multiplying fractions by whole numbers problem solving

Multiplying Fractions With Whole Numbers

For multiplying fractions with whole numbers, the whole number is written in the fraction form and then multiplied with the given fraction using the rules of multiplication of fractions. While multiplying fractions with whole numbers, it should also be remembered that the given fractions should be in the form of a proper fraction or an improper fraction. Let us learn more about multiplying fractions with whole numbers, along with some examples.

What is Multiplying Fractions With Whole Numbers?

Multiplying fractions with whole numbers is similar to repeated addition where the fraction is added the same number of times as the whole number. For multiplying fractions, we first multiply the numerators, then multiply the denominators, and finally, reduce the resultant fraction to its lowest terms. However, when we need to multiply fractions with whole numbers, we write the whole number in the form of a fraction by writing 1 as its denominator. After this step, we can multiply it using the same rules. For example, when we multiply the fraction a/b × c/d, we get (a × c) / (b × d). This rule is applicable while multiplying fractions with whole numbers as well.

How to Multiply Fractions With Whole Numbers?

Multiplying fractions with whole numbers is an easy concept. We just need to convert the whole number into a fraction by writing 1 as the denominator and writing the whole number as the numerator . Then it is multiplied by the given fraction. After multiplying these, the final result should be in the form of a proper fraction or a mixed fraction. If the result is in an improper fraction, we convert it to a mixed fraction. Let us understand the steps with the help of an example.

Example: Multiply 1/8 × 5

Solution: Here, 1/8 is the fraction and 5 is the whole number.

• Step 1: Convert the whole number to a fraction by writing 1 as the denominator. This means 5 is written as 5/1
• Step 2: Multiply the numerators. Here, 1 × 5 = 5
• Step 3: Multiply the denominators. Here, 8 × 1 = 8
• Step 4: Simplify and reduce the product, if needed. If the result is an improper fraction, we convert it to a mixed fraction. So, the product is 5/8

Let us look at another example to understand this better.

Example 2: Multiply 5 × 3/10.

Solution: Here, 5 is the whole number and 3/10 is the proper fraction.

• Step 1: We convert the whole number 5 into a fraction by writing 1 as the denominator. This means 5 is written as 5/1.
• Step 2: Multiply the numerators of both the fractions. 5/1 × 3/10 = 5 × 3 = 15.
• Step 3: Multiply the denominators of both the fractions. 5/1 × 3/10 = 1 × 10 = 10.
• Step 4: Simplify the fractions. 5/1 × 3/10 = 15/10. We can simplify this further as both 15 and 10 can be divided by 5. This means, (15 ÷ 5) / (10 ÷ 5) = 3/2. Therefore, 5 × 3/10 = 3/2 = \(1\dfrac{1}{2}\)

How to Multiply Mixed Fractions with Whole Numbers?

In order to multiply mixed fractions with whole numbers, we convert the mixed fraction to an improper fraction and then multiply it with the whole number.

Example: Multiply \(1\dfrac{2}{5}\) with 10.

Solution: Let us see how to multiply the given mixed fraction with a whole number.

• Step 1: First let us convert the mixed fraction to an improper fraction. This means \(1\dfrac{2}{5}\) = 7/5.
• Step 2: Then, convert the whole number 10 into a fraction. This means 10 = 10/1. This makes it 7/5 × 10/1
• Step 3: Multiply the numerators of both the fractions. 7 × 10 = 70. Multiply the denominators of both the fractions. This means 5 × 1 = 5.
• Step 4: Simplify and reduce the fraction, that is, 70/5 = (70 ÷ 5) / (5 ÷ 5) = 14/1. Therefore, \(1\dfrac{2}{5}\) × 10 = 14.

☛ Related Articles

• Multiplying Mixed Fractions
• Division of Fractions
• Reduce Fractions
• Addition and Subtraction of Fractions
• Fractions Formula

Examples on Multiplying Fractions With Whole Numbers

Example 1: Multiply the fraction with the whole number: 1/3 × 15

1/3 × 15 = 1/3 × 15/1 = (1 × 15) / (3 × 1) = 15/3 = 5.

Therefore, 1/3 × 15 = 5

Example 2: Find the product after multiplying the fraction with the whole number: 3/4 × 4

3/4 × 4 = 3/4 × 4/1 = (3 × 4) / (4 × 1) = 12/4 = 3.

Therefore, 3/4 × 4 = 3

Example 3: Find the product of the whole number 6 and the mixed fraction \(3\dfrac{4}{7}\)

Let us first convert the mixed fraction to a proper fraction.

