chose Saturday as their favorite day of the week. Add both the fractions 4/12 and 5/12 \(\frac { 4 }{ 12 } \) + \(\frac { 5 }{ 12 } \) = \(\frac { 9 }{ 12 } \)

Answer: \(\frac { 4}{ 10} \)

Explanation: the answer is 4/10 because 4/10 + 2/10= 6/10+ 4/10 = 10/10. a bit confusing 4 + 2 = 6 right the, 6 + 4 = 10 so 10/10.

Have you ever seen a stained glass window in a building or home? Artists have been designing stained glass windows for hundreds of years.

Help design the stained glass sail on the boat below.

Materials • color pencils

Look at the eight triangles in the sail. Use the guide below to color the triangles:

• \(\frac {2 }{8 } \) blue
• \(\frac {3 }{8 } \) red
• \(\frac { 2}{ 8} \) orange
• \(\frac {1 }{8 } \) yellow

Question 14: Write an Equation Write an equation that shows the fraction of triangles that are red or blue. Answer: \(\frac {3 }{8 } \) red

Question 15: What color is the greatest part of the sail? Write a fraction for that color. How do you know that fraction is greater than the other fractions? Explain. Answer: Red

Explanation: Among all the colors Red color has the greatest part of the sail.

Find the sum. Use fraction strips to help.

Answer: 3/6

Question 2: \(\frac { 4 }{ 10 } +\frac { 5 }{ 10 } =\frac { }{ } \)

Question 3: \(\frac { 1 }{ 3 } +\frac { 2 }{ 3 } =\frac { }{ } \)

Question 4: \(\frac { 2 }{ 4 } +\frac { 1 }{ 4 } =\frac { }{ } \)

Question 5: \(\frac { 2 }{ 12 } +\frac { 4 }{ 12 } =\frac { }{ } \)

Question 6: \(\frac { 1 }{ 6 } +\frac { 3 }{ 6 } =\frac { }{ } \)

Question 7: \(\frac { 3 }{ 12 } +\frac { 9 }{ 12 } =\frac { }{ } \)

Answer: 12/12

Question 8: \(\frac { 3 }{ 8 } +\frac { 4 }{ 8 } =\frac { }{ } \)

Question 9: \(\frac { 3 }{ 4 } +\frac { 1 }{ 4 } =\frac { }{ } \)

Question 9: \(\frac { 1 }{ 5 } +\frac { 2 }{ 5 } =\frac { }{ } \)

Answer: 3/5

Question 10: \(\frac { 6 }{ 10 } +\frac { 3 }{ 10 } =\frac { }{ } \)

Question 11: Lola walks \(\frac { 4 }{ 10} \) mile to her friend’s house. Then she walks \(\frac { 5 }{ 10 } \) mile to the store. How far does she walk in all?

Answer: \(\frac { 9 }{ 10 } \) mile

Explanation: Given, Lola walks \(\frac { 4 }{ 10} \) mile to her friend’s house. Then she walks \(\frac { 5 }{ 10 } \) mile to the store. \(\frac { 4 }{ 10} \) + \(\frac { 5 }{ 10 } \) = \(\frac { 9 }{ 10 } \) Therefore she walked \(\frac { 9 }{ 10 } \) mile in all.

Question 12: Evan eats \(\frac { 1 }{ 8 } \) of a pan of lasagna and his brother eats \(\frac { 2 }{ 8 } \) of it. What fraction of the pan of lasagna do they eat in all? Answer: \(\frac { 3 }{ 8 } \) of the pan

Explanation: Given, Evan eats \(\frac { 1 }{ 8 } \) of a pan of lasagna and his brother eats \(\frac { 2 }{ 8 } \) of it. \(\frac { 1 }{ 8 } \) + \(\frac { 2 }{ 8 } \) = \(\frac { 3 }{ 8 } \)

Question 13: Jacqueline buys \(\frac { 2 }{ 4 } \) yard of green ribbon and \(\frac { 1 }{ 4 } \) yard of pink ribbon. How many yards of ribbon does she buy in all?

Answer: \(\frac { 3 }{ 4 } \) yard

Explanation: Given, Jacqueline buys \(\frac { 2 }{ 4 } \) yard of green ribbon and \(\frac { 1 }{ 4 } \) yard of pink ribbon. \(\frac { 2 }{ 4 } \) + \(\frac { 1 }{ 4 } \) = \(\frac { 3 }{ 4 } \) Thus Jacqueline bought \(\frac { 3 }{ 4 } \) yards of ribbon in all.

Question 14: Shu mixes \(\frac { 2 }{ 3 } \) pound of peanuts with \(\frac { 1 }{ 3 } \) pound of almonds. How many pounds of nuts does Shu mix in all?

Answer: 3/3 pound

Explanation: Given, Shu mixes \(\frac { 2 }{ 3 } \) pound of peanuts with \(\frac { 1 }{ 3 } \) pound of almonds. \(\frac { 2 }{ 3 } \) + \(\frac { 1 }{ 3 } \) = \(\frac { 3 }{ 3 } \) Therefore Shu mix \(\frac { 3 }{ 3 } \) pounds of nuts in all.

Question 1: Mary Jane has \(\frac { 3 }{ 8 } \) of a medium pizza left. Hector has \(\frac { 2 }{ 8 } \) of another medium pizza left. How much pizza do they have altogether?

(a) \(\frac { 1 }{ 8 } \) (b) \(\frac { 4 }{ 8 } \) (c) \(\frac { 5 }{ 8 } \) (d) \(\frac { 6 }{ 8 } \)

Answer: \(\frac { 5 }{ 8 } \)

Explanation: Given, Mary Jane has \(\frac { 3 }{ 8 } \) of a medium pizza left. Hector has \(\frac { 2 }{ 8 } \) of another medium pizza left. To find how much pizza do they have altogether we have to add both the fractions. \(\frac { 3 }{ 8 } \) + \(\frac { 2 }{ 8 } \) = \(\frac { 5 }{ 8 } \) Therefore Mary Jane and Hector has \(\frac { 5 }{ 8 } \) pizza altogether. Thus the correct answer is option c.

Question 2: Jeannie ate \(\frac { 1 }{ 4 } \) of an apple. Kelly ate \(\frac { 2 }{ 4 } \) of the apple. How much did they eat in all?

(a) \(\frac { 1 }{ 8 } \) (b) \(\frac { 2 }{ 8 } \) (c) \(\frac { 3 }{ 8 } \) (d) \(\frac { 3 }{ 4 } \)

Explanation: Given, Jeannie ate \(\frac { 1 }{ 4 } \) of an apple. Kelly ate \(\frac { 2 }{ 4 } \) of the apple. \(\frac { 1 }{ 4 } \) + \(\frac { 2 }{ 4 } \) = \(\frac { 3 }{ 4 } \) Thus the correct answer is option d.

Question 3: Karen is making 14 different kinds of greeting cards. She is making 12 of each kind. How many greeting cards is she making?

(a) 120 (b) 132 (c) 156 (d) 168

Answer: 168

Explanation: Given, Karen is making 14 different kinds of greeting cards. She is making 12 of each kind. To find how many greeting cards she is making we have to multiply 14 and 12. 14 × 12 = 168. Thus the correct answer is option d.

Question 4: Jefferson works part-time and earns $1,520 in four weeks. How much does he earn each week?

(a) $305 (b) $350 (c) $380 (d) $385

Answer: $380

Explanation: Jefferson works part-time and earns $1,520 in four weeks. 1520 – 4 weeks ? – 1 week 1520/4 = $380 Thus the correct answer is option c.

Question 5: By installing efficient water fixtures, the average American can reduce water use to about 45 gallons of water per day. Using such water fixtures, about how many gallons of water would the average American use in December?

(a) about 1,200 gallons (b) about 1,500 gallons (c) about 1,600 gallons (d) about 2,000 gallons

Answer: about 1,500 gallons

Explanation: Given, By installing efficient water fixtures, the average American can reduce water use to about 45 gallons of water per day. 1 day – 45 gallons 31 days – ? 45 × 31 = 1395 gallons The number near to 1395 is 1500 gallons. Thus the correct answer is option b.

Question 6: Collin is making a bulletin board and note center. He is using square cork tiles and square dry-erase tiles. One of every 3 squares will be a cork square. If he uses 12 squares for the center, how many will be cork squares?

(a) 3 (b) 4 (c) 6 (d) 8

Explanation: Given that, Collin is making a bulletin board and note center. He is using square cork tiles and square dry-erase tiles. One of every 3 squares will be a cork square. 12/3 = 4 Thus the correct answer is option b.

Explanation: Given that, Lisa needs 4/5 pounds of shrimp to make shrimp salad. She has 1/5 pound of shrimp. The denominators have the same numbers and numerators have different numbers. 4/5 – 3/5 = 1/5 Thus Lisa needs 1/5 pounds more shrimp.

Use the model to find the difference.

Answer: 1/6

Explanation: Given two fractions 3/6 and 2/6 Denominators are same but the numerators are different. 3/6 – 2/6 = 1/6

Answer: 3/10

Explanation: Given two fractions 8/10 and 5/10 Denominators are the same but the numerators are different. 8/10 – 5/10 = 3/10

Subtract. Use models to help.

Question 4: \(\frac { 5 }{ 8 } – \frac { 2 }{ 8 } = \frac { }{ } \)

Answer: 3/8

Question 5: \(\frac { 7 }{ 12 } – \frac { 2 }{ 12 } = \frac { }{ } \)

Answer: 5/12

Question 6: \(\frac { 3 }{4 } – \frac { 2 }{ 4 } = \frac { }{ } \)

Answer: 1/4

Question 7: \(\frac { 2 }{ 3 } – \frac { 1 }{ 3 } = \frac { }{ } \)

Answer: 1/3

Question 8: \(\frac { 7 }{ 8 } – \frac { 5 }{ 8 } = \frac { }{ } \)

Answer: 2/8

Question 9: Explain how you could find the unknown addend in \(\frac { 2 }{ 6 } \) + _____ = 1 without using a model. Answer: 4/6

Explanation: 1 can be written in the fraction form as 6/6 2/6 + x = 6/6 x = 6/6 – 2/6 x = 4/6

Answer: 10/12

a. What do you need to know?

Answer: We need to find the fraction of the pie did they eat on the second night.

b. How can you find the number of pieces eaten on the second night?

Answer: We can find the number of pieces eaten on the second night by dividing the number of eaten pieces by the total number of pieces.

c. Explain the steps you used to solve the problem. Complete the sentences. After the first night, _______ pieces were left. After the second night, _______ pieces were left. So, _______ of the pie was eaten on the second night.

Answer: After the first night, 9 pieces were left. After the second night, 2 pieces were left. So, 10 of the pie was eaten on the second night.

Question 11: Make Connection Between Models Judi ate \(\frac { 7}{8} \) of a small pizza and Jack ate \(\frac { 2}{ 8 } \) of a second small pizza. How much more of a pizza did Judi eat? \(\frac { }{ } \) Answer: \(\frac {5}{8} \)

Explanation: Given, Make Connection Between Models Judi ate \(\frac { 7}{8} \) of a small pizza and Jack ate \(\frac { 2}{ 8 } \) of a second small pizza. \(\frac {7}{8} \) – \(\frac {2}{8} \) = \(\frac {5}{8} \) Therefore Judi eat \(\frac {5}{8} \) of a pizza.

Explanation: Given, Keiko sewed \(\frac { 3}{4} \) yard of lace on her backpack. Pam sewed \(\frac { 1}{4} \) yard of lace on her backpack. \(\frac {3}{4} \) – \(\frac {1}{4} \) = \(\frac {2}{4} \)

Explanation: Given the fraction, 4/5 and 1/5 The denominators of both the fractions are the same so subtract the numerators. 4/5 – 1/5 = 3/5

Question 2: \(\frac { 3}{ 4 } – \frac { 1}{ 4 } = \frac { —}{ — } \)

Answer: 2/4

Explanation: Given the fractions \(\frac { 3}{ 4 } \) and [/latex] \frac { 1}{ 4 } [/latex] The denominators of both the fractions are the same so subtract the numerators. \(\frac { 3}{ 4 } – \frac { 1}{ 4 } = \frac { 2 }{ 4 } \)

Question 3: \(\frac { 5}{ 6 } – \frac { 1}{ 6 } = \frac { —}{ — } \)

Answer: 4/6

Explanation: Given the fractions \(\frac { 5 }{ 6 } \) and [/latex] \frac { 1 }{ 6 } [/latex] The denominators of both the fractions are the same so subtract the numerators. \(\frac { 5}{ 6 } – \frac { 1}{ 6 } = \frac { 4 }{ 6 } \)

Question 4: \(\frac { 7}{ 8 } – \frac { 1}{ 8 } = \frac { —}{ — } \)

Answer: 6/8

Explanation: Given the fractions \(\frac { 7 }{ 8 } \) and [/latex] \frac { 1 }{ 8 } [/latex] The denominators of both the fractions are the same so subtract the numerators. \(\frac { 7}{ 8 } – \frac { 1}{ 8 } = \frac { 6 }{ 8 } \)

Question 5: \(\frac { 1}{ 3 } – \frac { 2}{ 3 } = \frac { —}{ — } \)

Explanation: Given the fractions \(\frac { 1 }{ 3 } \) and [/latex] \frac { 2 }{ 3 } [/latex] The denominators of both the fractions are the same so subtract the numerators. \(\frac { 1}{ 3 } – \frac { 2}{ 3 } = \frac { 1}{ 3 } \)

Question 6: \(\frac { 8}{ 10 } – \frac { 2}{ 10 } = \frac { —}{ — } \)

Answer: 6/10

Explanation: Given the fractions \(\frac { 8 }{ 10 } \) and [/latex] \frac { 2 }{ 10 } [/latex] The denominators of both the fractions are the same so subtract the numerators. \(\frac { 8}{ 10 } – \frac { 2}{ 10 } = \frac { 6 }{ 10 } \)

Question 7: \(\frac { 3}{ 4 } – \frac { 1}{ 4 } = \frac { —}{ — } \)

Explanation: Given the fractions \(\frac { 3 }{ 4 } \) and [/latex] \frac { 1 }{ 4 } [/latex] The denominators of both the fractions are the same so subtract the numerators. \(\frac { 3}{ 4 } – \frac { 1}{ 4 } = \frac { 2 }{ 4 } \)

Question 8: \(\frac { 7}{ 6 } – \frac {5}{ 6 } = \frac { —}{ — } \)

Answer: 2/6

Explanation: Given the fractions \(\frac { 7 }{ 6 } \) and [/latex] \frac { 5 }{ 6 } [/latex] The denominators of both the fractions are the same so subtract the numerators. \(\frac { 7}{ 6 } – \frac {5}{ 6 } = \frac { 2 }{ 6 } \)

Answer: 5/8 pound

Explanation: Given that, Ena is making trail mix. pretzels = 7/8 Raisins = 2/8 To find the number of more pounds of pretzels than raisins she buy we have to subtract both the fractions. 7/8 – 2/8 = 5/8

Question 10: How many more pounds of granola than banana chips does she buy? \(\frac { —}{ — } \)

Answer: 2/8 pound

Explanation: Granola = 5/8 Banana Chips = 3/8 To find How many more pounds of granola than banana chips does she buy we have to subtract both the fractions. 5/8 – 3/8 = 2/8 pounds

Question 1: Lee reads for \(\frac { 3}{ 4} \) hour in the morning and \(\frac {2}{ 4} \) hour in the afternoon. How much longer does Lee read in the morning than in the afternoon? (a) 5 hours (b) \(\frac { 5}{ 4} \) (c) \(\frac { 4}{ 4} \) (d) \(\frac { 1}{ 4} \)

Answer: \(\frac { 1}{ 4} \)

Explanation: Given, Lee reads for \(\frac { 3}{ 4} \) hour in the morning and \(\frac {2}{ 4} \) hour in the afternoon. \(\frac { 3}{ 4} \) – \(\frac {2}{ 4} \) = \(\frac { 1}{ 4} \) Lee read \(\frac { 1}{ 4} \) hour in the morning than in the afternoon. Thus the correct answer is option d.

Answer: \(\frac { 5}{ 6} – \frac { 3}{ 6} = \frac { 2}{ 6} \)

Explanation: From the above figure we can say that \(\frac { 5}{ 6} – \frac { 3}{ 6} = \frac { 2}{ 6} \) Thus the correct answer is option c.

Question 3: A city received 2 inches of rain each day for 3 days. The meteorologist said that if the rain had been snow, each inch of rain would have been 10 inches of snow. How much snow would that city have received in the 3 days?

(a) 20 inches (b) 30 inches (c) 50 inches (d) 60 inches

Answer: 60 inches

Explanation: Given, A city received 2 inches of rain each day for 3 days. 2 × 3 inches = 6 inches The meteorologist said that if the rain had been snow, each inch of rain would have been 10 inches of snow. 6 × 10 inches = 60 inches Therefore the city has received 60 inches of snow in 3 days. Thus the correct answer is option d.

Question 4: At a party there were four large submarine sandwiches, all the same size. During the party, \(\frac { 2}{ 3} \) of the chicken sandwich, \(\frac { 3}{ 4} \) of the tuna sandwich, \(\frac { 7}{ 12} \) of the roast beef sandwich, and \(\frac { 5}{ 6} \) of the veggie sandwich were eaten. Which sandwich had the least amount left?

(a) chicken (b) tuna (c) roast beef (d) veggie

Answer: veggie

Explanation: Given, At a party there were four large submarine sandwiches, all the same size. During the party, \(\frac { 2}{ 3} \) of the chicken sandwich, \(\frac { 3}{ 4} \) of the tuna sandwich, \(\frac { 7}{ 12} \) of the roast beef sandwich, and \(\frac { 5}{ 6} \) of the veggie sandwich were eaten. Compare the fractions \(\frac { 2}{ 3} \), \(\frac { 3}{ 4} \) , \(\frac { 7}{ 12} \) and \(\frac { 5}{ 6} \). Among all the fractions veggie has the least fraction. Thus the correct answer is option d.

Question 5: Deena uses \(\frac { 3}{ 8} \) cup milk and \(\frac { 2}{ 8} \) cup oil in a recipe. How much liquid does she use in all?

(a) \(\frac {1}{ 8} \) cup (b) \(\frac {5}{ 8} \) cup (c) \(\frac {6}{ 8} \) cup (d) 5 cups

Answer: \(\frac {5}{ 8} \) cup

Explanation: Given, Deena uses \(\frac { 3}{ 8} \) cup milk and \(\frac { 2}{ 8} \) cup oil in a recipe. \(\frac { 3}{ 8} \) + \(\frac { 2}{ 8} \) = \(\frac {5}{ 8} \) cup Therefore she used \(\frac {5}{ 8} \) cup of milk in all. Thus the correct answer is option b.

Question 6: In the car lot, \(\frac { 4}{ 12} \) of the cars are white and \(\frac { 3}{ 12} \) of the cars are blue. What fraction of the cars in the lot are either white or blue? (a) \(\frac { 1}{ 12} \) (b) \(\frac { 7}{ 24} \) (c) \(\frac { 7}{ 12} \) (d) 7

Answer: \(\frac { 7}{ 12} \)

Explanation: Given, In the car lot, \(\frac { 4}{ 12} \) of the cars are white and \(\frac { 3}{ 12} \) of the cars are blue. \(\frac { 4}{ 12} \) + \(\frac { 3}{ 12} \) = \(\frac { 7}{ 12} \) Thus the correct answer is option c.

