# How to multiply a fraction by another fraction

• Multiplying Fractions and Mixed Numbers
• Learning Objective(s)
• ·         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.

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 T. sugar

cups melted butter            tsp. vanilla

Suppose you only want to make 2 crumb piecrusts. You can multiply all the ingredients by , 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.

1. When you multiply a fraction by a fraction, you are finding a “fraction of a fraction.” Suppose you have  of a candy bar and you want to find  of the :
2.  By dividing each fourth in half, you can divide the candy bar into eighths.
3.
4. Then, choose half of those to get .
5.
6. In both of the above cases, to find the answer, you can multiply the numerators together and the denominators together.
 Multiplying Two Fractions Example: Multiplying More Than Two Fractions Example:
 Example Problem Multiply. Multiply the numerators and multiply the denominators. Simplify, if possible. This fraction is already in lowest terms. Answer

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

 Example Problem Multiply. Simplify the answer. Multiply the numerators and multiply the denominators. Simplify, if possible. Simplify by dividing the numerator and denominator by the common factor 2. Answer

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

 Example Problem Multiply. Simplify the answer. Reorder the numerators so that you can see a fraction that has a common factor. Simplify. Answer

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.

 Multiply. Simplify the answer. A) B) C) D) Show/Hide Answer A) Incorrect.  is an equivalent fraction to the correct answer , but it is not in lowest terms. You must divide numerator and denominator by the common factor 3. The correct answer is . B) Incorrect. You may have added numerators (3 + 1) and added denominators (4 + 3) instead of multiplying. The correct answer is . C) Correct. One way to find this answer is to multiply numerators and denominators , then simplify: . D) 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 .

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: , , and .

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 Example:

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 as a mixed number. See the examples below.

 Example Problem Multiply. Simplify the answer and write as a mixed number. Rewrite 7 as the improper fraction . Multiply the numerators and multiply the denominators. Rewrite as a mixed number. . Answer
 Example Problem Multiply. Simplify the answer and write as a mixed number. Rewrite 4 as the improper fraction . Multiply the numerators and multiply the denominators. Simplify. Answer 3
 Multiply. Simplify the answer and write it as a mixed number. A) B) C) D) Show/Hide Answer A) Incorrect. You may have added numerators and added denominators, to get , which is the mixed number . Make sure you multiply numerators and multiply denominators. Multiplying the two numbers gives , and since 15 ÷ 6 = 2R3, the mixed number is . The fractional part simplifies to . The correct answer is . B) Correct. Multiplying the two numbers gives , and since 15 ÷ 6 = 2R3, the mixed number is . The fractional part simplifies to . C) Incorrect. Multiplying the numerators and multiplying the denominators results in the improper fraction , but you need to express this as a mixed number. The correct answer is . D) 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 , and since 15 ÷ 6 = 2R3, the mixed number is . The fractional part simplifies to . The correct answer is .

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.

 Example Problem Multiply. Simplify the answer and write as a mixed number. Change  to an improper fraction. 5 • 2 + 1 = 11, and the denominator is 5. Change  to an improper fraction. 2 • 4 + 1 = 9, and the denominator is 2. Rewrite the multiplication problem, using the improper fractions. Multiply numerators and multiply denominators. Write as a mixed number. Answer
 Example Problem Multiply. Simplify the answer and write as a mixed number. Change  to an improper fraction. 3 • 3 + 1 = 10, and the denominator is 3. Rewrite the multiplication problem, using the improper fraction in place of the mixed number. Multiply numerators and multiply denominators. Rewrite as a mixed number. with a remainder of 4. Simplify the fractional part to lowest terms by dividing the numerator and denominator by the common factor 2. Answer

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

 Example Problem Multiply. Simplify the answer and write as a mixed number. Change  to an improper fraction. 5 • 1 + 3 = 8, and the denominator is 5. Change  to an improper fraction. 4 • 2 + 1 = 9, and the denominator is 4. Rewrite the multiplication problem using the improper fractions. Reorder the numerators so that you can see a fraction that has a common factor. Simplify. Multiply. Write as a mixed fraction. Answer

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 , and then you would need to simplify more to get your final answer.