\(3\dfrac{4}{7}\) = [(7 × 3) + 4] / 7 = 25/7

Let us convert the whole number to fraction,

Now let us multiply the fraction with the whole number,

6/1 × 25/7 = (6 × 25) / (1 × 7) = 150/7

Once the result is obtained we convert this into a mixed fraction.

150/7 = \(21\dfrac{3}{7}\).

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Practice Questions on Multiplying Fractions With Whole Numbers

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FAQs on Multiplying Fractions With Whole Numbers

What is meant by multiplying fractions with whole numbers.

Multiplying fractions with the whole numbers is considered as repeated addition where the fraction is added the same number of times as the whole number. The multiplication of fractions with whole numbers is done using the same multiplication rules, where the numerators are multiplied together, then the denominators are multiplied together and then they are reduced to get the product.

In order to multiply fractions with whole numbers, we use the following steps.

• Step 1: Convert the whole number into a fraction by writing 1 as its denominator.
• Step 2: After this, we have two fractions to multiply. So, we use the multiplication rule of fractions to multiply the fractions.
• Step 3: This means, first the numerators are multiplied together and then the denominators are multiplied together.
• Step 4: Finally, the product is simplified or reduced, if needed.

The following steps show how to multiply mixed fractions with whole numbers:

• Step 1: Convert the mixed fraction to an improper fraction.
• Step 2: Convert the whole number into a fraction with a denominator of 1.
• Step 3: Multiply the numerators.
• Step 4: Multiply the denominators.
• Step 5: Simplify the final result to its lowest terms.

How to Multiply Improper Fractions with Whole Numbers?

For multiplying improper fractions with whole numbers, we use the same rules of multiplication. This means the whole number is written in the form of a fraction and then multiplied with the improper fraction. The numerators are multiplied together, then the denominators are multiplied together and then they are simplified, if needed.

How to Multiply 3 Fractions with Whole Numbers?

In order to multiply 3 fractions with whole numbers, we use the following steps. Let us multiply 4/5 × 10/6 × 1/4 × 25.

• Step 1: Here, 25 is the whole number and the rest of them are fractions, so we will convert the whole number into a fraction by writing its denominator as 1. This means, 25 is written as 25/1
• Step 2: Now, we have 4 fractions to multiply. So, we use the multiplication rule of fractions to multiply all these fractions. 4/5 × 10/6 × 1/4 × 25/1
• Step 3: This means, first the numerators are multiplied together and then the denominators are multiplied together. Here, the product of the numerators will be 4 × 10 × 1 × 25 = 1000. The product of the denominators will be 5 × 6 × 4 × 1 = 120.
• Step 4: The fraction that we get as the product is 1000/120. Finally, the product is simplified or reduced, this means, 1000/120 = 25/3 = \(8\dfrac{1}{3}\)

How to Multiply Negative Fractions with Whole Numbers?

For multiplying negative fractions with whole numbers, we use the same rules of multiplication. This means the whole number is written in the form of a fraction and then multiplied with the negative fraction. The numerators are multiplied together, then the denominators are multiplied together and then they are simplified if needed. However, it should be remembered that the product will have the sign according to the sign given in the fraction. This means if a negative fraction is multiplied with a positive whole number, the product will have a negative sign. For example, -6/4 × 5 = -6/4 × 5/1. Now we can multiply the numerators and denominators to get -30/4 which will be further reduced to -15/2.

What is the Rule of Multiplying Fractions?

There are two simple steps for multiplying fractions. First, multiply the numerators, and then the denominators of both the fractions to obtain the resultant fraction. Then, we need to simplify the fraction obtained, to get the product. This can be further reduced, if required. This can be understood by a simple example. 2/6 × 4/7 = (2 × 4)/(6 × 7) = 8/42 = 4/21.

Multiplying Fractions

Multiply the tops, multiply the bottoms.

There are 3 simple steps to multiply fractions

1. Multiply the top numbers (the numerators ).

2. Multiply the bottom numbers (the denominators ).

3. Simplify the fraction if needed.

Example: 1 2 × 2 5

Step 1 . Multiply the top numbers:

1 2  ×  2 5   =   1 × 2     =   2  

Step 2 . Multiply the bottom numbers:

1 2  ×  2 5   =   1 × 2 2 × 5   =   2 10

Step 3 . Simplify the fraction :

Here you can see it with pizza ...

Do you see that half of two-fifths is two-tenths? Do you also see that two-tenths is simpler as one-fifth?