Question 1: 9 twelfth-size parts − 5 twelfth-size parts = \(\frac { —}{ — } \)

Answer: 4/12

Explanation: 9 twelfth-size parts − 5 twelfth-size parts 9 × \(\frac { 1 }{ 12 } \) = \(\frac { 9 }{ 12 } \) 5 × \(\frac { 1 }{ 12 } \) = \(\frac { 5 }{ 12 } \) The denominators of both the fractions are the same so subtract the numerators. \(\frac { 9 }{ 12 } \) – \(\frac { 5 }{ 12 } \) = \(\frac { 4 }{ 12 } \)

Question 2: \(\frac { 3}{ 12} + \frac {8}{ 12 } = \frac { —}{ — } \)

Answer: 11/12

Explanation: Given the fractions, \(\frac { 3 }{ 12 } \) and \(\frac { 8 }{ 12 } \) Add both the fractions The denominators of both the fractions are the same so add the numerators. \(\frac { 3}{ 12} + \frac {8}{ 12 } = \frac { 11 }{ 12 } \)

Question 3: \(\frac { 1}{ 3 } + \frac {1}{ 3 } = \frac { —}{ — } \)

Explanation: Given the fractions, \(\frac { 1 }{ 3 } \) and \(\frac { 1 }{ 3 } \) Add both the fractions The denominators of both the fractions are the same so add the numerators. \(\frac { 1}{ 3 } + \frac {1}{ 3 } = \frac { 2 }{ 3 } \)

Question 4: \(\frac { 3}{ 4 } – \frac {1}{ 4 } = \frac { —}{ — } \)

Explanation: Given the fractions, \(\frac { 3 }{ 4 } \) and \(\frac { 1 }{ 4 } \) Subtract both the fractions The denominators of both the fractions are the same so Subtract the numerators. \(\frac { 3}{ 4 } – \frac {1}{ 4 } = \frac { 2 }{ 4 } \)

Question 5: \(\frac { 2}{ 6 } + \frac {2}{ 6 } = \frac { —}{ — } \)

Explanation: Given the fractions, \(\frac { 2 }{ 6 } \) and \(\frac { 2 }{ 6 } \) Add both the fractions The denominators of both the fractions are the same so add the numerators. \(\frac { 2}{ 6 } + \frac {2}{ 6 } = \frac { 4 }{ 6 } \)

Question 6: \(\frac { 3}{ 8 } – \frac {1}{ 8 } = \frac { —}{ — } \)

Explanation: Given the fractions, \(\frac { 3 }{ 8 } \) and \(\frac { 1 }{ 8 } \) Subtract both the fractions The denominators of both the fractions are the same so Subtract the numerators. \(\frac { 3}{ 8 } – \frac {1}{ 8 } = \frac { 2 }{ 8 } \)

Question 7: \(\frac { 6}{ 10 } – \frac {2}{ 10 } = \frac { —}{ — } \)

Answer: 4/10

Explanation: Given the fractions, \(\frac { 6 }{ 10 } \) and \(\frac { 2 }{ 10 } \) Subtract both the fractions The denominators of both the fractions are the same so Subtract the numerators. \(\frac { 6}{ 10 } – \frac {2}{ 10 } = \frac { 4 }{ 10 } \)

Question 8: \(\frac { 1}{ 2 } – \frac {1}{2 } = \frac { —}{ — } \)

Explanation: Given the fractions, \(\frac { 1 }{ 2 } \) and \(\frac { 1 }{ 2 } \) Subtract both the fractions The denominators of both the fractions are the same so Subtract the numerators. \(\frac { 1}{ 2 } – \frac {1}{2 } \) = 0

Question 9: \(\frac {5}{ 6 } – \frac {4}{ 6 } = \frac { —}{ — } \)

Explanation: Given the fractions, \(\frac { 5 }{ 6 } \) and \(\frac { 4 }{ 6 } \) Subtract both the fractions The denominators of both the fractions are the same so Subtract the numerators. \(\frac {5}{ 6 } – \frac {4}{ 6 } = \frac { 1 }{ 6 } \)

Question 10: \(\frac { 4}{ 5 } – \frac {2}{ 5 } = \frac { —}{ — } \)

Answer: 2/5

Explanation: Given the fractions, \(\frac { 4 }{ 5 } \) and \(\frac { 2 }{ 5 } \) Subtract both the fractions The denominators of both the fractions are the same so Subtract the numerators. \(\frac { 4}{ 5 } – \frac {2}{ 5 } = \frac { 2 }{ 5 } \)

Question 11: \(\frac { 1}{ 4 } + \frac {1}{ 4 } = \frac { —}{ — } \)

Explanation: Given the fractions, \(\frac { 1 }{ 4 } \) and \(\frac { 1 }{ 4 } \) Add both the fractions The denominators of both the fractions are the same so add the numerators. \(\frac { 1}{ 4 } + \frac {1}{ 4 } = \frac { 2 }{ 4 } \)

Question 12: \(\frac { 9}{ 10 } – \frac {5}{ 10 } = \frac { —}{ — } \)

Explanation: Given the fractions, \(\frac { 9 }{ 10 } \) and \(\frac { 5 }{ 10 } \) Subtract both the fractions The denominators of both the fractions are the same so Subtract the numerators. \(\frac { 9}{ 10 } – \frac {5}{ 10 } = \frac { 4 }{ 10 } \)

Question 13: \(\frac { 1}{ 12 } + \frac {7}{ 12 } = \frac { —}{ — } \)

Answer: 8/12

Explanation: Given the fractions, \(\frac { 1 }{ 12 } \) and \(\frac { 7 }{ 12 } \) Add both the fractions The denominators of both the fractions are the same so add the numerators. \(\frac { 1}{ 12 } + \frac {7}{ 12 } = \frac { 8 }{ 12 } \)

Question 14: Christopher mixes \(\frac { 3}{ 8} \) gallon of red paint with \(\frac { 5}{ 8} \) gallon of blue paint to make purple paint. He uses \(\frac { 2}{8} \) gallon of the purple paint. How much purple paint is left? \(\frac { —}{ — } \) gallon

Answer: \(\frac { 6 }{ 8 } \) gallon

Explanation: Given, Christopher mixes \(\frac { 3}{ 8} \) gallon of red paint with \(\frac { 5}{ 8} \) gallon of blue paint to make purple paint. He uses \(\frac { 2}{8} \) gallon of the purple paint. \(\frac { 3}{ 8} \) + \(\frac { 5}{ 8} \) = \(\frac { 8 }{ 8 } \) \(\frac { 8 }{ 8 } \) – \(\frac { 2 }{ 8 } \) = \(\frac { 6 }{ 8 } \) gallon

Question 15: A city worker is painting a stripe down the center of Main Street. Main Street is \(\frac { 8}{ 10} \) mile long. The worker painted \(\frac { 4}{ 10} \) mile of the street. Explain how to find what part of a mile is left to paint. \(\frac { —}{ — } \) mile

Answer: \(\frac { 4 }{ 10 } \) mile

Explanation: Given, A city worker is painting a stripe down the center of Main Street. Main Street is \(\frac { 8}{ 10} \) mile long. The worker painted \(\frac { 4}{ 10} \) mile of the street. \(\frac { 8 }{ 10 } \) – \(\frac { 4 }{ 10 } \) = \(\frac { 4 }{ 10 } \) mile

Question 16: Sense or Nonsense? Brian says that when you add or subtract fractions with the same denominator, you can add or subtract the numerators and keep the same denominator. Is Brian correct? Explain.

Answer: correct

Explanation: The statement of Brian is correct because when you add or subtract fractions with the same denominator, you can add or subtract the numerators and keep the same denominator.

Question 17: The length of a rope was \(\frac { 6}{8} \) yard. Jeff cut the rope into 3 pieces. Each piece is a different length measured in eighths of a yard. What is the length of each piece of rope?

Answer: \(\frac { 2}{8} \)

Explanation: Given, The length of a rope was \(\frac { 6}{8} \) yard. Jeff cut the rope into 3 pieces. Each piece is a different length measured in eighths of a yard. Divide \(\frac { 6}{8} \) into 3 pieces. \(\frac { 6}{8} \) ÷ 3 = \(\frac { 2}{8} \)

Question 18: For 18a–18d, choose Yes or No to show if the sum or difference is correct.

a. \(\frac { 3}{ 5 } – \frac {1}{ 5 } = \frac {4 }{5 } \) (i) yes (ii) no

Explanation: The denominators of both the fractions are the same so Subtract the numerators. \(\frac { 3}{ 5 } – \frac {1}{ 5 } = \frac {2 }{5 } \) Thus the above statement is not correct.

b. \(\frac { 1}{ 4 } – \frac {2}{4 } = \frac {3 }{8 } \) (i) yes (ii) no

Explanation: The denominators of both the fractions are the same so Subtract the numerators. \(\frac { 1}{ 4 } – \frac {2}{4 } = \frac {1 }{4 } \) Thus the above statement is not correct.

c. \(\frac { 5}{ 8} – \frac {4}{ 8 } = \frac {1 }{8 } \) (i) yes (ii) no

Answer: yes

Explanation: The denominators of both the fractions are the same so Subtract the numerators. \(\frac { 5}{ 8} – \frac {4}{ 8 } = \frac {1 }{8 } \) Thus the above statement is correct.

d. \(\frac { 4}{ 9 } – \frac {2}{ 9 } = \frac {6 }{9 } \) (i) yes (ii) no Answer: no

Explanation: The denominators of both the fractions are the same so Subtract the numerators. d. \(\frac { 4}{ 9 } – \frac {2}{ 9 } = \frac {2 }{9 } \) Thus the above statement is not correct.

Answer: Jane’s Answer Makes Sense. Because the numerators are the same but the denominators are different. So, in order to add the fractions first, they have to make the denominators equal. 1/4 + 1/8 = 2/8 + 1/8 = 3/8

Find the sum or difference.

Explanation: The denominators of both the fractions are the same so add the numerators. \(\frac{4}{12}\) + \(\frac{8}{12}\) = \(\frac{12}{12}\)

Question 2. \(\frac{3}{6}-\frac{1}{6}\) = \(\frac{□}{□}\)

Explanation: The denominators of both the fractions are the same so Subtract the numerators. \(\frac{3}{6}\) – \(\frac{1}{6}\) = \(\frac{2}{6}\)

Question 3. \(\frac{4}{5}-\frac{3}{5}\) = \(\frac{□}{□}\)

Explanation: The denominators of both the fractions are the same so Subtract the numerators. \(\frac{4}{5}\) – \(\frac{3}{5}\) = \(\frac{1}{5}\)

Question 4. \(\frac{6}{10}+\frac{3}{10}\) = \(\frac{□}{□}\)

Explanation: The denominators of both the fractions are the same so add the numerators. \(\frac{6}{10}+\frac{3}{10}\) = \(\frac{9}{10}\)

Question 5. 1 – \(\frac{3}{8}\) = \(\frac{□}{□}\)

Answer: 5/8

Explanation: The denominators of both the fractions are the same so Subtract the numerators. 1 – \(\frac{3}{8}\) = \(\frac{8}{8}\) – \(\frac{3}{8}\) = \(\frac{5}{8}\)

Question 6. \(\frac{1}{4}+\frac{2}{4}\) = \(\frac{□}{□}\)

Explanation: The denominators of both the fractions are the same so add the numerators. \(\frac{1}{4}+\frac{2}{4}\) = \(\frac{3}{4}\)

Question 7. \(\frac{9}{12}-\frac{5}{12}\) = \(\frac{□}{□}\)

Explanation: The denominators of both the fractions are the same so Subtract the numerators. \(\frac{9}{12}-\frac{5}{12}\) = \(\frac{4}{12}\)

Question 8. \(\frac{5}{6}-\frac{2}{6}\) = \(\frac{□}{□}\)

Explanation: The denominators of both the fractions are the same so Subtract the numerators. \(\frac{5}{6}-\frac{2}{6}\) = \(\frac{3}{6}\)

Question 9. \(\frac{2}{3}+\frac{1}{3}\) = \(\frac{□}{□}\)

Answer: 3/3 = 1

Explanation: The denominators of both the fractions are the same so add the numerators. \(\frac{2}{3}+\frac{1}{3}\) = \(\frac{3}{3}\) = 1

Problem Solving

Question 10. Guy finds how far his house is from several locations and makes the table shown. How much farther away from Guy’s house is the library than the cafe? \(\frac{□}{□}\)

Answer: \(\frac{5}{10}\) mile

Explanation: The distance from Guy’s house to the library is \(\frac{9}{10}\) mile The distance from Guy’s house to the cafe is \(\frac{4}{10}\) mile To find how much farther away from Guy’s house is the library than the cafe subtract both the fractions. \(\frac{9}{10}\) – \(\frac{4}{10}\) = \(\frac{5}{10}\) mile

Question 11. If Guy walks from his house to school and back, how far does he walk? \(\frac{□}{□}\)

Answer: 10/10 mile

Explanation: The distance from Guy’s house to school = \(\frac{5}{10}\) mile From school to house \(\frac{5}{10}\) mile \(\frac{5}{10}\) + \(\frac{5}{10}\) = \(\frac{10}{10}\) mile

Question 1. Mr. Angulo buys \(\frac{5}{8}\) pound of red grapes and \(\frac{3}{8}\)pound of green grapes. How many pounds of grapes did Mr. Angulo buy in all? Options: a. \(\frac{1}{8}\) pound b. \(\frac{2}{8}\) pound c. 1 pound d. 2 pounds

Answer: 1 pound

Explanation: Given that, Mr. Angulo buys \(\frac{5}{8}\) pound of red grapes and \(\frac{3}{8}\)pound of green grapes. \(\frac{5}{8}\) + \(\frac{3}{8}\) = \(\frac{8}{8}\) = 1 Thus the correct answer is option c.

Answer: \(\frac{7}{8}\) – \(\frac{2}{8}\) = \(\frac{5}{8}\)

Explanation: By seeing the above figure we can say that, the equation of the model is \(\frac{7}{8}\) – \(\frac{2}{8}\) = \(\frac{5}{8}\) Thus the correct answer is option d.

Spiral Review

Question 3. There are 6 muffins in a package. How many packages will be needed to feed 48 people if each person has 2 muffins? Options: a. 4 b. 8 c. 16 d. 24

Explanation: There are 6 muffins in a package. Number of people = 48 48/6 = 8 Also given that each person gets 2 muffins. 8 × 2 = 16 Thus the correct answer is option c.

Question 4. Camp Oaks gets 32 boxes of orange juice and 56 boxes of apple juice. Each shelf in the cupboard can hold 8 boxes of juice. What is the least number of shelves needed for all the juice boxes? Options: a. 4 b. 7 c. 11 d. 88

Explanation: Given, Camp Oaks gets 32 boxes of orange juice and 56 boxes of apple juice. Each shelf in the cupboard can hold 8 boxes of juice. First, add the boxes of orange juice and apple juice. 32 + 56 = 88 boxes of juice Now divide 88 by 8 88/8 = 11 Thus the correct answer is option c.

Question 5. A machine makes 18 parts each hour. If the machine operates 24 hours a day, how many parts can it make in one day Options: a. 302 b. 332 c. 362 d. 432

Answer: 432

Explanation: Given, A machine makes 18 parts each hour. Multiply the number of parts with the number of hours. 18 × 24 = 432 parts in a day. Thus the correct answer is option d.

Answer: \(\frac{5}{6}\) – \(\frac{4}{6}\) = \(\frac{1}{6}\)

Explanation: By observing the figure we can say that the equation is \(\frac{5}{6}\) – \(\frac{4}{6}\) = \(\frac{1}{6}\). Thus the correct answer is option a.

Question 1. A ___________ always has a numerator of 1. ________________

Answer: unit fraction

Explanation: A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer.

Write the fraction as a sum of unit fractions.

Question 2. Type below: ____________

Answer: 1/3 + 1/3 + 1/3

Explanation: A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. The unit fraction of 3/3 is 1/3 + 1/3 + 1/3

Question 3. Type below: ____________

Answer: 1/12 + 1/12 + 1/12 + 1/12

A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. The unit fraction of 4/12 is 1/12 + 1/12 + 1/12 + 1/12.

Explanation: By using the above model we can write the equation 3/5 – 2/5 = 1/5

Explanation: By using the above model we can write the equation 5/6 – 1/6 = 4/6

Use the model to solve the equation.

Explanation: The denominators of both the fractions are the same so add the numerators. \(\frac{3}{8}+\frac{2}{8}\) = \(\frac{5}8}\)

Question 7. \(\frac{4}{10}+\frac{5}{10}\) = \(\frac{□}{□}\)

Explanation: The denominators of both the fractions are the same so add the numerators. \(\frac{4}{10}+\frac{5}{10}\) = \(\frac{9}{10}\)

Question 8. \(\frac{9}{12}-\frac{7}{12}\) = \(\frac{□}{□}\)

Answer: 2/12

Explanation: The denominators of both the fractions are the same so subtract the numerators. \(\frac{9}{12}-\frac{7}{12}\) = \(\frac{2}{12}\)

Answer: 3/3

Explanation: The denominators of both the fractions are the same so add the numerators. \(\frac{2}{3}+\frac{1}{3}\) = \(\frac{3}{3}\)

Question 10. \(\frac{1}{5}+\frac{3}{5}\) = \(\frac{□}{□}\)

Explanation: The denominators of both the fractions are the same so add the numerators. \(\frac{1}{5}+\frac{3}{5}\) = \(\frac{4}{5}\)

Question 11. \(\frac{2}{6}+\frac{2}{6}\) = \(\frac{□}{□}\)

Explanation: The denominators of both the fractions are the same so add the numerators. \(\frac{2}{6}+\frac{2}{6}\) = \(\frac{4}{6}\)

Question 12. \(\frac{4}{4}-\frac{2}{4}\) = \(\frac{□}{□}\)

Explanation: The denominators of both the fractions are the same so subtract the numerators. \(\frac{4}{4}-\frac{2}{4}\) = \(\frac{2}{4}\)

Question 13. \(\frac{7}{8}-\frac{4}{8}\) = \(\frac{□}{□}\)

Explanation: The denominators of both the fractions are the same so subtract the numerators. \(\frac{7}{8}-\frac{4}{8}\) = \(\frac{3}{8}\)

Question 14. Tyrone mixed \(\frac{7}{12}\) quart of red paint with \(\frac{1}{12}\) quart of yellow paint. How much paint does Tyrone have in the mixture? \(\frac{□}{□}\) quart

Answer: 8/12 quart

Explanation: Given that, Tyrone mixed \(\frac{7}{12}\) quart of red paint with \(\frac{1}{12}\) quart of yellow paint. Add both the fraction of paints. \(\frac{7}{12}\) + \(\frac{1}{12}\) = \(\frac{8}{12}\) quart Therefore Tyrone has \(\frac{8}{12}\) quart in the mixture.

Question 15. Jorge lives \(\frac{6}{8}\) mile from school and \(\frac{2}{8}\) mile from a ballpark. How much farther does Jorge live from school than from the ballpark? \(\frac{□}{□}\) mile

Answer: 4/8 mile

Explanation: Given, Jorge lives \(\frac{6}{8}\) mile from school and \(\frac{2}{8}\) mile from a ballpark. Subtract both the fractions. \(\frac{6}{8}\) – \(\frac{2}{8}\) = \(\frac{4}{8}\) Therefore Jorge live \(\frac{4}{8}\) mile from school than from the ballpark.