With Pen and Paper

And here is how to do it with a pen and paper (press the play button):

Another Example: 1 3 × 9 16

1 3  ×  9 16   =   1 × 9     =   9  

1 3  ×  9 16   =   1 × 9 3 × 16   =   9 48

Step 3 . Simplify the fraction:

9 48 = 3 16

(This time we simplified by dividing both top and bottom by 3)

♫ "Multiplying fractions: no big problem, Top times top over bottom times bottom. "And don't forget to simplify, Before it's time to say goodbye" ♫

Fractions and Whole Numbers

What about multiplying fractions and whole numbers?

Make the whole number a fraction, by putting it over 1.

Example: 5 is also 5 1

Then continue as before.

Example: 2 3  ×  5

Make 5 into 5 1 :

2 3  ×  5 1

Now just go ahead as normal.

Multiply tops and bottoms:

2 3  ×  5 1   =   2 × 5 3 × 1   =   10 3

The fraction is already as simple as it can be.

Answer = 10 3

Or you can just think of the whole number as being a "top" number:

Example: 3 ×  2 9

3    ×  2 9   =   3 × 2 9   =   6 9

Mixed Fractions

You can also read how to multiply mixed fractions

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How to Multiply Fractions With Whole Numbers

Last Updated: April 20, 2024 Fact Checked

This article was co-authored by Mario Banuelos, PhD and by wikiHow staff writer, Jessica Gibson . Mario Banuelos is an Assistant Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 2,284,817 times.

It's easy to multiply fractions by mixed fractions or whole numbers. Start by turning your mixed fractions or whole numbers into improper fractions. Then multiply the numerators of both improper fractions. Multiply the denominators and simplify your result.

Practice Problems

Multiplying Mixed Fractions by Mixed Fractions

• For example, if you start with 1 1/2 x 4 4/7, change them to improper fractions. 1 1/2 will become 3/2 and 4 4/7 will become 32/7. Your equation will now by 3/2 x 32/7.

• The numerator is always the top number in a fraction.
• For example, with 3/2 x 32/7, multiply 3 by 32 to get 96.

• For example, with 3/2 x 32/7, multiply the 2 by 7 to get 14.

• For example, if you got 96/14, see how many times 14 will go into 96. You'll get 6 with 12 left over. Place 12 over the denominator (14).
• Most instructors will want you to put the answer in the same form as the question. So if you started with mixed fractions, convert your answer to a mixed fraction.

• In this example, your final answer will be 6 6/7.

Multiplying Fractions by Whole Numbers

• For example, if you have 5 x 8/10, put the 5 over 1. You should now have 5/1 x 8/10.

• In the example, 5/1 x 8/10, multiply 5 by 8 to get 40.

• For example, if you're multiplying 5/1 x 8/10, multiply 1 by 10 to get 10. Place this below the line to get an answer of 40/10.

• To reduce 40/10, divide 40 by 10 to get 4 as your new answer.
• In many cases, you'll get a mixed number since the answer will have a remainder.

Community Q&A

• Remember that if you multiply a negative value by a positive value, the answer will be negative. However, if you have a negative value and a negative value, the negatives will cancel each other, giving you a positive result. Thanks Helpful 0 Not Helpful 0

You Might Also Like

• ↑ https://www.mathsisfun.com/mixed-fractions-multiply.html
• ↑ https://www.georgebrown.ca/sites/default/files/uploadedfiles/tlc/_documents/multiplying_and_dividing_fractions.pdf
• ↑ Mario Banuelos, PhD. Assistant Professor of Mathematics. Expert Interview. 11 December 2022.
• ↑ https://www.mathsisfun.com/fractions_multiplication.html
• ↑ https://www.cuemath.com/numbers/multiplying-fractions-with-whole-numbers/
• ↑ https://flexbooks.ck12.org/cbook/ck-12-middle-school-math-concepts-grade-6/section/7.1/primary/lesson/multiplication-of-fractions-and-whole-numbers-msm6/

About This Article

To multiply a fraction by a whole number, first rewrite the whole number as a fraction by putting it over a 1. For example, let’s say you’re trying to solve 5 x 8/10. You would start by rewriting 5 as a fraction. Now the equation looks like 5/1 x 8/10. Next you need to multiply the numerators, or top numbers, of the fractions together. In our example, 5 and 8 are the numerators, so you would multiply 5 by 8 and get 40. Now do the same thing with the denominators, or bottom numbers, of the fractions. One and 10 are the denominators, so you would multiply 1 by 10 and get 10. The new fraction is 40/10. If you can’t simplify the new fraction, you’re done and that’s your answer. If you can, simplify the fraction to the lowest terms. In our example we ended up with 40/10, which can be simplified by dividing the numerator and denominator by 10, which gives you 4/1. Therefore, 5 x 8/10 = 4/1, or 4. To learn how to multiply a normal fraction with a whole number, scroll down! Did this summary help you? Yes No

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Fraction Multiplication Word Problems Worksheets

Our printable worksheets on multiplying fractions word problems task grade 4 through grade 7 students with reading and solving realistic scenarios by performing fraction multiplication. The problems feature both common and uncommon denominators, so the budding problem-solving stars must follow the correct procedure to obtain the products. Equipped with answer keys, these pdf resources are available in customary and metric units. Try some multiplying fractions word problems worksheets for free!