Question 16. Su Ling started an art project with 1 yard of felt. She used \(\frac{2}{6}\) yard on Tuesday and \(\frac{3}{6}\) yard on Wednesday. How much felt does Su Ling have left? \(\frac{□}{□}\) yard

Answer: 1/6 yard

Explanation: Given, Su Ling started an art project with 1 yard of felt. She used \(\frac{2}{6}\) yard on Tuesday and \(\frac{3}{6}\) yard on Wednesday. \(\frac{3}{6}\) – \(\frac{2}{6}\) = \(\frac{1}{6}\) yard Therefore, Su Ling \(\frac{1}{6}\) yard left.

Question 17. Eloise hung artwork on \(\frac{2}{5}\) of a bulletin board. She hung math papers on \(\frac{1}{5}\) of the same bulletin board. What part of the bulletin board has artwork or math papers? \(\frac{□}{□}\)

Explanation: Given, Eloise hung artwork on \(\frac{2}{5}\) of a bulletin board. She hung math papers on \(\frac{1}{5}\) of the same bulletin board. \(\frac{2}{5}\) + \(\frac{1}{5}\) = \(\frac{3}{5}\) \(\frac{3}{5}\) part of the bulletin board has artwork or math papers.

Write the unknown numbers. Write mixed numbers above the number line and fractions greater than one below the number line.

Write the mixed number as a fraction.

Question 2. 1 \(\frac{1}{8}\) = \(\frac{□}{□}\)

Answer: 9/8

Explanation: Given the expression, 1 \(\frac{1}{8}\) Convert from the mixed fraction to the improper fraction. 1 \(\frac{1}{8}\) = (1 × 8 + 1)/8 = 9/8

Question 3. 1 \(\frac{3}{5}\) = \(\frac{□}{□}\)

Answer: \(\frac{8}{5}\)

Explanation: Given the expression, 1 \(\frac{3}{5}\) Convert from the mixed fraction to the improper fraction. 1 \(\frac{3}{5}\) = (5 × 1 + 3)/5 = \(\frac{8}{5}\)

Question 4. 1 \(\frac{2}{3}\) = \(\frac{□}{□}\)

Answer: 5/3

Explanation: Given the expression, 1 \(\frac{2}{3}\) Convert from the mixed fraction to the improper fraction. 1 \(\frac{2}{3}\) = (3 × 1 + 2)/3 = \(\frac{5}{3}\)

Write the fraction as a mixed number.

Question 5. \(\frac{11}{4}\) = _____ \(\frac{□}{□}\)

Answer: 2 \(\frac{3}{4}\)

Explanation: Given the expression, \(\frac{11}{4}\) Convert from the improper fraction to the mixed fraction. \(\frac{11}{4}\) = 2 \(\frac{3}{4}\)

Question 6. \(\frac{6}{5}\) = _____ \(\frac{□}{□}\)

Answer: 1 \(\frac{1}{5}\)

Explanation: Given the expression, \(\frac{6}{5}\) Convert from the improper fraction to the mixed fraction. \(\frac{6}{5}\) = 1 \(\frac{1}{5}\)

Question 7. \(\frac{13}{10}\) = _____ \(\frac{□}{□}\)

Answer: 1 \(\frac{3}{10}\)

Explanation: Given the expression, \(\frac{13}{10}\) Convert from the improper fraction to the mixed fraction. \(\frac{13}{10}\) = 1 \(\frac{3}{10}\)

Question 8. 2 \(\frac{7}{10}\) = \(\frac{□}{□}\)

Answer: \(\frac{27}{10}\)

Explanation: Given the expression, 2 \(\frac{7}{10}\) Convert from the mixed fraction to the improper fraction. 2 \(\frac{7}{10}\) = \(\frac{27}{10}\)

Question 9. 3 \(\frac{2}{3}\) = \(\frac{□}{□}\)

Answer: \(\frac{11}{3}\)

Explanation: Given the expression, 3 \(\frac{2}{3}\) Convert from the mixed fraction to the improper fraction. 3 \(\frac{2}{3}\) = \(\frac{11}{3}\)

Question 10. 4 \(\frac{2}{5}\) = \(\frac{□}{□}\)

Answer: \(\frac{22}{5}\)

Explanation: Given the expression, 4 \(\frac{2}{5}\) Convert from the mixed fraction to the improper fraction. 4 \(\frac{2}{5}\) = \(\frac{22}{5}\)

Use Repeated Reasoning Algebra Find the unknown numbers.

Question 11. \(\frac{13}{7}\) = 1 \(\frac{■}{7}\) ■ = _____

Answer: 1 \(\frac{6}{7}\)

Explanation: Given the expression, \(\frac{13}{7}\) Convert from the mixed fraction to the improper fraction. \(\frac{13}{7}\) = 1 \(\frac{6}{7}\)

Question 12. ■ \(\frac{5}{6}\) = \(\frac{23}{6}\) ■ = _____

Explanation: Given the expression, ■ \(\frac{5}{6}\) = \(\frac{23}{6}\) ■ \(\frac{5}{6}\) × 6 = 23 ■ ×  = 23 – 5 ■ = 18/6 ■ = 3

Question 13. \(\frac{57}{11}\) = ■ \(\frac{■}{11}\) _____ \(\frac{□}{□}\)

Answer: 5 \(\frac{2}{11}\)

Explanation: Given the expression, \(\frac{57}{11}\) = ■ \(\frac{■}{11}\) Convert from the improper fraction to the mixed fraction. \(\frac{57}{11}\) = 5 \(\frac{2}{11}\)

Question 14. Pen has \(\frac{1}{2}\)-cup and \(\frac{1}{8}\)-cup measuring cups. What are two ways he could measure out 1 \(\frac{3}{4}\) cups of flour? Type below: _________________

Answer: 3 \(\frac{1}{2}\)-cups and 2 \(\frac{1}{8}\)-cup

Explanation: Pen has \(\frac{1}{2}\)-cup and \(\frac{1}{8}\)-cup measuring cups. 1 \(\frac{3}{4}\) = \(\frac{1}{2}\) + \(\frac{1}{2}\) + \(\frac{1}{2}\) + \(\frac{1}{8}\) + \(\frac{1}{8}\) = 1 \(\frac{3}{4}\) = 3 \(\frac{1}{2}\)-cups + 2 \(\frac{1}{8}\)-cup

Question 15. Juanita is making bread. She needs 3 \(\frac{1}{2}\) cups of flour. Juanita only has a \(\frac{1}{4}\)-cup measuring cup. How many \(\frac{1}{4}\) cups of flour will Juanita use to prepare the bread? _____ \(\frac{1}{4}\) cups of flour

Answer: 14 \(\frac{1}{4}\) cups of flour

Explanation: Juanita is making bread. She needs 3 \(\frac{1}{2}\) cups of flour. Juanita only has a \(\frac{1}{4}\)-cup measuring cup. 3 \(\frac{1}{2}\) = \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) Therefore she needs 14 \(\frac{1}{4}\) cups of flour.

Question 16. Reason Quantitatively Cal is making energy squares. How many \(\frac{1}{2}\) cups of peanut butter are used in the recipe? _____ \(\frac{1}{2}\) cups of peanut butter

Answer: 3 \(\frac{1}{2}\) cups of peanut butter

Explanation: Given that 1 \(\frac{1}{2}\) cups of peanut butter are used in the recipe. We have to find how many \(\frac{1}{2}\) cups of peanut butter are used in the recipe. \(\frac{1}{2}\) + \(\frac{1}{2}\)  + \(\frac{1}{2}\) Therefore 3 \(\frac{1}{2}\) cups of peanut butter are used in the recipe.

Question 17. Suppose Cal wants to make 2 times as many energy squares as the recipe makes. How many cups of bran cereal should he use? Write your answer as a mixed number and as a fraction greater than 1 in simplest form. Type below: ____________

Answer: Take the amount of bran Cal is using and multiply it by 2 Given that 3 \(\frac{1}{4}\) cups of bran cereal is used in the recipe. 3 \(\frac{1}{4}\) × 2 = \(\frac{13}{4}\) × 2 = \(\frac{13}{2}\) = 6 \(\frac{1}{2}\) Thus 6 \(\frac{1}{2}\) cups of bran cereal he should use.

Question 18. Cal added 2 \(\frac{3}{8}\) cups of raisins. Write this mixed number as a fraction greater than 1 in the simplest form. \(\frac{□}{□}\)

Answer: \(\frac{19}{8}\)

Explanation: Given, Cal added 2 \(\frac{3}{8}\) cups of raisins. Convert from the mixed fraction to the improper fraction. 2 \(\frac{3}{8}\) = \(\frac{19}{8}\)

Question 19. Jenn is preparing brown rice. She needs 1 \(\frac{1}{2}\) cups of brown rice and 2 cups of water. Jenn has only a \(\frac{1}{8}\)– cup measuring cup. How many \(\frac{1}{8}\) cups each of rice and water will Jenn use to prepare the rice? brown rice: ________ \(\frac{1}{8}\) cups water: _________ \(\frac{1}{8}\) cups

Answer: Number of water cups = 16 Number of brown rice cups = 12

Explanation: Brown rice needed = 1 1/2 cups = 3/2 cups Water needed = 2 cups Measuring cups = 1/8 No. of cups used of water = 2/1/8 = 16 No. of cups used of rice = 3/2/1/8 = 12 cups

Question 2. 4 \(\frac{1}{3}\) \(\frac{□}{□}\)

Answer: \(\frac{13}{3}\)

Explanation: \(\frac{3}{3}\) + \(\frac{3}{3}\) + \(\frac{3}{3}\) + \(\frac{3}{3}\) + \(\frac{1}{3}\) = \(\frac{13}{3}\)

Question 3. 1 \(\frac{2}{5}\) \(\frac{□}{□}\)

Answer: \(\frac{7}{5}\)

Explanation: \(\frac{5}{5}\) + \(\frac{2}{5}\) = \(\frac{7}{5}\)

Question 4. 3 \(\frac{3}{2}\) \(\frac{□}{□}\)

Answer: \(\frac{9}{2}\)

Explanation: \(\frac{2}{2}\) + \(\frac{2}{2}\) + \(\frac{2}{2}\) + \(\frac{2}{2}\) + \(\frac{1}{2}\) = \(\frac{9}{2}\)

Question 5. 4 \(\frac{1}{8}\) \(\frac{□}{□}\)

Answer: \(\frac{33}{8}\)

Explanation: \(\frac{8}{8}\) + \(\frac{8}{8}\) + \(\frac{8}{8}\) + \(\frac{8}{8}\) + \(\frac{1}{8}\) = \(\frac{33}{8}\)

Question 6. 1 \(\frac{7}{10}\) \(\frac{□}{□}\)

Answer: \(\frac{17}{10}\)

Explanation: \(\frac{10}{10}\) + \(\frac{7}{10}\) = \(\frac{17}{10}\)

Question 7. 5 \(\frac{1}{2}\) \(\frac{□}{□}\)

Answer: \(\frac{11}{2}\)

Explanation: \(\frac{2}{2}\) + \(\frac{2}{2}\) + \(\frac{2}{2}\) + \(\frac{2}{2}\) + \(\frac{2}{2}\) + \(\frac{1}{2}\) = \(\frac{11}{2}\)

Question 8. 2 \(\frac{3}{8}\) \(\frac{□}{□}\)

Explanation: \(\frac{8}{8}\) + \(\frac{8}{8}\) + \(\frac{3}{8}\)

Question 9. \(\frac{31}{6}\) ______ \(\frac{□}{□}\)

Answer: 5 \(\frac{1}{6}\)

Explanation: \(\frac{6}{6}\) + \(\frac{6}{6}\) + \(\frac{6}{6}\) + \(\frac{6}{6}\) + \(\frac{6}{6}\) + \(\frac{1}{6}\) 1 + 1 + 1 + 1 + 1 + \(\frac{1}{6}\) = 5 \(\frac{1}{6}\)

Question 10. \(\frac{20}{10}\) ______ \(\frac{□}{□}\)

Explanation: \(\frac{10}{10}\) + \(\frac{10}{10}\) = 1 + 1 = 2

Question 11. \(\frac{15}{8}\) ______ \(\frac{□}{□}\)

Answer: 1 \(\frac{7}{8}\)

Explanation: \(\frac{8}{8}\) + \(\frac{7}{8}\) 1 + \(\frac{7}{8}\) = 1 \(\frac{7}{8}\)

Question 12. \(\frac{13}{6}\) ______ \(\frac{□}{□}\)

Answer: 2 \(\frac{1}{6}\)

Explanation: \(\frac{6}{6}\) + \(\frac{6}{6}\) + \(\frac{1}{6}\) = 1 + 1 + \(\frac{1}{6}\) = 2 \(\frac{1}{6}\)

Question 13. \(\frac{23}{10}\) ______ \(\frac{□}{□}\)

Answer: 2 \(\frac{3}{10}\)

Explanation: \(\frac{10}{10}\) + \(\frac{10}{10}\) + \(\frac{3}{10}\) 1 + 1 + \(\frac{3}{10}\) = 2 \(\frac{3}{10}\)

Question 14. \(\frac{19}{5}\) ______ \(\frac{□}{□}\)

Answer: 3 \(\frac{4}{5}\)

Explanation: \(\frac{5}{5}\) + \(\frac{5}{5}\) + \(\frac{5}{5}\) + \(\frac{4}{5}\) 1 + 1 + 1 + \(\frac{4}{5}\) = 3 \(\frac{4}{5}\)

Question 15. \(\frac{11}{3}\) ______ \(\frac{□}{□}\)

Answer: 3 \(\frac{2}{3}\)

Explanation: \(\frac{3}{3}\) + \(\frac{3}{3}\) + \(\frac{3}{3}\) + \(\frac{2}{3}\) = 1 + 1 + 1 \(\frac{2}{3}\) = 3 \(\frac{2}{3}\)

Question 16. \(\frac{9}{2}\) ______ \(\frac{□}{□}\)

Answer: 4 \(\frac{1}{2}\)

Explanation: \(\frac{2}{2}\) + \(\frac{2}{2}\) + \(\frac{2}{2}\) + \(\frac{2}{2}\) + \(\frac{1}{2}\) = 1 + 1 + 1 + 1 + \(\frac{1}{2}\) = 4 \(\frac{1}{2}\)

Question 17. A recipe calls for 2 \(\frac{2}{4}\) cups of raisins, but Julie only has a \(\frac{1}{4}\) -cup measuring cup. How many \(\frac{1}{4}\) cups does Julie need to measure out 2 \(\frac{2}{4}\) cups of raisins? She needs ______ \(\frac{1}{4}\) cups

Answer: 10 \(\frac{1}{4}\) cups

Explanation: Given, A recipe calls for 2 \(\frac{2}{4}\) cups of raisins, but Julie only has a \(\frac{1}{4}\) -cup measuring cup. \(\frac{4}{4}\) + \(\frac{4}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) = 10 \(\frac{1}{4}\) cups

Question 18. If Julie needs 3 \(\frac{1}{4}\) cups of oatmeal, how many \(\frac{1}{4}\) cups of oatmeal will she use? She will use ______ \(\frac{1}{4}\) cups of oatmeal

Answer: 13 \(\frac{1}{4}\) cups of oatmeal

Explanation: \(\frac{4}{4}\) + \(\frac{4}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) = 13 \(\frac{1}{4}\) Therefore Julie needs 13 \(\frac{1}{4}\) cups of oatmeal.

Question 1. Which of the following is equivalent to \(\frac{16}{3}\) ? Options: a. 3 \(\frac{1}{5}\) b. 3 \(\frac{2}{5}\) c. 5 \(\frac{1}{3}\) d. 5 \(\frac{6}{3}\)

Answer: 5 \(\frac{1}{3}\)

Explanation: Convert from improper fraction to the mixed fraction. \(\frac{16}{3}\) = \(\frac{3}{3}\) + \(\frac{3}{3}\) + \(\frac{3}{3}\) + \(\frac{3}{3}\) + \(\frac{3}{3}\) + \(\frac{1}{3}\) = 5 \(\frac{1}{3}\) Thus the correct answer is option c.

Question 2. Stacey filled her \(\frac{1}{2}\)cup measuring cup seven times to have enough flour for a cake recipe. How much flour does the cake recipe call for? Options: a. 3 cups b. 3 \(\frac{1}{2}\) cups c. 4 cups d. 4 \(\frac{1}{2}\) cups

Answer: 3 \(\frac{1}{2}\) cups

Explanation: Given, Stacey filled her \(\frac{1}{2}\)cup measuring cup seven times to have enough flour for a cake recipe. \(\frac{2}{2}\) + \(\frac{2}{2}\) + \(\frac{2}{2}\) + \(\frac{1}{2}\) 1 + 1 + 1 + \(\frac{1}{2}\) = 3 \(\frac{1}{2}\) cups Thus the correct answer is option b.

Question 3. Becki put some stamps into her stamp collection book. She put 14 stamps on each page. If she completely filled 16 pages, how many stamps did she put in the book? Options: a. 224 b. 240 c. 272 d. 275

Answer: 224

Explanation: Becki put some stamps into her stamp collection book. She put 14 stamps on each page. If she completely filled 16 pages Multiply 14 with 16 pages. 14 × 16 = 224 pages Thus the correct answer is option a.

Question 4. Brian is driving 324 miles to visit some friends. He wants to get there in 6 hours. How many miles does he need to drive each hour? Options: a. 48 miles b. 50 miles c. 52 miles d. 54 miles

Answer: 54 miles

Explanation: Brian is driving 324 miles to visit some friends. He wants to get there in 6 hours. Divide the number of miles by hours. 324/6 = 54 miles Thus the correct answer is option d.

Question 5. During a bike challenge, riders have to collect various colored ribbons. Each \(\frac{1}{2}\) mile they collect a red ribbon, each \(\frac{1}{8}\) mile they collect a green ribbon, and each \(\frac{1}{4}\) mile they collect a blue ribbon. Which colors of ribbons will be collected at the \(\frac{3}{4}\) mile marker? Options: a. red and green b. red and blue c. green and blue d. red, green, and blue

Answer: green and blue

Explanation: Given, During a bike challenge, riders have to collect various colored ribbons. Each \(\frac{1}{2}\) mile they collect a red ribbon, each \(\frac{1}{8}\) mile they collect a green ribbon, and each \(\frac{1}{4}\) mile they collect a blue ribbon. Green and Blue colors of ribbons will be collected at the \(\frac{3}{4}\) mile marker. Thus the correct answer is option c.

Question 6. Stephanie had \(\frac{7}{8}\) pound of bird seed. She used \(\frac{3}{8}\) pound to fill a bird feeder. How much bird seed does Stephanie have left? Options: a. \(\frac{3}{8}\) pound b. \(\frac{4}{8}\) pound c. 1 pound d. \(\frac{10}{8}\) pound

Answer: \(\frac{4}{8}\) pound

Explanation: Given, Stephanie had \(\frac{7}{8}\) pound of bird seed. She used \(\frac{3}{8}\) pound to fill a bird feeder. \(\frac{7}{8}\) – \(\frac{3}{8}\) = \(\frac{4}{8}\) pound Thus the correct answer is option b.

Write the sum as a mixed number with the fractional part less than 1.

Question 1. 1 \(\frac{1}{6}\) +3 \(\frac{3}{6}\) ———————– _______ \(\frac{□}{□}\)

Answer: 4 \(\frac{2}{3}\)

Explanation: 1 \(\frac{1}{6}\) +3 \(\frac{3}{6}\) 4 \(\frac{4}{6}\) = 4 \(\frac{2}{3}\)

Question 2. 1 \(\frac{4}{5}\) +7 \(\frac{2}{5}\) ———————– _______ \(\frac{□}{□}\)

Answer: 9 \(\frac{1}{5}\)

Explanation: 1 \(\frac{4}{5}\) +7 \(\frac{2}{5}\) 8 \(\frac{6}{5}\) = 9 \(\frac{1}{5}\)

Question 3. 2 \(\frac{1}{2}\) +3 \(\frac{1}{2}\) ———————– _______

Explanation: 2 \(\frac{1}{2}\) +3 \(\frac{1}{2}\) 5 \(\frac{2}{2}\) = 6

Find the difference.