Multiplying Fractions by Whole Numbers Word Problems

Impel the eager beavers in 4th grade and 5th grade to multiply fractions by whole numbers through word problems. Kids are required to multiply proper fractions and improper fractions by whole numbers correctly.

• Download the set

Multiplying Fractions by Cross-Cancelling Word Problems

Say a whole-hearted yes to our no-prep printable worksheets on fraction multiplication that have you fully covered! Interpret the word problems and multiply fractions with a special emphasis on cross cancelling.

Multiplying Mixed Numbers Word Problems

Multiplying mixed numbers may initially feel incomprehensible, but practice makes the pain melt away. Let grade 6 and grade 7 kids convert the mixed numbers into fractions and work out the products.

Multiplying Mixed Numbers and Fractions Word Problems

Let not the challenges of obtaining solutions for problems befall you! Our pdf worksheets on multiplying fractions word problems, equipped with answer key for a quick self-validation, help ease into the process.

Themed Fraction Multiplication Word Problems

Revive 5th grade, 6th grade, and 7th grade students' problem-solving fortunes with our themed word problems, featuring a good mix of like fractions, unlike fractions, whole numbers, and mixed numbers!

Related Worksheets

» Multiplying Fractions on a Number Line

» Multiplying Fractions by Whole Numbers

» Multiplying Fractions with Cross Cancelling

» Multiplying Mixed Numbers

» Fraction Word Problems

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Multiplying fractions word problems

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Multiplying Fractions, Whole Numbers, and Mixed Numbers Lesson Plan

Get the lesson materials.

Multiplying Fractions, Whole Numbers, Mixed Numbers Guided Notes Doodles Tiling

Ever wondered how to teach multiplying fractions, whole numbers, and mixed numbers in an engaging way to your 5th or 6th-grade students?

In this lesson plan, students will learn about fraction multiplication and its real-life applications. Through artistic and interactive guided notes, check for understanding activities, a doodle and color by number worksheet, and a maze worksheet, students will gain a comprehensive understanding of multiplying fractions, whole numbers, and mixed numbers.

The lesson culminates with a real-life example that explores how this math skill can be applied in practical situations. Students will have the opportunity to read and write about real-life uses of multiplying fractions, whole numbers, and mixed numbers, connecting their learning to the world around them.

• Standards : CCSS 5.NF.B.4 , CCSS 5.NF.B.4.a , CCSS 5.NF.B.4.b , CCSS 5.NF.B.5 , CCSS 5.NF.B.6
• Topic : Fractions
• Grades : 5th Grade , 6th Grade
• Type : Lesson Plans

Learning Objectives

After this lesson, students will be able to:

Multiply fractions by whole numbers using visual models and algorithms

Multiply fractions by fractions using visual models and algorithms

Multiply fractions and mixed numbers using visual models and algorithms

Explain the real life application of fraction multiplication

Prerequisites

Before this lesson, students should be familiar with:

Basic multiplication skills of whole numbers

Understanding of the concept of numerators vs. denominators

Basic understanding of mixed numbers and improper fractions

Colored pencils or markers

Multiplying Fractions, Whole Numbers, Mixed Numbers Guided Notes

Key Vocabulary

Multiplying fractions

Whole numbers

Mixed numbers

Visual models

Introduction

As a hook, ask students why multiplying fractions, whole numbers, and mixed numbers is an important skill in real life. Refer to the real-life application page included in this resource for ideas. You can also ask questions like "Have you ever had to multiply a fraction and a whole number?" or "Can you think of a situation where you would need to multiply mixed numbers?" This will help activate their prior knowledge and engage them in the lesson.

Use the first page of the guided notes to introduce the concept of multiplying fractions by whole numbers. Walk through the key points of this topic, including how to set up the multiplication problem, multiply the numerator and the whole number, and simplify the fraction if necessary. Also, take some time to go through how to multiply fractions by whole numbers (bottom of page 1 of guided notes). Emphasize the visual models provided in the guided notes to help students understand the concept.

Introduce how to multiply fractions by mixed numbers using the second page of the guided notes. Introduce visual diagrams and tiling (bottom of page 2 of guided notes).

Based on student responses, reteach concepts that students need extra help with. If your class has a wide range of proficiency levels, you can pull out students for reteaching, and have more advanced students begin work on the practice exercises.