Question 4. 3 \(\frac{7}{12}\) -2 \(\frac{5}{12}\) ———————– _______ \(\frac{□}{□}\)

Answer: 1 \(\frac{1}{6}\)

Explanation: 3 \(\frac{7}{12}\) -2 \(\frac{5}{12}\) 1 \(\frac{2}{12}\) = 1 \(\frac{1}{6}\)

Question 5. 4 \(\frac{2}{3}\) -3 \(\frac{1}{3}\) ———————– _______ \(\frac{□}{□}\)

Answer: 1 \(\frac{1}{3}\)

Explanation: 4 \(\frac{2}{3}\) -3 \(\frac{1}{3}\) 1 \(\frac{1}{3}\)

Question 6. 6 \(\frac{9}{10}\) -3 \(\frac{7}{10}\) ———————– _______ \(\frac{□}{□}\)

Answer: 3 \(\frac{1}{5}\)

Explanation: 6 \(\frac{9}{10}\) -3 \(\frac{7}{10}\) 3 \(\frac{2}{10}\)

Question 7. 7 \(\frac{4}{6}\) +4 \(\frac{3}{6}\) ———————– _______ \(\frac{□}{□}\)

Answer: 12 \(\frac{1}{6}\)

Explanation: 7 \(\frac{4}{6}\) +4 \(\frac{3}{6}\) 12 \(\frac{1}{6}\)

Question 8. 8 \(\frac{1}{3}\) +3 \(\frac{2}{3}\) ———————– _______ \(\frac{□}{□}\)

Explanation: 8 \(\frac{1}{3}\) +3 \(\frac{2}{3}\) 11 \(\frac{3}{3}\) = 12

Question 9. 5 \(\frac{4}{8}\) +3 \(\frac{5}{8}\) ———————– _______ \(\frac{□}{□}\)

Answer: 9 \(\frac{1}{8}\)

Explanation: 5 \(\frac{4}{8}\) +3 \(\frac{5}{8}\) 9 \(\frac{1}{8}\)

Question 10. 5 \(\frac{5}{12}\) +4 \(\frac{2}{12}\) ———————– _______ \(\frac{□}{□}\)

Answer: 9 \(\frac{7}{12}\)

Explanation: 5 \(\frac{5}{12}\) +4 \(\frac{2}{12}\) 9 \(\frac{7}{12}\)

Question 11. 5 \(\frac{7}{8}\) -2 \(\frac{3}{8}\) ———————– _______ \(\frac{□}{□}\)

Answer: 3 \(\frac{1}{2}\)

Explanation: 5 \(\frac{7}{8}\) -2 \(\frac{3}{8}\) 3 \(\frac{1}{2}\)

Question 12. 5 \(\frac{7}{12}\) -4 \(\frac{1}{12}\) ———————– _______ \(\frac{□}{□}\)

Answer: 1 \(\frac{1}{2}\)

Explanation: 5 \(\frac{7}{12}\) -4 \(\frac{1}{12}\) 1 \(\frac{1}{2}\)

Question 13. 3 \(\frac{5}{10}\) -1 \(\frac{3}{10}\) ———————– _______ \(\frac{□}{□}\)

Answer: 2 \(\frac{1}{5}\)

Explanation: 3 \(\frac{5}{10}\) -1 \(\frac{3}{10}\) 2 \(\frac{1}{5}\)

Question 14. 7 \(\frac{3}{4}\) -2 \(\frac{2}{4}\) ———————– _______ \(\frac{□}{□}\)

Answer: 5 \(\frac{1}{4}\)

Explanation: 7 \(\frac{3}{4}\) -2 \(\frac{2}{4}\) 5 \(\frac{1}{4}\)

Practice: Copy and Solve Find the sum or difference.

Question 15. \(1 \frac{3}{8}+2 \frac{7}{8}\) = _______ \(\frac{□}{□}\)

Answer: 4 \(\frac{1}{4}\)

Explanation: First add the whole numbers 1 + 2 = 3 3/8 + 7/8 = 10/8 Convert from improper fraction to the mixed fraction 10/8 = 5/4 = 1 1/4 3 + 1 1/4 = 4 1/4

Question 16. \(6 \frac{5}{8}\) – 4 = _______ \(\frac{□}{□}\)

Answer: 2 \(\frac{5}{8}\)

Explanation: \(6 \frac{5}{8}\) – 4 Subtract the whole numbers 6 – 4 = 2 = 2 \(\frac{5}{8}\)

Question 17. \(9 \frac{1}{2}+8 \frac{1}{2}\) = _______

Explanation: 9 \(\frac{1}{2}\) + 8 \(\frac{1}{2}\) 18

Question 18. \(6 \frac{3}{5}+4 \frac{3}{5}\) = _______ \(\frac{□}{□}\)

Answer: 11 \(\frac{1}{5}\)

Explanation: 6 \(\frac{3}{5}\) + 4 \(\frac{3}{5}\) 11 \(\frac{1}{5}\)

Question 19. \(8 \frac{7}{10}-\frac{4}{10}\) = _______ \(\frac{□}{□}\)

Answer: 8 \(\frac{3}{10}\)

Explanation: 8 \(\frac{7}{10}\)  – \(\frac{4}{10}\) 8 \(\frac{3}{10}\)

Question 20. \(7 \frac{3}{5}-6 \frac{3}{5}\) = _______

Explanation: 7 \(\frac{3}{5}\) + 6 \(\frac{3}{5}\) 1

Solve. Write your answer as a mixed number.

Question 21. Make Sense of Problems The driving distance from Alex’s house to the museum is 6 \(\frac{7}{10}\) miles. What is the round-trip distance? _______ \(\frac{□}{□}\) miles

Answer: 13 \(\frac{2}{5}\) miles

Explanation: Given that, The driving distance from Alex’s house to the museum is 6 \(\frac{7}{10}\) miles. To find the round-trip distance we have to multiply the driving distance with 2. 6 \(\frac{7}{10}\) × 2 = 13 \(\frac{4}{10}\) = 13 \(\frac{2}{5}\) miles

Question 22. The driving distance from the sports arena to Kristina’s house is 10 \(\frac{9}{10}\) miles. The distance from the sports arena to Luke’s house is 2 \(\frac{7}{10}\) miles. How much greater is the driving distance between the sports arena and Kristina’s house than between the sports arena and Luke’s house? _______ \(\frac{□}{□}\) miles

Answer: 8 \(\frac{1}{5}\) miles

Explanation: Given, The driving distance from the sports arena to Kristina’s house is 10 \(\frac{9}{10}\) miles. The distance from the sports arena to Luke’s house is 2 \(\frac{7}{10}\) miles. 10 \(\frac{9}{10}\) –  2 \(\frac{7}{10}\) First, subtract the whole numbers and then subtract the fractions 10 – 2 = 8 \(\frac{9}{10}\) – \(\frac{7}{10}\) = \(\frac{1}{5}\) = 8 \(\frac{1}{5}\) miles

Question 23. Pedro biked from his house to the nature preserve, a distance of 23 \(\frac{4}{5}\) miles. Sandra biked from her house to the lake, a distance of 12 \(\frac{2}{5}\) miles. How many miles less did Sandra bike than Pedro? _______ \(\frac{□}{□}\) miles

Answer: 11 \(\frac{2}{5}\) miles

Explanation: Pedro biked from his house to the nature preserve, a distance of 23 4/5 miles. Converting 23 4/5 miles to an improper fraction, it becomes 119/5 miles. Sandra biked from her house to the lake, a distance of 12 2/5 miles. Converting 12 2/5 miles to an improper fraction, it becomes 62/5 miles. Therefore, the difference in the number of miles biked by Sandra and Pedro is 119/5 – 62/5 = 57/5 = 11 2/5 miles

Question 24. During the Martinez family trip, they drove from home to a ski lodge, a distance of 55 \(\frac{4}{5}\) miles, and then drove an additional 12 \(\frac{4}{5}\) miles to visit friends. If the family drove the same route back home, what was the distance traveled during their trip? _______ \(\frac{□}{□}\) miles

Answer: 68 \(\frac{3}{5}\) miles

Explanation: Given, During the Martinez family trip, they drove from home to a ski lodge, a distance of 55 \(\frac{4}{5}\) miles, and then drove an additional 12 \(\frac{4}{5}\) miles to visit friends. 55 \(\frac{4}{5}\) + 12 \(\frac{4}{5}\) = 67 \(\frac{8}{5}\) = 68 \(\frac{3}{5}\) miles

Question 25. For 25a–25d, select True or False for each statement. a. 2 \(\frac{3}{8}\) + 1 \(\frac{6}{8}\) is equal to 4 \(\frac{1}{8}\). i. True ii. False

Answer: True

Explanation: Given the statement 2 \(\frac{3}{8}\) + 1 \(\frac{6}{8}\) is equal to 4 \(\frac{1}{8}\). First add the whole numbers 2 + 1 = 3 \(\frac{3}{8}\) + \(\frac{6}{8}\) = \(\frac{9}{8}\) Convert the improper fraction to the mixed fraction. \(\frac{9}{8}\) = 1 \(\frac{1}{8}\) 3 +1 \(\frac{1}{8}\) = 4 \(\frac{1}{8}\). Thus the above statement is true.

Question 25. b. 1 \(\frac{1}{6}\) + 1 \(\frac{4}{12}\) is equal to 2 \(\frac{2}{12}\). i. True ii. False

Answer: False

Explanation: 1 \(\frac{1}{6}\) + 1 \(\frac{4}{12}\) is equal to 2 \(\frac{2}{12}\). First add the whole numbers 1 + 1 = 2 \(\frac{1}{6}\) = \(\frac{2}{12}\)

\(\frac{2}{12}\) + \(\frac{4}{12}\) = \(\frac{6}{12}\) = 2 \(\frac{6}{12}\) Thus the above statement is false.

Question 25. c. 5 \(\frac{5}{6}\) – 2 \(\frac{4}{6}\) is equal to 1 \(\frac{3}{6}\). i. True ii. False

Explanation: 5 \(\frac{5}{6}\) – 2 \(\frac{4}{6}\) is equal to 1 \(\frac{3}{6}\). 5 – 2 = 3 \(\frac{5}{6}\) – \(\frac{4}{6}\) = \(\frac{1}{6}\) = 3 \(\frac{1}{6}\) Thus the above statement is false.

Question 25. d. 5 \(\frac{5}{8}\) – 3 \(\frac{2}{8}\) is equal to 2 \(\frac{3}{8}\). i. True ii. False

Explanation: 5 \(\frac{5}{8}\) – 3 \(\frac{2}{8}\) is equal to 2 \(\frac{3}{8}\) First, subtract the whole numbers 5 – 3 = 2 \(\frac{5}{8}\) – \(\frac{2}{8}\) = \(\frac{3}{8}\) = 2 \(\frac{3}{8}\) Thus the above statement is true.

Find the sum. Write the sum as a mixed number, so the fractional part is less than 1.

Question 2. 4 \(\frac{1}{2}\) +2 \(\frac{1}{2}\) _______ \(\frac{□}{□}\)

4 \(\frac{1}{2}\) +2 \(\frac{1}{2}\) 6 \(\frac{2}{2}\) = 6 + 1 = 7

Question 3. 2 \(\frac{2}{3}\) +3 \(\frac{2}{3}\) _______ \(\frac{□}{□}\)

Answer: 6 \(\frac{1}{3}\)

Explanation: 2 \(\frac{2}{3}\) +3 \(\frac{2}{3}\) 5 \(\frac{4}{3}\) = 5 + 1 \(\frac{1}{3}\) = 6 \(\frac{1}{3}\)

Question 4. 6 \(\frac{4}{5}\) +7 \(\frac{4}{5}\) _______ \(\frac{□}{□}\)

Answer: 14 \(\frac{3}{5}\)

Explanation: 6 \(\frac{4}{5}\) +7 \(\frac{4}{5}\) 13 \(\frac{8}{5}\) 13 + 1 \(\frac{3}{5}\) = 14 \(\frac{3}{5}\)

Question 5. 9 \(\frac{3}{6}\) +2 \(\frac{2}{6}\) _______ \(\frac{□}{□}\)

Answer: 11 \(\frac{5}{6}\)

Explanation: 9 \(\frac{3}{6}\) +2 \(\frac{2}{6}\) 11 \(\frac{5}{6}\)

Question 6. 8 \(\frac{4}{12}\) +3 \(\frac{6}{12}\) _______ \(\frac{□}{□}\)

Answer: 11 \(\frac{10}{12}\)

Explanation: 8 \(\frac{4}{12}\) +3 \(\frac{6}{12}\) 11 \(\frac{10}{12}\)

Question 7. 4 \(\frac{3}{8}\) +1 \(\frac{5}{8}\) _______ \(\frac{□}{□}\)

Explanation: 4 \(\frac{3}{8}\) +1 \(\frac{5}{8}\) 5 \(\frac{8}{8}\) = 5 + 1 = 6

Question 8. 9 \(\frac{5}{10}\) +6 \(\frac{3}{10}\) _______ \(\frac{□}{□}\)

Answer: 15 \(\frac{8}{10}\)

Explanation: 9 \(\frac{5}{10}\) +6 \(\frac{3}{10}\) 15 \(\frac{8}{10}\)

Question 9. 6 \(\frac{7}{8}\) -4 \(\frac{3}{8}\) _______ \(\frac{□}{□}\)

Answer: 2 \(\frac{4}{8}\)

Explanation: 6 \(\frac{7}{8}\) -4 \(\frac{3}{8}\) 2 \(\frac{4}{8}\)

Question 10. 4 \(\frac{2}{3}\) -3 \(\frac{1}{3}\) _______ \(\frac{□}{□}\)

Question 11. 6 \(\frac{4}{5}\) -3 \(\frac{3}{5}\) _______ \(\frac{□}{□}\)

Explanation: 6 \(\frac{4}{5}\) -3 \(\frac{3}{5}\) 3 \(\frac{1}{5}\)

Question 12. 7 \(\frac{3}{4}\) -2 \(\frac{1}{4}\) _______ \(\frac{□}{□}\)

Answer: 5 \(\frac{1}{2}\)

Explanation: 7 \(\frac{3}{4}\) -2 \(\frac{1}{4}\) 5 \(\frac{2}{4}\) = 5 \(\frac{1}{2}\)

Question 13. James wants to send two gifts by mail. One package weighs 2 \(\frac{3}{4}\) pounds. The other package weighs 1 \(\frac{3}{4}\) pounds. What is the total weight of the packages? _______ \(\frac{□}{□}\)

Explanation: 2 \(\frac{3}{4}\) + 1 \(\frac{3}{4}\) 4 \(\frac{1}{2}\)

Question 14. Tierra bought 4 \(\frac{3}{8}\) yards blue ribbon and 2 \(\frac{1}{8}\) yards yellow ribbon for a craft project. How much more blue ribbon than yellow ribbon did Tierra buy? _______ \(\frac{□}{□}\)

Answer: 2 \(\frac{1}{4}\)

Explanation: Given, 4 \(\frac{3}{8}\) -2 \(\frac{1}{8}\)  2 \(\frac{1}{4}\)

Question 1. Brad has two lengths of copper pipe to fit together. One has a length of 2 \(\frac{5}{12}\) feet and the other has a length of 3 \(\frac{7}{12}\) feet. How many feet of pipe does he have in all? Options: a. 5 feet b. 5 \(\frac{6}{12}\) feet c. 5 \(\frac{10}{12}\) feet d. 6 feet

Answer: 5 feet

Explanation: Given, Brad has two lengths of copper pipe to fit together. One has a length of 2 \(\frac{5}{12}\) feet and the other has a length of 3 \(\frac{7}{12}\) feet. Add both the lengths 2 \(\frac{5}{12}\) + 3 \(\frac{7}{12}\) = 5 \(\frac{12}{12}\) = 5 feet Thus the correct answer is option a.

Question 2. A pattern calls for 2 \(\frac{1}{4}\) yards of material and 1 \(\frac{1}{4}\) yards of lining. How much total fabric is needed? Options: a. 2 \(\frac{2}{4}\) yards b. 3 yards c. 3 \(\frac{1}{4}\) yards d. 3 \(\frac{2}{4}\) yards

Answer: 3 \(\frac{2}{4}\) yards

Explanation: Given, A pattern calls for 2 \(\frac{1}{4}\) yards of material and 1 \(\frac{1}{4}\) yards of lining. 2 \(\frac{1}{4}\) + 1 \(\frac{1}{4}\) = 3 + \(\frac{1}{4}\) + \(\frac{1}{4}\) = 3 \(\frac{2}{4}\) yards Thus the correct answer is option d.

Question 3. Shanice has 23 baseball trading cards of star players. She agrees to sell them for $16 each. How much will she get for the cards? Options: a. $258 b. $358 c. $368 d. $468

Answer: $368

Explanation: Given, Shanice has 23 baseball trading cards of star players. She agrees to sell them for $16 each. To find how much will she get for the cards 23 × 16 = 368 Therefore she will get $368 for the cards. Thus the correct answer is option c.

Question 4. Nanci is volunteering at the animal shelter. She wants to spend an equal amount of time playing with each dog. She has 145 minutes to play with all 7 dogs. About how much time can she spend with each dog? Options: a. about 10 minutes b. about 20 minutes c. about 25 minutes d. about 26 minutes

Answer: about 20 minutes

Explanation: Given, Nanci is volunteering at the animal shelter. She wants to spend an equal amount of time playing with each dog. She has 145 minutes to play with all 7 dogs. 145/7 = 20.7 Therefore she can spend about 20 minutes with each dog. Thus the correct answer is option b.

Question 5. Frieda has 12 red apples and 15 green apples. She is going to share the apples equally among 8 people and keep any extra apples for herself. How many apples will Frieda keep for herself? Options: a. 3 b. 4 c. 6 d. 7

Explanation: Given, Frieda has 12 red apples and 15 green apples. She is going to share the apples equally among 8 people and keep any extra apples for herself. 12 + 15 = 27 27/8 27 – 24 = 3 Thus Frieda keep for herself 3 apples. Thus the correct answer is option a.

Question 6. The Lynch family bought a house for $75,300. A few years later, they sold the house for $80,250. How much greater was the selling price than the purchase price? Options: a. $4,950 b. $5,050 c. $5,150 d. $5,950

Answer: $4,950

Explanation: Given, The Lynch family bought a house for $75,300. A few years later, they sold the house for $80,250. $80,250 – $75,300 = $4,950 Thus the correct answer is option a.

Question 1. Rename both mixed numbers as fractions. Find the difference. 3 \(\frac{3}{6}\) = \(\frac{■}{6}\) −1 \(\frac{4}{6}\) = – \(\frac{■}{6}\) —————————————- _______ \(\frac{□}{□}\)

Answer: 1 \(\frac{5}{6}\)

Explanation: Convert from mixed fractions to the improper fractions. 3 \(\frac{3}{6}\) = \(\frac{21}{6}\) 1 \(\frac{4}{6}\) = \(\frac{10}{6}\) \(\frac{21}{6}\) – \(\frac{10}{6}\) \(\frac{11}{6}\) Convert from improper fractions to the mixed fractions. \(\frac{11}{6}\) = 1 \(\frac{5}{6}\)

Question 2. 1 \(\frac{1}{3}\) − \(\frac{2}{3}\) ——————— \(\frac{□}{□}\)

Answer: \(\frac{2}{3}\)

Explanation: Convert from mixed fractions to improper fractions. 1 \(\frac{1}{3}\) = \(\frac{4}{3}\) \(\frac{4}{3}\) – \(\frac{2}{3}\) \(\frac{2}{3}\)

Question 3. 4 \(\frac{7}{10}\) − 1 \(\frac{9}{10}\) ——————— ______ \(\frac{□}{□}\)

Answer: 2 \(\frac{8}{10}\)

Explanation: Convert from mixed fractions to improper fractions. 4 \(\frac{7}{10}\) = \(\frac{47}{10}\) 1 \(\frac{9}{10}\) = \(\frac{19}{10}\) \(\frac{47}{10}\) – \(\frac{19}{10}\) \(\frac{28}{10}\) = 2 \(\frac{8}{10}\)

Question 4. 3 \(\frac{5}{12}\) − \(\frac{8}{12}\) ——————— _____ \(\frac{□}{□}\)

Answer: 2 \(\frac{9}{12}\)

Explanation: Convert from mixed fractions to improper fractions. 3 \(\frac{5}{12}\) = \(\frac{41}{12}\) \(\frac{41}{12}\) − \(\frac{8}{12}\) 2 \(\frac{9}{12}\)

Question 5. 8 \(\frac{1}{10}\) − 2 \(\frac{9}{10}\) ——————— \(\frac{□}{□}\)

Answer: 5 \(\frac{1}{5}\)

Explanation: Convert from mixed fractions to improper fractions. 8 \(\frac{1}{10}\) = \(\frac{81}{10}\) 2 \(\frac{9}{10}\) = \(\frac{29}{10}\) \(\frac{81}{10}\) –\(\frac{29}{10}\) \(\frac{52}{10}\) = 5 \(\frac{1}{5}\)

Question 6. 2 − 1 \(\frac{1}{4}\) ——————— \(\frac{□}{□}\)

Answer: \(\frac{3}{4}\)

Explanation: Convert from mixed fractions to improper fractions. 1 \(\frac{1}{4}\) = \(\frac{5}{4}\) 2 − 1 \(\frac{1}{4}\) \(\frac{3}{4}\)

Question 7. 4 \(\frac{1}{5}\) − 3 \(\frac{2}{5}\) ——————— \(\frac{□}{□}\)

Answer: \(\frac{4}{5}\)

Explanation: Convert from mixed fractions to improper fractions. 4 \(\frac{1}{5}\) = \(\frac{21}{5}\) 3 \(\frac{2}{5}\) = \(\frac{17}{5}\) \(\frac{21}{5}\) –\(\frac{17}{5}\) \(\frac{4}{5}\)

Practice: Copy and Solve Find the difference.

Question 8. \(4 \frac{1}{6}-2 \frac{5}{6}\) _____ \(\frac{□}{□}\)

Explanation: Convert from mixed fractions to improper fractions. 4 \(\frac{1}{6}\) = \(\frac{25}{6}\) 2 \(\frac{5}{6}\) = \(\frac{17}{6}\) \(\frac{25}{6}\) –\(\frac{17}{6}\) \(\frac{8}{6}\) = 1 \(\frac{1}{3}\)

Question 9. \(6 \frac{9}{12}-3 \frac{10}{12}\) _____ \(\frac{□}{□}\)

Answer: 2 \(\frac{11}{12}\)

Explanation: Convert from mixed fractions to improper fractions. 6 \(\frac{9}{12}\) – 3 \(\frac{10}{12}\) 2 \(\frac{11}{12}\)

Question 10. \(3 \frac{3}{10}-\frac{7}{10}\) _____ \(\frac{□}{□}\)

Answer: 2 \(\frac{3}{5}\)

Explanation: Convert from mixed fractions to improper fractions. 3 \(\frac{3}{10}\) = \(\frac{33}{10}\) \(\frac{33}{10}\) – \(\frac{7}{10}\) 2 \(\frac{3}{5}\)

Question 11. 4 – 2 \(\frac{3}{5}\) _____ \(\frac{□}{□}\)

Answer: 1 \(\frac{2}{5}\)

Explanation: Convert from mixed fractions to improper fractions. 2 \(\frac{3}{5}\) = \(\frac{13}{5}\) 4 –\(\frac{13}{5}\)  1 \(\frac{2}{5}\)

Question 12. Lisa mixed 4 \(\frac{2}{6}\) cups of orange juice with 3 \(\frac{1}{6}\) cups of pineapple juice to make fruit punch. She and her friends drank 3 \(\frac{4}{6}\) cups of the punch. How much of the fruit punch is left? _____ \(\frac{□}{□}\) cups

Answer: 3 \(\frac{5}{6}\) cups

Explanation: Given, Lisa mixed 4 \(\frac{2}{6}\) cups of orange juice with 3 \(\frac{1}{6}\) cups of pineapple juice to make fruit punch. She and her friends drank 3 \(\frac{4}{6}\) cups of the punch. Convert from mixed fractions to improper fractions. 4 \(\frac{2}{6}\) + 3 \(\frac{1}{6}\) 7 \(\frac{3}{6}\) Now subtract 3 \(\frac{4}{6}\) from 7 \(\frac{3}{6}\). 7 \(\frac{3}{6}\) -3 \(\frac{4}{6}\) 3 \(\frac{5}{6}\)

Rename the fractions to solve.

Question 13. Analyze Relationships Trumpets and cornets are brass instruments. Fully stretched out, the length of a trumpet is 5 \(\frac{1}{4}\) feet and the length of a cornet is 4 \(\frac{2}{4}\) feet. The trumpet is how much longer than the cornet? \(\frac{□}{□}\) feet

Answer: \(\frac{3}{4}\) feet

Explanation: Given, Trumpets and cornets are brass instruments. Fully stretched out, the length of a trumpet is 5 \(\frac{1}{4}\) feet and the length of a cornet is 4 \(\frac{2}{4}\) feet. 5 \(\frac{1}{4}\) – 4 \(\frac{2}{4}\) First subtract the whole numbers 5 – 4 = 1 \(\frac{1}{4}\) – \(\frac{2}{4}\) = \(\frac{1}{4}\) 1 – \(\frac{1}{4}\) = \(\frac{3}{4}\) feet Therefore trumpet is \(\frac{3}{4}\) feet longer than the cornet.

Question 14. Tubas, trombones, and French horns are brass instruments. Fully stretched out, the length of a tuba is 18 feet, the length of a trombone is 9 \(\frac{11}{12}\) feet, and the length of a French horn is 17 \(\frac{1}{12}\) feet. The tuba is how much longer than the French horn? The French horn is how much longer than the trombone? Type below: _____________

Answer: First, convert the fractions to decimals making the trombone 8.93 feet and the french horn 17.21 feet. The tuba would be 0.79 feet longer than the french horn, and the french horn would be 8.23 feet longer than the trombone. However, if you need the answer to remain a fraction, the tuba would be 11/14 feet longer than a french horn, and a french horn would be 8 3/14 feet longer than a trombone.

Question 15. The pitch of a musical instrument is related to its length. In general, the greater the length of a musical instrument, the lower its pitch. Order the brass instruments identified on this page from lowest pitch to the highest pitch. ____________ ____________ ____________

Answer: Tuba French Horn Trombone

Explanation: By seeing the above answer we can write the order of the brass instruments from the lowest pitch to the highest pitch. The order is tuba, french horn, and trombone.

Question 16. Alicia had 3 \(\frac{1}{6}\)yards of fabric. After making a tablecloth, she had 1 \(\frac{3}{6}\) yards of fabric. Alicia said she used 2 \(\frac{3}{6}\) yards of fabric for the tablecloth. Do you agree? Explain. ______

Answer: Yes

Explanation: An easier way to do this is to make the fractions improper fractions. 3 1/6 can be rewritten as 19/6. 1 4/6 can be rewritten as 10/6. Multiply the denominator by the number at its side, and add it to the numerator. 2 3/6 is 15/6. Subtract 10/6 from 19/6. 19/6-10/6=9/6. 9/6 is not 15/6, therefore she did not use 2 3/6 yards of fabric.

Question 2. 6 – 3 \(\frac{2}{5}\) _______ \(\frac{□}{□}\)

Explanation: First subtract the whole numbers 6 – 3 = 3 Next subtract the fractions, 3 – \(\frac{2}{5}\) = 2 \(\frac{3}{5}\)

Question 3. 5 \(\frac{1}{4}\) – 2 \(\frac{3}{4}\) _______ \(\frac{□}{□}\)

Answer: 2 \(\frac{1}{2}\)

Explanation: First subtract the whole numbers 5 – 2 = 3 Next subtract the fractions, \(\frac{1}{4}\) – \(\frac{3}{4}\) = – \(\frac{1}{2}\) 3 – \(\frac{1}{2}\) = 2 \(\frac{1}{2}\)

Question 4. 9 \(\frac{3}{8}\) – 8 \(\frac{7}{8}\) _______ \(\frac{□}{□}\)

Answer: \(\frac{1}{2}\)

Explanation: First subtract the whole numbers 9 – 8 = 1 Next subtract the fractions, \(\frac{3}{8}\) – \(\frac{7}{8}\) = – \(\frac{4}{8}\) = – \(\frac{1}{2}\) = 1 – \(\frac{1}{2}\) = \(\frac{1}{2}\)

Question 5. 12 \(\frac{3}{10}\) – 7 \(\frac{7}{10}\) _______ \(\frac{□}{□}\)

Answer: 4 \(\frac{3}{5}\)

Explanation: First subtract the whole numbers 12 – 7 = 5 Next subtract the fractions, \(\frac{3}{10}\) – \(\frac{7}{10}\) = – \(\frac{4}{10}\) 5 – \(\frac{4}{10}\) 5 – \(\frac{2}{5}\) = 4 \(\frac{3}{5}\)

Question 6. 8 \(\frac{1}{6}\) – 3 \(\frac{5}{6}\) _______ \(\frac{□}{□}\)

Answer: 4 \(\frac{1}{3}\)

Explanation: First subtract the whole numbers 8 – 3 = 5 Next subtract the fractions, \(\frac{1}{6}\) – \(\frac{5}{6}\) = – \(\frac{2}{3}\) 5 – \(\frac{2}{3}\) = 4 \(\frac{1}{3}\)

Question 7. 7 \(\frac{3}{5}\) – 4 \(\frac{4}{5}\) _______ \(\frac{□}{□}\)

Answer: 2 \(\frac{4}{5}\)

Explanation: First subtract the whole numbers 7 – 4 = 3 Next subtract the fractions, \(\frac{3}{5}\) – \(\frac{4}{5}\) = – \(\frac{1}{5}\) 3 – \(\frac{1}{5}\) = 2 \(\frac{4}{5}\)

Question 8. 10 \(\frac{1}{2}\) – 8 \(\frac{1}{2}\) _______ \(\frac{□}{□}\)

Explanation: First subtract the whole numbers 10 – 8 = 2 \(\frac{1}{2}\) – \(\frac{1}{2}\) = 0

Question 9. 7 \(\frac{1}{6}\) – 2 \(\frac{5}{6}\) _______ \(\frac{□}{□}\)

Explanation: First subtract the whole numbers 7 – 2 = 5 Next subtract the fractions, \(\frac{1}{6}\) – \(\frac{5}{6}\) = – \(\frac{4}{6}\) 5 – \(\frac{4}{6}\) = 4 \(\frac{1}{3}\)

Question 10. 9 \(\frac{3}{12}\) – 4 \(\frac{7}{12}\) _______ \(\frac{□}{□}\)

Answer: 2 \(\frac{2}{3}\)

Explanation: First subtract the whole numbers 9 – 4 = 5 Next subtract the fractions, \(\frac{3}{12}\) – \(\frac{7}{12}\) = – \(\frac{4}{12}\) = – \(\frac{1}{3}\) 5 – \(\frac{1}{3}\) = 2 \(\frac{2}{3}\)

Question 11. 9 \(\frac{1}{10}\) – 8 \(\frac{7}{10}\) _______ \(\frac{□}{□}\)

Answer: \(\frac{2}{5}\)

Explanation: First subtract the whole numbers 9 – 8 = 1 Next subtract the fractions, \(\frac{1}{10}\) – \(\frac{7}{10}\) = – \(\frac{6}{10}\) 1 – \(\frac{3}{5}\) = \(\frac{2}{5}\)

Question 12. 9 \(\frac{1}{3}\) – \(\frac{2}{3}\) _______ \(\frac{□}{□}\)

Answer: 8 \(\frac{2}{3}\)

Explanation: 9 \(\frac{1}{3}\) – \(\frac{2}{3}\) 8 \(\frac{2}{3}\)

Question 13. 3 \(\frac{1}{4}\) – 1 \(\frac{3}{4}\) _______ \(\frac{□}{□}\)

3 \(\frac{1}{4}\) – 1 \(\frac{3}{4}\) 1 \(\frac{1}{2}\)

Question 14. 4 \(\frac{5}{8}\) – 1 \(\frac{7}{8}\) _______ \(\frac{□}{□}\)

Explanation: First subtract the whole numbers 4 – 1 = 3 Next subtract the fractions, \(\frac{5}{8}\) – \(\frac{7}{8}\) = – \(\frac{1}{4}\) 3 – \(\frac{1}{4}\) = 2 \(\frac{3}{4}\)

Question 15. 5 \(\frac{1}{12}\) – 3 \(\frac{8}{12}\) _______ \(\frac{□}{□}\)

Answer: 1 \(\frac{5}{12}\)

Explanation: First subtract the whole numbers 5 – 3 = 2 Next subtract the fractions, \(\frac{1}{12}\) – \(\frac{8}{12}\) = – \(\frac{7}{12}\) 2 – \(\frac{7}{12}\) = 1 \(\frac{5}{12}\)

Question 16. 7 – 1 \(\frac{3}{5}\) _______ \(\frac{□}{□}\)

Answer: 5 \(\frac{2}{5}\)

Explanation: 7 – 1 \(\frac{3}{5}\) 5 \(\frac{2}{5}\)

Question 17. Alicia buys a 5-pound bag of rocks for a fish tank. She uses 1 \(\frac{1}{8}\) pounds for a small fish bowl. How much is left? _______ \(\frac{□}{□}\)

Answer: 3 \(\frac{7}{8}\)

Explanation: Given, Alicia buys a 5-pound bag of rocks for a fish tank. She uses 1 \(\frac{1}{8}\) pounds for a small fish bowl. First subtract the whole numbers 5 – 1 = 4 4 – 1 \(\frac{1}{8}\) = 3 \(\frac{7}{8}\)

Question 18. Xavier made 25 pounds of roasted almonds for a fair. He has 3 \(\frac{1}{2}\) pounds left at the end of the fair. How many pounds of roasted almonds did he sell at the fair? _______ \(\frac{□}{□}\)

Answer: 21 \(\frac{1}{2}\)

Explanation: Given, Xavier made 25 pounds of roasted almonds for a fair. He has 3 \(\frac{1}{2}\) pounds left at the end of the fair. First subtract the whole numbers 25 – 3 = 22 22 – \(\frac{1}{2}\) = 21 \(\frac{1}{2}\)

Question 1. Reggie is making a double-layer cake. The recipe for the first layer calls for 2 \(\frac{1}{4}\) cups sugar. The recipe for the second layer calls for 1 \(\frac{1}{4}\) cups sugar. Reggie has 5 cups of sugar. How much will he have left after making both recipes? Options: a. 1 \(\frac{1}{4}\) cups b. 1 \(\frac{2}{4}\) cups c. 2 \(\frac{1}{4}\) cups d. 2 \(\frac{2}{4}\) cups

Answer: 1 \(\frac{2}{4}\) cups

Explanation: Given, Reggie is making a double-layer cake. The recipe for the first layer calls for 2 \(\frac{1}{4}\) cups sugar. The recipe for the second layer calls for 1 \(\frac{1}{4}\) cups sugar. Reggie has 5 cups of sugar. 2 \(\frac{1}{4}\) + 1 \(\frac{1}{4}\) = 3 \(\frac{1}{2}\) 5 – 3 \(\frac{1}{2}\) = 1 \(\frac{2}{4}\) cups Thus the correct answer is option b.

Question 2. Kate has 4 \(\frac{3}{8}\) yards of fabric and needs 2 \(\frac{7}{8}\) yards to make a skirt. How much extra fabric will she have left after making the skirt? Options: a. 2 \(\frac{4}{8}\) yards b. 2 \(\frac{2}{8}\) yards c. 1 \(\frac{4}{8}\) yards d. 1 \(\frac{2}{8}\) yards

Answer: 1 \(\frac{4}{8}\) yards

Explanation: Given, Kate has 4 \(\frac{3}{8}\) yards of fabric and needs 2 \(\frac{7}{8}\) yards to make a skirt. First, subtract the whole numbers 4 – 2 = 2 Next, subtract the fractions, \(\frac{3}{8}\) – \(\frac{7}{8}\) = – \(\frac{4}{8}\) 2 – \(\frac{4}{8}\) = 1 \(\frac{4}{8}\) yards Thus the correct answer is option c.

Question 3. Paulo has 128 glass beads to use to decorate picture frames. He wants to use the same number of beads on each frame. If he decorates 8 picture frames, how many beads will he put on each frame? Options: a. 6 b. 7 c. 14 d. 16

Explanation: Given, Paulo has 128 glass beads to use to decorate picture frames. He wants to use the same number of beads on each frame 128/8 = 16 Thus the correct answer is option d.

Question 4. Madison is making party favors. She wants to make enough favors so each guest gets the same number of favors. She knows there will be 6 or 8 guests at the party. What is the least number of party favors Madison should make? Options: a. 18 b. 24 c. 30 d. 32

Explanation: Given, Madison is making party favors. She wants to make enough favors so each guest gets the same number of favors. She knows there will be 6 or 8 guests at the party. To find the least number of party favors, we have to consider the number of guests. In this case, there are two possibilities—6 or 8. For 6: 6, 12, 18, 24 (Add 6 to each number) For 8: 8, 16, 24 (Add 8 to each number) Now in both series, the least number (that is in common) is 24. Hence, Madison should make at least 24 party favors. Thus the correct answer is option b.

Question 5. A shuttle bus makes 4 round-trips between two shopping centers each day. The bus holds 24 people. If the bus is full on each one-way trip, how many passengers are carried by the bus each day? Options: a. 96 b. 162 c. 182 d. 192

Explanation: Given, A shuttle bus makes 4 round-trips between two shopping centers each day. The bus holds 24 people. 4 × 24 = 96 Thus the correct answer is option a.

Question 6. To make a fruit salad, Marvin mixes 1 \(\frac{3}{4}\) cups of diced peaches with 2 \(\frac{1}{4}\) cups of diced pears. How many cups of peaches and pears are in the fruit salad? Options: a. 4 cups b. 3 \(\frac{2}{4}\) cups c. 3 \(\frac{1}{4}\) cups d. 3 cups

Answer: 4 cups

Explanation: Given, To make a fruit salad, Marvin mixes 1 \(\frac{3}{4}\) cups of diced peaches with 2 \(\frac{1}{4}\) cups of diced pears. 1 \(\frac{3}{4}\) + 2 \(\frac{1}{4}\) = 4 cups Thus the correct answer is option a.