Have students practice multiplying fractions, whole numbers, and mixed numbers using the maze activity (page 3 of guided notes). Walk around to answer student questions.

Fast finishers can dive into the color by number activity for extra practice (page 4 of guided notes). You can assign it as homework for the remainder of the class.

Real-Life Application

Use the last page of the guided notes to bring the class back together, and introduce the concept of real-life applications of multiplying fractions, whole numbers, and mixed numbers. Explain to the students that understanding how to multiply these types of numbers can be useful in various real-life situations.

For example, when baking or cooking, we often need to adjust a recipe to make more or less food. Sometimes, the recipe is written for a certain number of servings, and we need to double or halve the ingredients. By knowing how to multiply fractions, whole numbers, and mixed numbers, we can easily scale the recipe to fit our needs.

Another example is when building or constructing. Architects, engineers, and carpenters often need to calculate measurements and dimensions when designing and building structures. By knowing how to multiply fractions and whole numbers, they can accurately determine the amount of materials needed.

Refer to the FAQ section for more ideas on how to teach the real-life application of multiplying fractions, whole numbers, and mixed numbers.

Additional Self-Checking Digital Practice

If you're looking for digital practice for multiplying fractions by whole numbers, fractions, and mixed numbers, try my Pixel Art activities in Google Sheets. Every answer is automatically checked, and correct answers unlock parts of a mystery picture. It's incredibly fun, and a powerful tool for differentiation.

Here are 2 activities to explore:

Multiplying Fractions by Tiling Visual Models Pixel Art Google Sheets

(FREE!) Multiplying Fractions by Whole Numbers and Fractions Pixel Art | Google Sheets

Multiplying and Dividing Fractions Digital Pixel Art BUNDLE

What is multiplying fractions by whole numbers? Open

Multiplying fractions by whole numbers is the process of finding the product when a whole number is multiplied by a fraction. The whole number can be expressed as a fraction with a denominator of 1. To multiply a whole number by a fraction, we multiply the numerator of the fraction by the whole number.

How do you multiply fractions by whole numbers? Open

To multiply a fraction by a whole number, follow these steps:

Write the whole number as a fraction with a denominator of 1.

Multiply the numerators of the fractions.

Multiply the denominators of the fractions.

Simplify the resulting fraction, if necessary.

What are visual models for multiplying fractions? Open

Visual models for multiplying fractions are diagrams or drawings that help illustrate the process of multiplying fractions. There are different types of visual models, such as area models and number line models, that can be used to represent fractions and their multiplication.

How can I use visual models to multiply fractions by whole numbers? Open

To use visual models to multiply fractions by whole numbers, you can:

Use an area model: Draw a rectangle or square to represent the whole number and divide it into equal parts to represent the fraction. Shade the appropriate fraction of the whole number and count the shaded parts.

Use a number line model: Draw a number line and place the whole number and the fraction on it. Mark and count the jumps from the whole number to find the product.

What is the algorithm for multiplying fractions by whole numbers? Open

The algorithm for multiplying fractions by whole numbers is as follows:

What are some real-life applications of multiplying fractions by whole numbers? Open

Some real-life applications of multiplying fractions by whole numbers include:

Calculating the total cost of a certain number of items when the price is given as a fraction of a dollar.

Scaling recipes when cooking or baking.

Determining distances or lengths when given a fractional scale.

How can I link multiplying fractions by whole numbers to real-life math skills? Open

To link multiplying fractions by whole numbers to real-life math skills, you can:

Provide real-life word problems that require the multiplication of fractions by whole numbers.

Discuss and analyze real-life situations where multiplying fractions by whole numbers is necessary.

Encourage students to create their own real-life scenarios and solve them using the concepts learned in this lesson.

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Multiplying Fractions By Whole Numbers Song

Multiplying Fractions by Whole Numbers shows up in your daily life all the time, but perhaps you just don't know it yet. In this song we'll create and solve word problems from real life scenarios to hammer that point home.

Perhaps you've run halfway to the big box store and know that the store is 8 exactly miles away. But how far have you ran already? Don't worry, we can solve that! Now you are just one multiplying a fraction by a whole number problem away from knowing how far you've already run. 1/2 x 8 miles = 4 miles.

Song Lyrics:

I was running 8 miles away. I had already run 3/4 of the way. To find how far I'd traveled from my front door. I did 8 times 3 and got 24. Then I divided by 4 and I got 6. Six miles I had run in my brand new kicks!

Multiplying whole numbers by fractions is awesome. Yea! Well here's a similar problem. We'll not have forgotten. Multiply by the top. Divide by the bottom. Multiply by the top. Divide by the bottom.