Question 1. Complete. Name the property used. \(\left(3 \frac{4}{10}+5 \frac{2}{10}\right)+\frac{6}{10}\) ______ \(\frac{□}{□}\)

Answer: The property used is associative property. 9 \(\frac{2}{10}\)

Explanation: The associative property states that you can add or multiply regardless of how the numbers are grouped. Given, \(\left(3 \frac{4}{10}+5 \frac{2}{10}\right)+\frac{6}{10}\) First add the whole numbers in the group. (3 \(\frac{4}{10}\) + 5 \(\frac{2}{10}\)) + \(\frac{6}{10}\) 3 + 5 = 8 8 + \(\frac{4}{10}\) + \(\frac{2}{10}\) + \(\frac{6}{10}\) Now add the fractions 8 + \(\frac{6}{10}\) + \(\frac{6}{10}\) 8 + \(\frac{12}{10}\) Convert from improper fractions to the mixed fractions. \(\frac{12}{10}\) = 1 \(\frac{2}{10}\) 8 + 1 \(\frac{2}{10}\) = 9 \(\frac{2}{10}\) Thus \(\left(3 \frac{4}{10}+5 \frac{2}{10}\right)+\frac{6}{10}\) = 9 \(\frac{2}{10}\)

Use the properties and mental math to find the sum.

Question 2. \(\left(2 \frac{7}{8}+3 \frac{2}{8}\right)+1 \frac{1}{8}\) ______ \(\frac{□}{□}\)

Answer: 7 \(\frac{1}{4}\)

Explanation: The associative property states that you can add or multiply regardless of how the numbers are grouped. Given \(\left(2 \frac{7}{8}+3 \frac{2}{8}\right)+1 \frac{1}{8}\) First add the whole numbers in the group. (2 \(\frac{7}{8}\) + 3 \(\frac{2}{8}\)) + 1 \(\frac{1}{8}\) 2 + 3 = 5 5 + \(\frac{7}{8}\) + \(\frac{2}{8}\) + 1 \(\frac{1}{8}\) 5 + \(\frac{9}{8}\) + 1 \(\frac{1}{8}\) 6 + \(\frac{10}{8}\) = 7 \(\frac{1}{4}\) Thus \(\left(2 \frac{7}{8}+3 \frac{2}{8}\right)+1 \frac{1}{8}\) = 7 \(\frac{1}{4}\)

Question 3. \(1 \frac{2}{5}+\left(1+\frac{3}{5}\right)\) ______

Explanation: The associative property states that you can add or multiply regardless of how the numbers are grouped. Given, \(1 \frac{2}{5}+\left(1+\frac{3}{5}\right)\) First add the whole numbers in the group. 1 + \(\frac{3}{5}\) = 1 \(\frac{3}{5}\) 1 \(\frac{2}{5}\) + 1 \(\frac{3}{5}\) 1 + 1 + \(\frac{5}{5}\) 1 + 1 + 1 = 3 Thus \(1 \frac{2}{5}+\left(1+\frac{3}{5}\right)\) = 3

Question 4. \(5 \frac{3}{6}+\left(5 \frac{5}{6}+4 \frac{3}{6}\right)\) ______ \(\frac{□}{□}\)

Answer: 15 \(\frac{5}{6}\)

Explanation: The associative property states that you can add or multiply regardless of how the numbers are grouped. Given, \(5 \frac{3}{6}+\left(5 \frac{5}{6}+4 \frac{3}{6}\right)\) First add the whole numbers in the group. 5 + 4 = 9 \(\frac{5}{6}\) + \(\frac{3}{6}\) = \(\frac{8}{6}\) 5 \(\frac{3}{6}\) + 9 \(\frac{8}{6}\) 5 \(\frac{3}{6}\) + 10 \(\frac{2}{6}\) = 15 \(\frac{5}{6}\) Thus \(5 \frac{3}{6}+\left(5 \frac{5}{6}+4 \frac{3}{6}\right)\) = 15 \(\frac{5}{6}\)

Question 5. \(\left(1 \frac{1}{4}+1 \frac{1}{4}\right)+2 \frac{3}{4}\) ______ \(\frac{□}{□}\)

Explanation: The associative property states that you can add or multiply regardless of how the numbers are grouped. Given, \(\left(1 \frac{1}{4}+1 \frac{1}{4}\right)+2 \frac{3}{4}\) First add the whole numbers in the group. (1 \(\frac{1}{4}\) + 1 \(\frac{1}{4}\)) + 2 \(\frac{3}{4}\) 1 + 1 = 2 2 \(\frac{1}{4}\) + \(\frac{1}{4}\) + 2 \(\frac{3}{4}\) 2 \(\frac{1}{2}\) + 2 \(\frac{3}{4}\) Add the whole numbers 2 + 2 = 4 4 \(\frac{1}{2}\) + \(\frac{3}{4}\) = 5 \(\frac{1}{4}\) Thus \(\left(1 \frac{1}{4}+1 \frac{1}{4}\right)+2 \frac{3}{4}\) = 5 \(\frac{1}{4}\)

Question 6. \(\left(12 \frac{4}{9}+1 \frac{2}{9}\right)+3 \frac{5}{9}\) ______ \(\frac{□}{□}\)

Answer: 17 \(\frac{2}{9}\)

Explanation: The associative property states that you can add or multiply regardless of how the numbers are grouped. Given, \(\left(12 \frac{4}{9}+1 \frac{2}{9}\right)+3 \frac{5}{9}\) First add the whole numbers in the group. 12 + 1 = 13 Add the fraction in the group. \(\frac{4}{9}\) + \(\frac{2}{9}\) + 3 \(\frac{5}{9}\) = 13 \(\frac{6}{9}\) + 3 \(\frac{5}{9}\) = 16 \(\frac{11}{9}\) = 17 \(\frac{2}{9}\) Thus \(\left(12 \frac{4}{9}+1 \frac{2}{9}\right)+3 \frac{5}{9}\) = 17 \(\frac{2}{9}\)

Question 7. \(\left(\frac{3}{12}+1 \frac{8}{12}\right)+\frac{9}{12}\) ______ \(\frac{□}{□}\)

Explanation: The associative property states that you can add or multiply regardless of how the numbers are grouped. Given, \(\left(\frac{3}{12}+1 \frac{8}{12}\right)+\frac{9}{12}\) First add the fractions in the group. \(\frac{3}{12}\) + \(\frac{8}{12}\) = \(\frac{11}{12}\) 1 \(\frac{11}{12}\) + \(\frac{9}{12}\) = 1 \(\frac{20}{12}\) = 2 \(\frac{2}{3}\) Thus \(\left(\frac{3}{12}+1 \frac{8}{12}\right)+\frac{9}{12}\) = 2 \(\frac{2}{3}\)

Question 8. \(\left(45 \frac{1}{3}+6 \frac{1}{3}\right)+38 \frac{2}{3}\) ______ \(\frac{□}{□}\)

Answer: 90 \(\frac{1}{3}\)

Explanation: Given, \(\left(45 \frac{1}{3}+6 \frac{1}{3}\right)+38 \frac{2}{3}\) First add the whole numbers in the group. 45 + 6 = 51 (51 \(\frac{1}{3}\) + \(\frac{1}{3}\)) + 38 \(\frac{2}{3}\) 51 \(\frac{2}{3}\) + 38 \(\frac{2}{3}\) = 89 \(\frac{4}{3}\) = 90 \(\frac{1}{3}\) Thus \(\left(45 \frac{1}{3}+6 \frac{1}{3}\right)+38 \frac{2}{3}\) = 90 \(\frac{1}{3}\)

Question 9. \(\frac{1}{2}+\left(103 \frac{1}{2}+12\right)\) ______ \(\frac{□}{□}\)

Answer: 116

Explanation: Given, \(\frac{1}{2}+\left(103 \frac{1}{2}+12\right)\) First add the whole numbers in the group. 103 + \(\frac{1}{2}\) + 12 = 115 \(\frac{1}{2}\) 115 \(\frac{1}{2}\) + \(\frac{1}{2}\) = 116 Thus \(\frac{1}{2}+\left(103 \frac{1}{2}+12\right)\) = 116

Question 10. \(\left(3 \frac{5}{10}+10\right)+11 \frac{5}{10}\) ______

Explanation: Given, \(\left(3 \frac{5}{10}+10\right)+11 \frac{5}{10}\) First add the whole numbers in the group. 3 + 10 = 13 13 + \(\frac{5}{10}\) + 11 \(\frac{5}{10}\) Add the whole numbers 13 + 11 = 24 24 + \(\frac{5}{10}\) + \(\frac{5}{10}\) = 25 Thus \(\left(3 \frac{5}{10}+10\right)+11 \frac{5}{10}\) = 25

Question 11. Pablo is training for a marathon. He ran 5 \(\frac{4}{8}\) miles on Friday, 6 \(\frac{5}{8}\) miles on Saturday, and 7 \(\frac{4}{8}\) miles on Sunday. How many miles did he run on all three days? ______ \(\frac{□}{□}\) miles

Answer: 19 \(\frac{5}{8}\) miles

Explanation: Given, Pablo is training for a marathon. He ran 5 \(\frac{4}{8}\) miles on Friday, 6 \(\frac{5}{8}\) miles on Saturday, and 7 \(\frac{4}{8}\) miles on Sunday. Add all the fractions to find how many miles he runs on all three days. 5 \(\frac{4}{8}\) + 6 \(\frac{5}{8}\) + 7 \(\frac{4}{8}\) First add the whole numbers 5 + 6 + 7 = 18 18 + \(\frac{4}{8}\) + \(\frac{5}{8}\) + \(\frac{4}{8}\) = 18 + \(\frac{13}{8}\) = 19 \(\frac{5}{8}\) miles Therefore Pablo runs 19 \(\frac{5}{8}\) miles on all three days.

Question 12. At lunchtime, Dale’s Diner served a total of 2 \(\frac{2}{6}\) pots of vegetable soup, 3 \(\frac{5}{6}\) pots of chicken soup, and 4 \(\frac{3}{6}\) pots of tomato soup. How many pots of soup were served in all? ______ \(\frac{□}{□}\) pots

Answer: 10 \(\frac{2}{3}\) pots

Explanation: Given, At lunchtime, Dale’s Diner served a total of 2 \(\frac{2}{6}\) pots of vegetable soup, 3 \(\frac{5}{6}\) pots of chicken soup, and 4 \(\frac{3}{6}\) pots of tomato soup. 2 \(\frac{2}{6}\) + 3 \(\frac{5}{6}\) + 4 \(\frac{3}{6}\) First add the whole numbers 2 + 3 + 4 = 9 Next add the fractions. \(\frac{2}{6}\) + \(\frac{5}{6}\) + \(\frac{3}{6}\) = \(\frac{10}{6}\) 9 + \(\frac{10}{6}\) = 10 \(\frac{2}{3}\) pots Therefore 10 \(\frac{2}{3}\) pots of soup were served in all.

Question 13. Which property of addition would you use to regroup the addends in Expression A? ______ property

Answer: Associative Property

Explanation: The associative property states that you can add or multiply regardless of how the numbers are grouped. Expression A is \(\frac{1}{8}\) + (\(\frac{7}{8}\) + \(\frac{4}{8}\)) The denominators of all three fractions are the same. So, the property for expression A is Associative Property.

Question 14. Which two expressions have the same value? ________ and _________

Answer: A and C

Explanation: Expression A is \(\frac{1}{8}\) + (\(\frac{7}{8}\) + \(\frac{4}{8}\)) \(\frac{1}{8}\) + (\(\frac{11}{8}\) = \(\frac{12}{8}\) Expression B is 1/2 + 2 1/2 + 4/2 = 5/2 Expression C is \(\frac{3}{7}\) + (\(\frac{1}{2}\) + \(\frac{4}{7}\)) \(\frac{1}{2}\) + \(\frac{4}{7}\) = \(\frac{7}{14}\) + \(\frac{8}{14}\) = \(\frac{15}{14}\) \(\frac{15}{14}\) + \(\frac{3}{7}\) = \(\frac{15}{14}\) + \(\frac{6}{14}\) = \(\frac{21}{14}\) Thus the expressions A and C has the same value.

Question 16. Costumes are being made for the high school musical. The table at the right shows the amount of fabric needed for the costumes of the male and female leads. Alice uses the expression \(7 \frac{3}{8}+1 \frac{5}{8}+2 \frac{4}{8}\) to find the total amount of fabric needed for the costume of the female lead. To find the value of the expression using mental math, Alice used the properties of addition. \(7 \frac{3}{8}+1 \frac{5}{8}+2 \frac{4}{8}=\left(7 \frac{3}{8}+1 \frac{5}{8}\right)+2 \frac{4}{8}\) Alice added 7 + 1 and was able to quickly add \(\frac{3}{8}\) and \(\frac{5}{8}\) to the sum of 8 to get 9. She added 2 \(\frac{4}{8}\) to 9, so her answer was 11 \(\frac{4}{8}\). So, the amount of fabric needed for the costume of the female lead actor is 11 \(\frac{4}{8}\) yards. Write a new problem using the information for the costume for the male lead actor. Pose a Problem                     Solve your problem. Check your solution. Type below: _____________

Answer: Alice used the expressions 1 2/8 + 2 3/8 + 5 6/8 to find the total amount of frabric needed for the costume of the male lead. What is the total amount of fabric needed for the costume? Answer: Alice wrote the expressions as (1 2/8 + 5 6/8) + 2 3/8 and simplified it by adding the whole number parts and the fraction parts in the parentheses. Then she added the mixed number: 1 + 5 + 1 + 2 3/8 = 9 3/8. So, the male leads costume needed 9 3/8 yards of fabric.

Question 16. Identify Relationships Explain how using the properties of addition makes both problems easier to solve. Type below: ____________

Answer: The properties make the properties the easier to solve because you can rearrange the mixed numbers so that their fraction parts add to 1.

Question 2. \(10 \frac{1}{8}+\left(3 \frac{5}{8}+2 \frac{7}{8}\right)\) _______ \(\frac{□}{□}\)

Answer: 16 \(\frac{5}{8}\)

Explanation: Given, \(10 \frac{1}{8}+\left(3 \frac{5}{8}+2 \frac{7}{8}\right)\) First add the whole numbers in the bracket. 3 + 2 = 5 10 \(\frac{1}{8}\) + 5 + \(\frac{5}{8}\) + \(\frac{7}{8}\) 10 \(\frac{1}{8}\) + 5 + \(\frac{12}{8}\) 10 + 5 = 15 15 + \(\frac{1}{8}\) + \(\frac{12}{8}\) 15 + \(\frac{13}{8}\) 16 \(\frac{5}{8}\) \(10 \frac{1}{8}+\left(3 \frac{5}{8}+2 \frac{7}{8}\right)\) = 16 \(\frac{5}{8}\)

Question 3. \(8 \frac{1}{5}+\left(3 \frac{2}{5}+5 \frac{4}{5}\right)\) _______ \(\frac{□}{□}\)

Answer: 17 \(\frac{2}{5}\)

Explanation: \(8 \frac{1}{5}+\left(3 \frac{2}{5}+5 \frac{4}{5}\right)\) 8 \(\frac{1}{5}\) + 3 \(\frac{2}{5}\) + 5 \(\frac{4}{5}\) 3 + 5 = 8 8 \(\frac{1}{5}\) + 8 + \(\frac{2}{5}\) + \(\frac{4}{5}\) 8 \(\frac{1}{5}\) + 8 + \(\frac{6}{5}\) 8 + 8 = 16 16 + \(\frac{1}{5}\) + \(\frac{6}{5}\) 16 + \(\frac{7}{5}\) 17 \(\frac{2}{5}\) \(8 \frac{1}{5}+\left(3 \frac{2}{5}+5 \frac{4}{5}\right)\) = 17 \(\frac{2}{5}\)

Question 4. \(6 \frac{3}{4}+\left(4 \frac{2}{4}+5 \frac{1}{4}\right)\) _______ \(\frac{□}{□}\)

Answer: 16 \(\frac{1}{2}\)

Explanation: \(6 \frac{3}{4}+\left(4 \frac{2}{4}+5 \frac{1}{4}\right)\) First add the whole numbers in the bracket. 6 \(\frac{3}{4}\) + 4 \(\frac{2}{4}\) + 5 \(\frac{1}{4}\) 4 + 5 = 9 6 \(\frac{3}{4}\) + 9 \(\frac{3}{4}\) 6 + 9 = 15 15 + \(\frac{3}{4}\) + \(\frac{3}{4}\) 16 \(\frac{1}{2}\) \(6 \frac{3}{4}+\left(4 \frac{2}{4}+5 \frac{1}{4}\right)\) = 16 \(\frac{1}{2}\)

Question 5. \(\left(6 \frac{3}{6}+10 \frac{4}{6}\right)+9 \frac{2}{6}\) _______ \(\frac{□}{□}\)

Answer: 26 \(\frac{3}{6}\)

Explanation: \(\left(6 \frac{3}{6}+10 \frac{4}{6}\right)+9 \frac{2}{6}\) 6 \(\frac{3}{6}\) + 10 \(\frac{4}{6}\) + 9 \(\frac{2}{6}\) First add the whole numbers in the bracket. 6 + 10 = 16 16 + \(\frac{3}{6}\) + \(\frac{4}{6}\) + 9 \(\frac{2}{6}\) 16 + \(\frac{7}{6}\) + 9 \(\frac{2}{6}\) 16 + 9 = 25 25 + \(\frac{7}{6}\) + \(\frac{2}{6}\) 25 + \(\frac{9}{6}\) = 26 \(\frac{3}{6}\) \(\left(6 \frac{3}{6}+10 \frac{4}{6}\right)+9 \frac{2}{6}\) = 26 \(\frac{3}{6}\)

Question 6. \(\left(6 \frac{2}{5}+1 \frac{4}{5}\right)+3 \frac{1}{5}\) _______ \(\frac{□}{□}\)

Answer: 11 \(\frac{2}{5}\)

Explanation: \(\left(6 \frac{2}{5}+1 \frac{4}{5}\right)+3 \frac{1}{5}\) 6 \(\frac{2}{5}\) + 1 \(\frac{4}{5}\) + 3 \(\frac{1}{5}\) First add the whole numbers in the bracket. 6 + 1 = 7 7 \(\frac{2}{5}\) + \(\frac{4}{5}\) + 3 \(\frac{1}{5}\) 7 + \(\frac{6}{5}\) + 3 \(\frac{1}{5}\) 7 + 3 = 10 10 + \(\frac{6}{5}\) + \(\frac{1}{5}\) 10 + \(\frac{7}{5}\) = 11 \(\frac{2}{5}\) Therefore \(\left(6 \frac{2}{5}+1 \frac{4}{5}\right)+3 \frac{1}{5}\) = 11 \(\frac{2}{5}\)

Question 7. \(7 \frac{7}{8}+\left(3 \frac{1}{8}+1 \frac{1}{8}\right)\) _______ \(\frac{□}{□}\)

Answer: 12 \(\frac{1}{8}\)

Explanation: \(7 \frac{7}{8}+\left(3 \frac{1}{8}+1 \frac{1}{8}\right)\) 7 \(\frac{7}{8}\) + 3 \(\frac{1}{8}\) + 1 \(\frac{1}{8}\) First add the whole numbers in the bracket. 3 + 1 = 4 7 \(\frac{7}{8}\) + 4 + \(\frac{1}{8}\) + \(\frac{1}{8}\) 7 \(\frac{7}{8}\) + 4 +\(\frac{2}{8}\) 7 + 4 = 11 11 + \(\frac{7}{8}\) + \(\frac{2}{8}\) 11 + \(\frac{9}{8}\) = 12 \(\frac{1}{8}\) Thus \(7 \frac{7}{8}+\left(3 \frac{1}{8}+1 \frac{1}{8}\right)\) = 12 \(\frac{1}{8}\)