I was riding to the video game shop. I had ridden 5/6 of 12 city blocks. How far I had gone is what I wanted to find. 12 Times 5 is what I multiplied. I got 60 and then divided by 6. I had ridden 10 blocks and did a cool bike trick.

A girl used repeated addition to see.. if she could use the strategy to solve 1/2 times 3. She thought fast, and then it came like a FLASH! That 1/2 + 1/2 + 1/2 is 3 halves. She converted 3 halves to 1 and a half. She had a crush that would last. It was a crush on MATH!

This song targets TEKS and Common Core learning standards from 4th Grade to 6th Grade. Look into the relevant standards here , or dig deeper into multiplying fractions here.

If you are interested in getting ideas on how to plan a robust standards-aligned Multiply Fractions by Whole Numbers lesson, we recommend checking out Instructure's recommendations for common core standards 4.NF.4 , and 5.NF.4 . These pages help break down standard language, lay out the grade-appropriate level of rigor for each concept, and offer a variety of suggestions for activities (lesson seeds) that help students achieve their learning targets.

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Course: 4th grade   >   Unit 9

• Equivalent fraction and whole number multiplication problems
• Multiplying unit fractions and whole numbers
• Multiply unit fractions and whole numbers

Multiply fractions and whole numbers

• Equivalent whole number and fraction multiplication expressions
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

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2.2.1: Multiplying Fractions and Mixed Numbers

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Learning Objectives

• Multiply two or more fractions.
• Multiply a fraction by a whole number.
• Multiply two or more mixed numbers.
• Solve application problems that require multiplication of fractions or mixed numbers.

Introduction

Just as you add, subtract, multiply, and divide when working with whole numbers, you also use these operations when working with fractions. There are many times when it is necessary to multiply fractions and mixed numbers . For example, this recipe will make 4 crumb piecrusts:

5 cups graham crackers

8 tablespoons sugar

\(\ 1 \frac{1}{2}\) cups melted butter

\(\ \frac{1}{4}\) teaspoon vanilla

Suppose you only want to make 2 crumb piecrusts. You can multiply all the ingredients by \(\ \frac{1}{2}\), since only half of the number of piecrusts are needed. After learning how to multiply a fraction by another fraction, a whole number or a mixed number, you should be able to calculate the ingredients needed for 2 piecrusts.

Multiplying Fractions

When you multiply a fraction by a fraction, you are finding a “fraction of a fraction.” Suppose you have \(\ \frac{3}{4}\) of a candy bar and you want to find \(\ \frac{1}{2}\) of the \(\ \frac{3}{4}\):

By dividing each fourth in half, you can divide the candy bar into eighths.

Then, choose half of those to get \(\ \frac{3}{8}\).

In both of the above cases, to find the answer, you can multiply the numerators together and the denominators together.

Multiplying Two Fractions

\(\ \frac{a}{b} \cdot \frac{c}{d}=\frac{a \cdot c}{b \cdot d}=\frac{\text { product of the numerators }}{\text { product of the denominators }}\)

\(\ \frac{3}{4} \cdot \frac{1}{2}=\frac{3 \cdot 1}{4 \cdot 2}=\frac{3}{8}\)

Multiplying More Than Two Fractions

\(\ \frac{a}{b} \cdot \frac{c}{d} \cdot \frac{e}{f}=\frac{a \cdot c \cdot e}{b \cdot d \cdot f}\)

\(\ \frac{1}{3} \cdot \frac{2}{4} \cdot \frac{3}{5}=\frac{1 \cdot 2 \cdot 3}{3 \cdot 4 \cdot 5}=\frac{6}{60}\)

\(\ \frac{8}{15}\)

If the resulting product needs to be simplified to lowest terms, divide the numerator and denominator by common factors.

\(\ \frac{2}{3} \cdot \frac{1}{4}=\frac{1}{6}\)

You can also simplify the problem before multiplying, by dividing common factors.

You do not have to use the “simplify first” shortcut, but it could make your work easier because it keeps the numbers in the numerator and denominator smaller while you are working with them.

\(\ \frac{3}{4} \cdot \frac{1}{3}\) Multiply. Simplify the answer.

• \(\ \frac{3}{12}\)
• \(\ \frac{4}{7}\)
• \(\ \frac{1}{4}\)
• \(\ \frac{36}{144}\)
• Incorrect. \(\ \frac{3}{12}\) is an equivalent fraction to the correct answer \(\ \frac{1}{4}\), but it is not in lowest terms. You must divide numerator and denominator by the common factor 3. The correct answer is \(\ \frac{1}{4}\).
• Incorrect. You may have added numerators (3+1) and added denominators (4+3) instead of multiplying. The correct answer is \(\ \frac{1}{4}\).
• Correct. One way to find this answer is to multiply numerators and denominators, \(\ \frac{3 \cdot 1}{4 \cdot 3}=\frac{3}{12}\), then simplify: \(\ \frac{3 \div 3}{12 \div 3}=\frac{1}{4}\).
• Incorrect. You probably found a common denominator, multiplied correctly, but then forgot to simplify. Finding a common denominator is not necessary and makes the multiplication harder because you are working with greater than necessary numbers. The correct answer is \(\ \frac{1}{4}\).

Multiplying a Fraction by a Whole Number

When working with both fractions and whole numbers, it is useful to write the whole number as an improper fraction (a fraction where the numerator is greater than or equal to the denominator). All whole numbers can be written with a "1" in the denominator. For example: \(\ 2=\frac{2}{1}\), \(\ 5=\frac{5}{1}\), and \(\ 100=\frac{100}{1}\). Remember that the denominator tells how many parts there are in one whole, and the numerator tells how many parts you have.

Multiplying a Fraction and a Whole Number

\(\ a \cdot \frac{b}{c}=\frac{a}{1} \cdot \frac{b}{c}\)

\(\ 4 \cdot \frac{2}{3}=\frac{4}{1} \cdot \frac{2}{3}=\frac{8}{3}\)

Often when multiplying a whole number and a fraction, the resulting product will be an improper fraction. It is often desirable to write improper fractions as a mixed number for the final answer. You can simplify the fraction before or after rewriting it as a mixed number. See the examples below.

\(\ 7 \cdot \frac{3}{5}=4 \frac{1}{5}\)

\(\ 4 \cdot \frac{3}{4}=3\)

\(\ 3 \cdot \frac{5}{6}\) Multiply. Simplify the answer and write it as a mixed number.

• \(\ 1 \frac{1}{7}\)
• \(\ 2 \frac{1}{2}\)
• \(\ \frac{5}{2}\)
• \(\ \frac{8}{6}\)
• Incorrect. You may have added the numerators and added the denominators to get \(\ \frac{8}{7}\), which is the mixed number \(\ 1 \frac{1}{7}\). Make sure you multiply numerators and multiply denominators. Multiplying the two numbers gives you \(\ \frac{15}{6}\), and since \(\ 15 \div 6=2 \mathrm{R} 3\), the mixed number is \(\ 2 \frac{3}{6}\). The fractional part simplifies to \(\ \frac{1}{2}\). The correct answer is \(\ 2 \frac{1}{2}\).
• Correct. Multiplying the two numbers gives \(\ \frac{15}{6}\), and since \(\ 15 \div 6=2 \mathrm{R} 3\), the mixed number is \(\ 2 \frac{3}{6}\). The fractional part simplifies to \(\ \frac{1}{2}\).
• Incorrect. Multiplying the numerators and multiplying the denominators results in the improper fraction \(\ \frac{5}{2}\), but you need to express this as a mixed number. The correct answer is \(\ 2 \frac{1}{2}\).
• Incorrect. You may have added numerators and placed it over the denominator of 6. Make sure you multiply numerators and multiply denominators. Multiplying the two numbers gives \(\ \frac{15}{6}\), and since \(\ 15 \div 6=2 \mathrm{R} 3\), the mixed number is \(\ 2 \frac{3}{6}\). The fractional part simplifies to \(\ \frac{1}{2}\). The correct answer is \(\ 2 \frac{1}{2}\).

Multiplying Mixed Numbers

If you want to multiply two mixed numbers, or a fraction and a mixed number, you can again rewrite any mixed number as an improper fraction.

So, to multiply two mixed numbers, rewrite each as an improper fraction and then multiply as usual. Multiply numerators and multiply denominators and simplify. And, as before, when simplifying, if the answer comes out as an improper fraction, then convert the answer to a mixed number.

\(\ 2 \frac{1}{5} \cdot 4 \frac{1}{2}=9 \frac{9}{10}\)

\(\ \frac{1}{2} \cdot 3 \frac{1}{3}=1 \frac{2}{3}\)

As you saw earlier, sometimes it’s helpful to look for common factors in the numerator and denominator before you simplify the products.

\(\ 1 \frac{3}{5} \cdot 2 \frac{1}{4}=3 \frac{3}{5}\)

In the last example, the same answer would be found if you multiplied numerators and multiplied denominators without removing the common factor. However, you would get \(\ \frac{72}{20}\), and then you would need to simplify more to get your final answer.