Question 8. \(14 \frac{1}{10}+\left(20 \frac{2}{10}+15 \frac{7}{10}\right)\) _______ \(\frac{□}{□}\)

Explanation: \(14 \frac{1}{10}+\left(20 \frac{2}{10}+15 \frac{7}{10}\right)\) First add the whole numbers in the bracket. 14 \(\frac{1}{10}\) + 20 \(\frac{2}{10}\) + 15 \(\frac{7}{10}\) 20 + 15 = 35 14 \(\frac{1}{10}\) + 35 + \(\frac{2}{10}\) + \(\frac{7}{10}\) 14 \(\frac{1}{10}\) + 35 \(\frac{9}{10}\) 49 \(\frac{1}{10}\) + \(\frac{9}{10}\) 49 + 1 = 50 Thus \(14 \frac{1}{10}+\left(20 \frac{2}{10}+15 \frac{7}{10}\right)\) = 50

Question 9. \(\left(13 \frac{2}{12}+8 \frac{7}{12}\right)+9 \frac{5}{12}\) _______ \(\frac{□}{□}\)

Answer: 31 \(\frac{2}{12}\)

Explanation: \(\left(13 \frac{2}{12}+8 \frac{7}{12}\right)+9 \frac{5}{12}\) 13 \(\frac{2}{12}\) + 8 \(\frac{7}{12}\) + 9 \(\frac{5}{12}\) First add the whole numbers in the bracket. 13 + 8 = 21 21 + \(\frac{2}{12}\) + \(\frac{7}{12}\) + 9 \(\frac{5}{12}\) 21 + \(\frac{9}{12}\) + 9 \(\frac{5}{12}\) 30 + \(\frac{9}{12}\) + \(\frac{5}{12}\) = 31 \(\frac{2}{12}\) Thus \(\left(13 \frac{2}{12}+8 \frac{7}{12}\right)+9 \frac{5}{12}\) = 31 \(\frac{2}{12}\)

Question 10. Nate’s classroom has three tables of different lengths. One has a length of 4 \(\frac{1}{2}\) feet, another has a length of 4 feet, and a third has a length of 2 \(\frac{1}{2}\) feet. What is the length of all three tables when pushed end to end? _______ \(\frac{□}{□}\)

Explanation: Given, Nate’s classroom has three tables of different lengths. One has a length of 4 \(\frac{1}{2}\) feet, another has a length of 4 feet, and a third has a length of 2 \(\frac{1}{2}\) feet. 4 \(\frac{1}{2}\) + 4 + 2 \(\frac{1}{2}\) 4 + 4 + 2 = 10 \(\frac{1}{2}\) + \(\frac{1}{2}\) = 1 10 + 1 = 11 Therefore the length of all three tables when pushed end to end is 11 feet.

Question 11. Mr. Warren uses 2 \(\frac{1}{4}\) bags of mulch for his garden and another 4 \(\frac{1}{4}\) bags for his front yard. He also uses \(\frac{3}{4}\) bag around a fountain. How many total bags of mulch does Mr. Warren use? _______ \(\frac{□}{□}\)

Explanation: Given, Mr. Warren uses 2 \(\frac{1}{4}\) bags of mulch for his garden and another 4 \(\frac{1}{4}\) bags for his front yard. He also uses \(\frac{3}{4}\) bag around a fountain. 2 \(\frac{1}{4}\) + 4 \(\frac{1}{4}\) + \(\frac{3}{4}\) 2 + 4 = 6 6 + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{3}{4}\) = 7 \(\frac{1}{4}\)

Question 1. A carpenter cut a board into three pieces. One piece was 2 \(\frac{5}{6}\) feet long. The second piece was 3 \(\frac{1}{6}\) feet long. The third piece was 1 \(\frac{5}{6}\) feet long. How long was the board? Options: a. 6 \(\frac{5}{6}\) feet b. 7 \(\frac{1}{6}\) feet c. 7 \(\frac{5}{6}\) feet d. 8 \(\frac{1}{6}\) feet

Answer: c. 7 \(\frac{5}{6}\) feet

Explanation: Given, A carpenter cut a board into three pieces. One piece was 2 \(\frac{5}{6}\) feet long. The second piece was 3 \(\frac{1}{6}\) feet long. The third piece was 1 \(\frac{5}{6}\) feet long. Add three pieces. 2 \(\frac{5}{6}\) + 3 \(\frac{1}{6}\) = 5 + \(\frac{6}{6}\) = 5 + 1 = 6 6 + 1 \(\frac{5}{6}\) = 7 \(\frac{5}{6}\) feet Thus the correct answer is option c.

Question 2. Harry works at an apple orchard. He picked 45 \(\frac{7}{8}\) pounds of apples on Monday. He picked 42 \(\frac{3}{8}\) pounds of apples on Wednesday. He picked 54 \(\frac{1}{8}\) pounds of apples on Friday. How many pounds of apples did Harry pick those three days? Options: a. 132 \(\frac{3}{8}\) pounds b. 141 \(\frac{3}{8}\) pounds c. 142 \(\frac{1}{8}\) pounds d. 142 \(\frac{3}{8}\) pounds

Answer: 142 \(\frac{3}{8}\) pounds

Explanation: Given, Harry works at an apple orchard. He picked 45 \(\frac{7}{8}\) pounds of apples on Monday. He picked 42 \(\frac{3}{8}\) pounds of apples on Wednesday. He picked 54 \(\frac{1}{8}\) pounds of apples on Friday. 45 \(\frac{7}{8}\) + 42 \(\frac{3}{8}\) + 54 \(\frac{1}{8}\) Add the whole numbers first 45 + 42 + 54 = 141 141 + \(\frac{7}{8}\) + \(\frac{3}{8}\) + \(\frac{1}{8}\) 141 + 1 \(\frac{3}{8}\) = 142 \(\frac{3}{8}\) pounds Thus the correct answer is option d.

Question 3. There were 6 oranges in the refrigerator. Joey and his friends ate 3 \(\frac{2}{3}\) oranges. How many oranges were left? Options: a. 2 \(\frac{1}{3}\) oranges b. 2 \(\frac{2}{3}\) oranges c. 3 \(\frac{1}{3}\) oranges d. 9 \(\frac{2}{3}\) oranges

Answer: 9 \(\frac{2}{3}\) oranges

Explanation: Given, There were 6 oranges in the refrigerator. Joey and his friends ate 3 \(\frac{2}{3}\) oranges. 6 + 3 \(\frac{2}{3}\) = 9 \(\frac{2}{3}\) oranges Thus the correct answer is option d.

Question 4. Darlene was asked to identify which of the following numbers is prime. Which number should she choose? Options: a. 2 b. 12 c. 21 d. 39

Explanation: A prime number is an integer, or whole number, that has only two factors 1 and itself. In the above options, all are composite numbers except 2. Therefore 2 is a prime number. Thus the correct answer is option a.

Question 5. A teacher has 100 chairs to arrange for an assembly. Which of the following is NOT a way the teacher could arrange the chairs? Options: a. 10 rows of 10 chairs b. 8 rows of 15 chairs c. 5 rows of 20 chairs d. 4 rows of 25 chairs

Answer: 8 rows of 15 chairs

Explanation: A teacher has 100 chairs to arrange for an assembly. 15 × 8 = 120 So, 8 rows of 15 chairs are not the way to arrange the chairs. Thus the correct answer is option b.

Question 6. Nic bought 28 folding chairs for $16 each. How much money did Nic spend on chairs? Options: a. $196 b. $348 c. $448 d. $600

Answer: c. $448

Explanation: Given, Nic bought 28 folding chairs for $16 each. 28 × 16 = 448 Thus the correct answer is option c.

Question 1. Last week, Sia ran 1 \(\frac{1}{4}\) miles each day for 5 days and then took 2 days off. Did she run at least 6 miles last week? First, model the problem. Describe your model. Type below: _________

Answer: I will model the problem using fraction strips. I need a 1 strip for the whole and a 1/4 part for each of the 5 days. My model has a total of five 1 strops and five 1/4 parts.

Question 1. Then, regroup the parts in the model to find the number of whole miles Sia ran. Sia ran ___________ whole miles and ___________ mile. Finally, compare the total number of miles she ran to 6 miles. So, Sia ___________ run at least 6 miles last week. 6 \(\frac{1}{4}\) miles _____ 6 miles

Answer: Sia ran 6 whole miles and 1/4 mile. So, Sia did run at least 6 miles last week. 6 \(\frac{1}{4}\) miles > 6 miles

Question 2. What if Sia ran only \(\frac{3}{4}\) mile each day. Would she have run at least 6 miles last week? Explain. _____

Explanation: She would have run \(\frac{3}{4}\) + \(\frac{3}{4}\) + \(\frac{3}{4}\) + \(\frac{3}{4}\) + \(\frac{3}{4}\) + \(\frac{3}{4}\) = \(\frac{15}{4}\) or 3 \(\frac{3}{4}\) miles.

Question 3. A quarter is \(\frac{1}{4}\) dollar. Noah has 20 quarters. How much money does he have? Explain. $ _____

Explanation: Since each quarter is 1/4 dollar, each group of 4 quarters is 1 dollar. Since 4/4 + 4/4 + 4/4 + 4/4 + 4/4 = 20/4, Noah has 1 + 1 + 1 + 1 + 1 = 5 dollars

Question 4. How many \(\frac{2}{5}\) parts are in 2 wholes? _____

Explanation: \(\frac{2}{5}\)/2 = 5

Question 5. A company shipped 15,325 boxes of apples and 12,980 boxes of oranges. How many more boxes of apples than oranges did the company ship? _____ boxes

Answer: 2345 boxes

Explanation: Given, A company shipped 15,325 boxes of apples and 12,980 boxes of oranges. Subtract 12,980 from 15,325 boxes 15,325 – 12,980 = 2,345 boxes.

Question 6. Analyze A fair sold a total of 3,300 tickets on Friday and Saturday. It sold 100 more on Friday than on Saturday. How many tickets did the fair sell on Friday? _____ tickets

Answer: 1700 tickets

Explanation: Given, Analyze A fair sold a total of 3,300 tickets on Friday and Saturday. It sold 100 more on Friday than on Saturday. 3,300 – 100 = 3,200 tickets 3200/2 = 1,600 tickets It sold 1600 tickets on saturday and 1700 tickets on Friday.

Question 7. Emma walked \(\frac{1}{4}\) mile on Monday, \(\frac{2}{4}\) mile on Tuesday, and \(\frac{3}{4}\) mile on Wednesday. If the pattern continues, how many miles will she walk on Friday? Explain how you found the number of miles. \(\frac{□}{□}\) miles

Answer: \(\frac{5}{4}\) miles

Explanation: I made a table that shows each day and the distance she walked. Then I looked for a pattern. The pattern showed that she walked 1/4 mile more each day. I continued the pattern to show she walked 4/4 mile on Thursday and 5/4 miles on Friday.

Question 8. Jared painted a mug \(\frac{5}{12}\) red and \(\frac{4}{12}\) blue. What part of the mug is not red or blue? \(\frac{□}{□}\)

Answer: \(\frac{3}{12}\)

Explanation: Given, Jared painted a mug \(\frac{5}{12}\) red and \(\frac{4}{12}\) blue. We have to find What part of the mug is not red or blue that means \(\frac{3}{12}\) part is neither red nor blue.

Explanation: Given, Each day, Mrs. Hewes knits \(\frac{1}{3}\) of a scarf in the morning and \(\frac{1}{3}\) of a scarf in the afternoon. \(\frac{1}{3}\) + \(\frac{1}{3}\) = \(\frac{2}{3}\) Thus it takes 3 days to knit 2 scarves.

Read each problem and solve.

Question 2. Val walks 2 \(\frac{3}{5}\) miles each day. Bill runs 10 miles once every 4 days. In 4 days, who covers the greater distance? _________

Answer: Val

Explanation: Given, Val walks 2 \(\frac{3}{5}\) miles each day. Bill runs 10 miles once every 4 days. 2 \(\frac{3}{5}\) × 4 Convert from mixed fraction to the improper fraction. 2 \(\frac{3}{5}\) = \(\frac{13}{5}\) × 4 = 10.4 10.4 > 10 Thus Val covers the greater distance.

Question 3. Chad buys peanuts in 2-pound bags. He repackages them into bags that hold \(\frac{5}{6}\) pound of peanuts. How many 2-pound bags of peanuts should Chad buy so that he can fill the \(\frac{5}{6}\) -pound bags without having any peanuts left over? _________ 2-pound bags

Explanation: Given, Chad buys peanuts in 2-pound bags. He repackages them into bags that hold \(\frac{5}{6}\) pound of peanuts. \(\frac{5}{6}\) + \(\frac{5}{6}\) + \(\frac{5}{6}\) + \(\frac{5}{6}\) + \(\frac{5}{6}\) Thus 5 2-pound bags of peanuts are left.

Question 4. A carpenter has several boards of equal length. He cuts \(\frac{3}{5}\) of each board. After cutting the boards, the carpenter notices that he has enough pieces left over to make up the same length as 4 of the original boards. How many boards did the carpenter start with? _________

Explanation: Given, A carpenter has several boards of equal length. He cuts \(\frac{3}{5}\) of each board. After cutting the boards, the carpenter notices that he has enough pieces left over to make up the same length as 4 of the original boards. 4 of the original boards have a summed length of 20 units. 5 x 4 = 20. Since 2/5 is left from each board, you simply add them until the 2’s add to 20. So, 2 x 10 = 20. Hence, there are 10 2/5 boards. That’s just 4 of the boards that the 2/5 make up, but that should also mean that there are 10 3/5 boards as well. 30/5 + 20/5 = 50/5 = 10

Question 1. Karyn cuts a length of ribbon into 4 equal pieces, each 1 \(\frac{1}{4}\) feet long. How long was the ribbon? Options: a. 4 feet b. 4 \(\frac{1}{4}\) feet c. 5 feet d. 5 \(\frac{1}{4}\) feet

Explanation: Given, Karyn cuts a length of ribbon into 4 equal pieces, each 1 \(\frac{1}{4}\) feet long. 1 \(\frac{1}{4}\) × 4 Convert from the mixed fraction to the improper fraction. 1 \(\frac{1}{4}\) = \(\frac{5}{4}\) \(\frac{5}{4}\) × 4 = 5 feet Thus the correct answer is option c.

Question 2. Several friends each had \(\frac{2}{5}\) of a bag of peanuts left over from the baseball game. They realized that they could have bought 2 fewer bags of peanuts between them. How many friends went to the game? Options: a. 6 b. 5 c. 4 d. 2

Explanation: Given, Several friends each had \(\frac{2}{5}\) of a bag of peanuts left over from the baseball game. They realized that they could have bought 2 fewer bags of peanuts between them 2 ÷ \(\frac{2}{5}\) = 5 Thus the correct answer is option b.

Question 3. A frog made three jumps. The first was 12 \(\frac{5}{6}\) inches. The second jump was 8 \(\frac{3}{6}\) inches. The third jump was 15 \(\frac{1}{6}\) inches. What was the total distance the frog jumped? Options: a. 35 \(\frac{3}{6}\) inches b. 36 \(\frac{1}{6}\) inches c. 36 \(\frac{3}{6}\) inches d. 38 \(\frac{1}{6}\) inches

Answer: 36 \(\frac{3}{6}\) inches

Explanation: Given, A frog made three jumps. The first was 12 \(\frac{5}{6}\) inches. The second jump was 8 \(\frac{3}{6}\) inches. The third jump was 15 \(\frac{1}{6}\) inches. First add the whole numbers 12 + 8 + 15 = 35 Next add the fractions, \(\frac{5}{6}\) + \(\frac{3}{6}\) + \(\frac{1}{6}\) = 1 \(\frac{3}{6}\) 35 + \(\frac{3}{6}\) = 36 \(\frac{3}{6}\) inches Thus the correct answer is option c.

Question 4. LaDanian wants to write the fraction \(\frac{4}{6}\) as a sum of unit fractions. Which expression should he write? Options: a. \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\) b. \(\frac{2}{6}+\frac{2}{6}\) c. \(\frac{3}{6}+\frac{1}{6}\) d. \(\frac{1}{6}+\frac{1}{6}+\frac{2}{6}\)

Answer: \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\)

Explanation: Given, LaDanian wants to write the fraction \(\frac{4}{6}\) as a sum of unit fractions. The unit fraction for \(\frac{4}{6}\) is \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\) Thus the correct answer is option a.

Question 5. Greta made a design with squares. She colored 8 out of the 12 squares blue. What fraction of the squares did she color blue? Options: a. \(\frac{1}{4}\) b. \(\frac{1}{3}\) c. \(\frac{2}{3}\) d. \(\frac{3}{4}\)

Explanation: Given, Greta made a design with squares. She colored 8 out of the 12 squares blue. \(\frac{8}{12}\) = \(\frac{2}{3}\) Thus the correct answer is option c.

Question 6. The teacher gave this pattern to the class: the first term is 5 and the rule is add 4, subtract 1. Each student says one number. The first student says 5. Victor is tenth in line. What number should Victor say? Options: a. 17 b. 19 c. 20 d. 21

Answer: given a=5 d=4-1=3 to find t10 tn=a + (n-1) d t10=5 + (10-1) 3 t10=5 + 27 t10 = 32 victor is tenth in line,therefore he should say the number 32

Explanation: Given, A painter mixed \(\frac{1}{4}\) quart of red paint with \(\frac{3}{4}\) blue paint to make purple paint. \(\frac{1}{4}\) + \(\frac{3}{4}\) = \(\frac{4}{4}\) or 1.

Question 2. Ivan biked 1 \(\frac{2}{3}\) hours on Monday, 2 \(\frac{1}{3}\) hours on Tuesday, and 2 \(\frac{2}{3}\) hours on Wednesday. What is the total number of hours Ivan spent biking? Ivan spen _______ hours biking. _____ \(\frac{□}{□}\)

Answer: 6 \(\frac{2}{3}\)

Explanation: Given, Ivan biked 1 \(\frac{2}{3}\) hours on Monday, 2 \(\frac{1}{3}\) hours on Tuesday, and 2 \(\frac{2}{3}\) hours on Wednesday. 1 \(\frac{2}{3}\) + 2 \(\frac{1}{3}\) + 2 \(\frac{2}{3}\) First add the whole numbers, 1 + 2 + 2 = 5 2/3 + 1/3 + 2/3 = 5/3 Convert from improper fraction to the mixed fraction. 5/3 = 1 2/3 5 + 1 1/3  = 6 \(\frac{2}{3}\)

Question 3. Tricia had 4 \(\frac{1}{8}\) yards of fabric to make curtains. When she finished she had 2 \(\frac{3}{8}\) yards of fabric left. She said she used 2 \(\frac{2}{8}\) yards of fabric for the curtains. Do you agree? Explain. ______

Explanation: When I subtract 2 \(\frac{3}{8}\) and 4 \(\frac{1}{8}\), the answer is not 2 \(\frac{2}{8}\). The mixed number 4 \(\frac{1}{8}\) needs to be regrouped as a mixed number with a fraction greater than 1. 4 \(\frac{1}{8}\) = 3 \(\frac{9}{8}\) So, 3 \(\frac{9}{8}\) – 2 \(\frac{3}{8}\) = 1 \(\frac{6}{8}\) or 1 \(\frac{3}{4}\)

Answer: \(\frac{8}{10}\)

Explanation: Given, Miguel’s class went to the state fair. The fairground is divided into sections. Rides are in \(\frac{6}{10}\) of the fairground. Games are in \(\frac{2}{10}\) of the fairground. \(\frac{6}{10}\) + \(\frac{2}{10}\) = \(\frac{8}{10}\)

Question 4. Part B How much greater is the part of the fairground with rides than with farm exhibits? Explain how the model could be used to find the answer. \(\frac{□}{□}\)

Answer: \(\frac{5}{10}\)

Explanation: I could shade 6 sections to represent the section with the rides, and then I could cross out 1 section to represent the farm exhibits. This leaves 5 sections, so the part of the fairground with rides is 5/10 or 1/2 greater than the part with farm exhibits.