\(\ 1 \frac{3}{5} \cdot 3 \frac{1}{3}\)

• \(\ \frac{80}{15}\)
• \(\ 5 \frac{5}{15}\)
• \(\ 4 \frac{14}{15}\)
• \(\ 5 \frac{1}{3}\)
• Incorrect. You probably wrote both mixed numbers as improper fractions correctly. You probably also correctly multiplied numerators and denominators. However, this improper fraction still needs to be rewritten as a mixed number and simplified. Dividing \(\ 80 \div 15=5\) with a remainder of 5 or \(\ 5 \frac{5}{15}\), then simplifying the fractional part, the correct answer is \(\ 5 \frac{1}{3}\).
• Incorrect. You probably wrote both mixed numbers as improper fractions correctly. You probably also correctly multiplied numerators and denominators, and wrote the answer as a mixed number. However, the mixed number is not in lowest terms. \(\ \frac{5}{15}\) can be simplified to \(\ \frac{1}{3}\) by dividing numerator and denominator by the common factor 5. The correct answer is \(\ 5 \frac{1}{3}\).
• Incorrect. This is the result of adding the two numbers. To multiply, rewrite each mixed number as an improper fraction: \(\ 1 \frac{3}{5}=\frac{8}{5}\) and \(\ 3 \frac{1}{3}=\frac{10}{3}\). Next, multiply numerators and multiply denominators: \(\ \frac{8}{5} \cdot \frac{10}{3}=\frac{80}{15}\). Then, write the resulting improper fraction as a mixed number: \(\ \frac{80}{15}=5 \frac{5}{15}\). Finally, simplify the fractional part by dividing both numerator and denominator by the common factor, 5. The correct answer is \(\ 5 \frac{1}{3}\).
• Correct. First, rewrite each mixed number as an improper fraction: \(\ 1 \frac{3}{5}=\frac{8}{5}\) and \(\ 3 \frac{1}{3}=\frac{10}{3}\). Next, multiply numerators and multiply denominators: \(\ \frac{8}{5} \cdot \frac{10}{3}=\frac{80}{15}\). Then write as a mixed fraction \(\ \frac{80}{15}=5 \frac{5}{15}\). Finally, simplify the fractional part by dividing both numerator and denominator by the common factor 5.

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Activities to Teach Students to Multiply Fractions by Whole Numbers Using Models

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Multiplying fractions with whole numbers is one of the most important skills that students need to learn in math. However, it can be tricky for students to visualize how to do this, especially when it involves fractions. Luckily, there are a variety of activities that teachers can use to help students understand and learn how to multiply fractions by whole numbers using models.

1. Fraction Strips

Fraction strips are a great tool that can be used to help students visualize the multiplication of fractions with whole numbers. To do this activity, students are given a set of fraction strips. They are then asked to multiply a whole number by a fraction. For example, if the question is 3 x 1/2, students will use three whole fraction strips to represent the whole number, and then use one-half of a fraction strip to represent the fraction. From there, they can count the number of shaded strips to get their final answer. 2. Area Models

Area models are another helpful tool that can be used to teach students how to multiply fractions by whole numbers. An area model is a rectangular grid that is divided into smaller squares. The whole number is represented by the number of rows in the grid, while the fraction is represented by the number of squares shaded within each row. Students can then count the number of shaded squares to get their final answer.

3. Tape Diagrams

Tape diagrams, also known as bar models, are another visual representation that can be used to help students understand how to multiply fractions with whole numbers. In a tape diagram, students use bars to represent the whole number and fractions. For example, if the question is 5 x 3/4, students would draw five bars and then divide each bar into four equal parts. They would then shade in three of those parts to represent the fraction. From there, they can count the total number of shaded parts to find their final answer.

4. Real-world Examples

It can be helpful to use real-world examples to teach students how to multiply fractions with whole numbers. For example, if you are baking a cake and need to multiply the recipe by 1/2, you could ask students to figure out how much flour, sugar, or eggs they would need if they were using the recipe to make half as many cakes. This helps students see the purpose behind learning how to multiply fractions with whole numbers and can make the concept more tangible.

5. Interactive Games There are many interactive games and online activities that can be used to help students practice multiplying fractions with whole numbers using models. For example, the website Math Playground has a variety of games where students can practice visualizing and multiplying fractions with whole numbers, including games like “Fraction Fling” and “Fraction Palooza.” These games are engaging and can help students practice the skill in a fun and interactive way.

In conclusion, teaching students to multiply fractions with whole numbers using models can be challenging, but with the help of hands-on activities and visual aids, it can become a much easier concept to understand. Utilizing fraction strips, area models, tape diagrams, real-world examples, and interactive games can go a long way in deepening students’ understanding and solidifying their grasp of the concept.

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