Question 5. Rita is making chili. The recipe calls for 2 \(\frac{3}{4}\) cups of tomatoes. How many cups of tomatoes, written as a fraction greater than one, are used in the recipe? _____ cups

Answer: 11/4 cups

Explanation: Given, Rita is making chili. The recipe calls for 2 \(\frac{3}{4}\) cups of tomatoes. Convert from the mixed fraction to the improper fraction. 2 \(\frac{3}{4}\) = 11/4 cups

Question 6. Lamar’s mom sells sports equipment online. She sold \(\frac{9}{10}\) of the sports equipment. Select a way \(\frac{9}{10}\) can be written as a sum of fractions. Mark all that apply. Options: a. \(\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+\frac{2}{10}\) b. \(\frac{3}{10}+\frac{2}{10}+\frac{3}{10}+\frac{1}{10}\) c. \(\frac{2}{10}+\frac{2}{10}+\frac{2}{10}+\frac{2}{10}\) d. \(\frac{4}{10}+\frac{1}{10}+\frac{1}{10}+\frac{3}{10}\) e. \(\frac{4}{10}+\frac{3}{10}+\frac{1}{10}+\frac{1}{10}+\frac{1}{10}\) f. \(\frac{2}{10}+\frac{2}{10}+\frac{2}{10}+\frac{3}{10}\)

Answer: \(\frac{3}{10}+\frac{2}{10}+\frac{3}{10}+\frac{1}{10}\)

Explanation: a. \(\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+\frac{2}{10}\) = 6/10 ≠ 9/10 b. \(\frac{3}{10}+\frac{2}{10}+\frac{3}{10}+\frac{1}{10}\) = 9/10 c. \(\frac{2}{10}+\frac{2}{10}+\frac{2}{10}+\frac{2}{10}\) = 8/10 d. \(\frac{4}{10}+\frac{1}{10}+\frac{1}{10}+\frac{3}{10}\) = 9/10 e. \(\frac{4}{10}+\frac{3}{10}+\frac{1}{10}+\frac{1}{10}+\frac{1}{10}\) = 10/10 ≠ 9/10 f. \(\frac{2}{10}+\frac{2}{10}+\frac{2}{10}+\frac{3}{10}\) = 9/10 Thus the suitable answers are b, d, f.

Question 7. Bella brought \(\frac{8}{10}\) gallon of water on a hiking trip. She drank \(\frac{6}{10}\) gallon of water. How much water is left? \(\frac{□}{□}\) gallons

Answer: \(\frac{2}{10}\) gallons

Explanation: Given, Bella brought \(\frac{8}{10}\) gallon of water on a hiking trip. She drank \(\frac{6}{10}\) gallon of water. To find how much water is left we have to subtract the two fractions. \(\frac{8}{10}\) – \(\frac{6}{10}\) = \(\frac{2}{10}\) gallons

Answer: \(\frac{7}{10}\)

Explanation: Given, In a survey, \(\frac{6}{10}\) of the students chose Saturday and \(\frac{1}{10}\) chose Monday as their favorite day of the week. \(\frac{6}{10}\) + \(\frac{1}{10}\) = \(\frac{7}{10}\)

Question 8. Part B How are the numerator and denominator of your answer related to the model? Explain. Type below: ___________

Answer: The numerator shows the number of parts shaded. The denominator shows the size of the parts.

Question 10. For numbers 10a–10e, select Yes or No to show if the sum or difference is correct. (a) \(\frac{2}{8}+\frac{1}{8}=\frac{3}{8}\) i. yes ii. no

Explanation: Denominators are the same but the numerators are different. So, add the numerators. \(\frac{2}{8}+\frac{1}{8}=\frac{3}{8}\) Thus the above statement is true.

Question 10. (b) \(\frac{4}{5}+\frac{1}{5}=\frac{5}{5}\) i. yes ii. no

Explanation: Denominators are the same but the numerators are different. So, add the numerators. \(\frac{4}{5}+\frac{1}{5}=\frac{5}{5}\) Thus the above statement is true.

Question 10. (c) \(\frac{4}{6}+\frac{1}{6}=\frac{5}{12}\) i. yes ii. no

Explanation: Denominators are the same but the numerators are different. So, add the numerators. \(\frac{4}{6}+\frac{1}{6}=\frac{5}{6}\) Thus the above statement is false.

Question 10. (d) \(\frac{6}{12}-\frac{4}{12}=\frac{2}{12}\) i. yes ii. no

Explanation: Denominators are the same but the numerators are different. So, subtract the numerators. \(\frac{6}{12}-\frac{4}{12}=\frac{2}{12}\) Thus the above statement is true.

Question 10. (e) \(\frac{7}{9}-\frac{2}{9}=\frac{9}{9}\) i. yes ii. no

Explanation: Denominators are the same but the numerators are different. So, subtract the numerators. \(\frac{7}{9}-\frac{2}{9}=\frac{5}{9}\) Thus the above statement is false.

Question 11. Gina has 5 \(\frac{2}{6}\) feet of silver ribbon and 2 \(\frac{4}{6}\) of gold ribbon. How much more silver ribbon does Gina have than gold ribbon? ______ \(\frac{□}{□}\) feet more silver ribbon.

Answer: 2 \(\frac{4}{6}\) feet more silver ribbon.

Explanation: Given, Gina has 5 \(\frac{2}{6}\) feet of silver ribbon and 2 \(\frac{4}{6}\) of gold ribbon. 5 \(\frac{2}{6}\) – 2 \(\frac{4}{6}\) = \(\frac{32}{6}\) – \(\frac{16}{6}\) = \(\frac{16}{6}\) Convert from improper fraction to the mixed fraction. 2 \(\frac{4}{6}\) feet more silver ribbon Therefore Gina has 2 \(\frac{4}{6}\) feet more silver ribbon than gold ribbon.

Question 12. Jill is making a long cape. She needs 4 \(\frac{1}{3}\) yards of blue fabric for the outside of the cape. She needs 3 \(\frac{2}{3}\) yards of purple fabric for the lining of the cape. Part A Jill incorrectly subtracted the two mixed numbers to find how much more blue fabric than purple fabric she should buy. Her work is shown below. \(4 \frac{1}{3}-3 \frac{2}{3}=\frac{12}{3}-\frac{9}{3}=\frac{3}{3}\) Why is Jill’s work incorrect? Type below: __________________

Answer: Jill changed only the whole number parts of the mixed number to thirds. She forgot to add the fraction part of the mixed number.

Question 12. Part B How much more blue fabric than purple fabric should Jill buy? Show your work. \(\frac{□}{□}\)

Answer: 4 \(\frac{1}{3}\) – 3 \(\frac{2}{3}\) = \(\frac{13}{3}\) – \(\frac{11}{3}\) = \(\frac{2}{3}\) Jill should buy \(\frac{2}{3}\) yard more blue fabric than purple fabric.

Explanation: Given, Russ has two jars of glue. One jar is \(\frac{1}{5}\) full. The other jar is \(\frac{2}{5}\) full. \(\frac{1}{5}\) + \(\frac{2}{5}\) = \(\frac{3}{5}\)

Answer: \(\frac{1}{4}\)

Explanation: Given that, Gertie ran \(\frac{3}{4}\) mile during physical education class. Sarah ran \(\frac{2}{4}\) mile during the same class. \(\frac{3}{4}\) – \(\frac{2}{4}\) = \(\frac{1}{4}\)

Question 15. Teresa planted marigolds in \(\frac{2}{8}\) of her garden and petunias in \(\frac{3}{8}\) of her garden. What fraction of the garden has marigolds and petunias? \(\frac{□}{□}\)

Answer: \(\frac{5}{8}\)

Explanation: Given, Teresa planted marigolds in \(\frac{2}{8}\) of her garden and petunias in \(\frac{3}{8}\) of her garden. Add both the fractions 2/8 and 3/8 to find the fraction of the garden has marigolds and petunias. \(\frac{2}{8}\) + \(\frac{3}{8}\) = \(\frac{5}{8}\)

Explanation: Each day she eats 1/2 cups of rice. But we want to know how long it will take to each 2 cups worth. so lets make an equation. 1/2 × x = 2 x = 4 Thus It will take 4 days to eat 2 cups of rice cereal.

Question 18. Three girls are selling cases of popcorn to earn money for a band trip. In week 1, Emily sold 2 \(\frac{3}{4}\) cases, Brenda sold 4 \(\frac{1}{4}\) cases, and Shannon sold 3 \(\frac{1}{2}\) cases. Part A How many cases of popcorn have the girls sold in all? Explain how you found your answer. ______ \(\frac{□}{□}\)

Answer: 10 \(\frac{1}{2}\) cases

Explanation: Given, Three girls are selling cases of popcorn to earn money for a band trip. In week 1, Emily sold 2 \(\frac{3}{4}\) cases, Brenda sold 4 \(\frac{1}{4}\) cases, and Shannon sold 3 \(\frac{1}{2}\) cases. First I add the whole numbers 2 + 4 + 3 = 9 cases. Then I add the fractions by combining 3/4 + 1/4 into one whole. So, 9 + 1 + 1/2 = 10 \(\frac{1}{2}\) cases

Question 18. Part B The girls must sell a total of 35 cases in order to have enough money for the trip. Suppose they sell the same amount in week 2 and week 3 of the sale as in week 1. Will the girls have sold enough cases of popcorn to go on the trip? Explain. ______

Explanation: Given, The girls must sell a total of 35 cases in order to have enough money for the trip. Suppose they sell the same amount in week 2 and week 3 of the sale as in week 1. If I add the sales from the 3 weeks, or 10 1/2 + 10 1/2 + 10 1/2, the sum is only 31 1/2 cases of popcorn. Thus is less than 35 cases.

Question 19. Henry ate \(\frac{3}{8}\) of a sandwich. Keith ate \(\frac{4}{8}\) of the same sandwich. How much more of the sandwich did Keith eat than Henry? \(\frac{□}{□}\) of the sandwich

Answer: \(\frac{1}{8}\) of the sandwich

Explanation: Given, Henry ate \(\frac{3}{8}\) of a sandwich. Keith ate \(\frac{4}{8}\) of the same sandwich. \(\frac{4}{8}\) – \(\frac{3}{8}\) = \(\frac{1}{8}\) of the sandwich

Question 20. For numbers 20a–20d, choose True or False for each sentence. a. \(1 \frac{4}{9}+2 \frac{6}{9}\) is equal to 4 \(\frac{1}{9}\) i. True ii. False

Explanation: \(1 \frac{4}{9}+2 \frac{6}{9}\) = 4 \(\frac{1}{9}\) First add the whole numbers 1 + 2 = 3 4/9 + 6/9 = 10/9 Convert it into the mixed fractions 10/9 = 1 \(\frac{1}{9}\) 3 + 1 \(\frac{1}{9}\) = 4 \(\frac{1}{9}\) Thus the above statement is true.

Question 20. b. \(3 \frac{5}{6}+2 \frac{3}{6}\) is equal to 5 \(\frac{2}{6}\) i. True ii. False

Explanation: First add the whole numbers 3 + 2 = 5 5/6 + 3/6 = 8/6 Convert it into the mixed fractions 8/6 = 1 \(\frac{2}{6}\) 5 + 1 \(\frac{2}{6}\) = 6 \(\frac{2}{6}\) Thus the above statement is false.

Question 20. c. \(4 \frac{5}{8}-2 \frac{4}{8}\) is equal to 2 \(\frac{3}{8}\) i. True ii. False

Explanation: \(4 \frac{5}{8}-2 \frac{4}{8}\) First subtract the whole numbers 4 – 2 = 2 5/8 – 4/8 = 1/8 = 2 \(\frac{1}{8}\) Thus the above statement is false.

Question 20. d. \(5 \frac{5}{8}-3 \frac{2}{8}\) is equal to 2 \(\frac{3}{8}\) i. True ii. False

Explanation: \(5 \frac{5}{8}-3 \frac{2}{8}\) 5 – 3 = 2 5/8 – 2/8 = 3/8 = 2 \(\frac{3}{8}\) \(5 \frac{5}{8}-3 \frac{2}{8}\) = 2 \(\frac{3}{8}\) Thus the above statement is true.

Question 21. Justin lives 4 \(\frac{3}{5}\) miles from his grandfather’s house. Write the mixed number as a fraction greater than one. 4 \(\frac{3}{5}\) = \(\frac{□}{□}\)

Answer: \(\frac{23}{5}\)

Explanation: Justin lives 4 \(\frac{3}{5}\) miles from his grandfather’s house. Convert from mixed fractions to an improper fraction. 4 \(\frac{3}{5}\) = \(\frac{23}{5}\)

Explanation: \(\frac{3}{4}\) The unit fraction of \(\frac{3}{4}\) is \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) \(\frac{3}{4}\) = 3 × \(\frac{1}{4}\) Thus the whole number is 3.

Write the fraction as a product of a whole number and a unit fraction.

Question 2. \(\frac{4}{5}\) = ______ × \(\frac{1}{5}\)

Explanation: The unit fraction for \(\frac{4}{5}\) is \(\frac{1}{5}\) + \(\frac{1}{5}\) + \(\frac{1}{5}\) + \(\frac{1}{5}\) \(\frac{4}{5}\) = 4 × \(\frac{1}{5}\) Thus the whole number is 4.

Question 3. \(\frac{3}{10}\) = ______ × \(\frac{1}{10}\)

Explanation: The unit fraction for \(\frac{3}{10}\) is \(\frac{1}{10}\) + \(\frac{1}{10}\) + \(\frac{1}{10}\) \(\frac{3}{10}\) = 3 × \(\frac{1}{10}\) Thus the whole number is 3.

Question 4. \(\frac{8}{3}\) = ______ × \(\frac{1}{3}\)

Explanation: The unit fraction for \(\frac{8}{3}\) is \(\frac{1}{3}\) + \(\frac{1}{3}\) + \(\frac{1}{3}\) + \(\frac{1}{3}\) + \(\frac{1}{3}\) + \(\frac{1}{3}\) + \(\frac{1}{3}\) + \(\frac{1}{3}\) \(\frac{8}{3}\) = 8 × \(\frac{1}{3}\) Thus the whole number is 8.

List the next four multiples of the unit fraction.

Question 5. \(\frac{1}{6}\) , Type below: ___________

Answer: 2/6, 3/6, 4/6, 5/6

Explanation: The next four multiples of \(\frac{1}{6}\) is \(\frac{2}{6}\) , \(\frac{3}{6}\) , \(\frac{4}{6}\) , \(\frac{5}{6}\)

Question 6. \(\frac{1}{3}\) , Type below: ___________

Answer: 2/3, 3/3, 4/3, 5/3

Explanation: The next four multiples of \(\frac{1}{3}\) is \(\frac{2}{3}\), \(\frac{3}{3}\), \(\frac{4}{3}\) and \(\frac{5}{3}\)

Question 7. \(\frac{5}{6}\) = ______ × \(\frac{1}{6}\)

Explanation: The unit fraction for \(\frac{5}{6}\) is \(\frac{1}{6}\) + \(\frac{1}{6}\) + \(\frac{1}{6}\) + \(\frac{1}{6}\) + \(\frac{1}{6}\) \(\frac{5}{6}\) = 5 × \(\frac{1}{6}\) Thus the whole number is 5.

Question 8. \(\frac{9}{4}\) = ______ × \(\frac{1}{4}\)

Explanation: The unit fraction for \(\frac{9}{4}\) is \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{1}{4}\) \(\frac{9}{4}\) = 9 × \(\frac{1}{4}\) Thus the whole number is 9.

Question 9. \(\frac{3}{100}\) = ______ × \(\frac{1}{100}\)

Explanation: The unit fraction for \(\frac{3}{100}\) is \(\frac{1}{100}\) + \(\frac{1}{100}\) + \(\frac{1}{100}\) \(\frac{3}{100}\) = 3 × \(\frac{1}{100}\) Thus the whole number is 3.

Question 10. \(\frac{1}{10}\) , Type below: ___________

Answer: 2/10, 3/10, 4/10, 5/10

Explanation: The next four multiples of \(\frac{1}{10}\) is 2/10, 3/10, 4/10, 5/10

Question 11. \(\frac{1}{8}\) , Type below: ___________

Answer: 2/8, 3/8, 4/8, 5/8

Explanation: The next four multiples of \(\frac{1}{8}\) is 2/8, 3/8, 4/8, 5/8.

Question 12. Robyn uses \(\frac{1}{2}\) cup of blueberries to make each loaf of blueberry bread. Explain how many loaves of blueberry bread she can make with 2 \(\frac{1}{2}\) cups of blueberries. _____ loaves of blueberry bread

Answer: 5 loaves of blueberry bread

Explanation: Given, Robyn uses \(\frac{1}{2}\) cup of blueberries to make each loaf of blueberry bread. The unit fraction for 2 \(\frac{1}{2}\) is \(\frac{1}{2}\) + \(\frac{1}{2}\) + \(\frac{1}{2}\) + \(\frac{1}{2}\) + \(\frac{1}{2}\) =  5 loaves of blueberry bread

Question 13. Nigel cut a loaf of bread into 12 equal slices. His family ate some of the bread and now \(\frac{5}{12}\) of the loaf is left. Nigel wants to put each of the leftover slices in its own bag. How many bags does Nigel need? _____ bags

Answer: 5 bags

Explanation: Given, Nigel cut a loaf of bread into 12 equal slices. His family ate some of the bread and now \(\frac{5}{12}\) of the loaf is left. Nigel wants to put each of the leftover slices in its own bag. \(\frac{5}{12}\) = \(\frac{1}{12}\) + \(\frac{1}{12}\) + \(\frac{1}{12}\) + \(\frac{1}{12}\) + \(\frac{1}{12}\) = 5 bags

Question 14. Which fraction is a multiple of \(\frac{1}{5}\)? Mark all that apply. Options: a. \(\frac{4}{5}\) b. \(\frac{5}{7}\) c. \(\frac{5}{9}\) d. \(\frac{3}{5}\)

Answer: \(\frac{4}{5}\), \(\frac{3}{5}\)

Explanation: The multiples of the \(\frac{1}{5}\) is \(\frac{4}{5}\), \(\frac{3}{5}\).

Sense or Nonsense?

Answer: The boy’s statement makes sense. Because 4/5 is not the multiple of 1/4.

Question 15. For the statement that is nonsense, write a new statement that makes sense. Type below: _________________

Answer: 4/5 is the multiple of 1/5.

Conclusion:

Just click on the links available above and practice the concepts of add and subtract fractions for homework help & standard tests. Help students to practice all chapter 7 questions from Go Math Answer Key to write the answers perfectly. For more questions just go with our Go Math Grade 4 Answer Key Chapter 7 Add and Subtract Fractions Homework Practice FL pdf article.

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