 Multiply. Simplify the answer and write as a mixed number. A) B) C) D) Show/Hide Answer A) 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 ÷15 = 5 with a remainder of 5 or , then simplifying the fractional part, the correct answer is . B) 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.  can be simplified to  by dividing numerator and denominator by the common factor 5. The correct answer is . C) Incorrect. This is the result of adding the two numbers. To multiply, rewrite each mixed number as an improper fraction:  and . Next, multiply numerators and multiply denominators: . Then, write the resulting improper fraction as a mixed number: . Finally, simplify the fractional part by dividing both numerator and denominator by the common factor, 5. The correct answer is . D) Correct. First, rewrite each mixed number as an improper fraction:  and . Next, multiply numerators and multiply denominators: . Then write as a mixed fraction  . Finally, simplify the fractional part by dividing both numerator and denominator by the common factor 5.

Solving Problems by Multiplying Fractions and Mixed Numbers

Now that you know how to multiply a fraction by another fraction, by a whole number, or by a mixed number, you can use this knowledge to solve problems that involve multiplication and fractional amounts. For example, you can now calculate the ingredients needed for the 2 crumb piecrusts.

 Example Problem 5 cups graham crackers        8 T. sugar  cups melted butter            tsp. vanilla The recipe at left makes 4 piecrusts. Find the ingredients needed to make only 2 piecrusts. Since the recipe is for 4 piecrusts, you can multiply each of the ingredients by  to find the measurements for just 2 piecrusts. cups of graham crackers are needed. 5 cups graham crackers: Since the result is an improper fraction, rewrite  as the improper fraction . 4 T. sugar is needed. 8 T. sugar: This is another example of a whole number multiplied by a fraction. cup melted butter is needed. cups melted butter: You need to multiply a mixed number by a fraction. So, first rewrite as the improper fraction :  2 • 1 + 1, and the denominator is 2. Then, rewrite the multiplication problem, using the improper fraction in place of the mixed number. Multiply. tsp. vanilla is needed. tsp. vanilla: Here, you multiply a fraction by a fraction. Answer The ingredients needed for 2 pie crusts are: cups graham crackers 4 T. sugar  cup melted butter  tsp. vanilla

## Multiplying Fractions

To multiply fractions is as easy as following the 3 suggested steps below. It’s understood that no fraction can have a denominator of color{red}0 because it will be an undefined term.

### Steps in Multiplying Fractions

Given two fractions with nonzero denominators :

Step 1: Multiply the numerators.

• This will be the numerator of the “new” fraction.

Step 2: Multiply the denominators.

• This will be the denominator of the “new” fraction.

Step 3: Simplify the resulting fraction by reducing it to the lowest term, if needed.

Before we go over some examples, there are other ways to mean multiplication.

• Dot symbol as a multiplication operator

• Parenthesis as a multiplication operator

### Examples of How to Multiply Fractions

Example 1: Multiply.

Multiply the numerators of the fractions.

Similarly, multiply the denominators together.

The resulting fraction after multiplication is already in its reduced form since the Greatest Common Divisor of the numerator and denominator is color{blue}+1. This becomes our final answer!

Example 2: Multiply.

Step 1: Multiply the top numbers.

Step 2: Multiply the bottom numbers.

Step 3: Simplify the answer by reducing to the lowest term.

Divide the top and bottom by its Greatest Common Factor (GCF) which is 10.

Example 3: Multiply.

• You may encounter a problem where you will be asked to multiply three fractions.

## Multiplying fractions

The purpose of this series of lessons is to develop understanding of the multiplication of fractions.

Specific Learning Outcomes

• Record in words, the actions and results of finding a fraction of a fraction.
• Record and respond to written multiplication equations.
• Use arrays to model and solve multiplication equations that involve subdividing the unit.
• Notice, explain and generalise what is happening to the numbers in a multiplication algorithm.
• Pose and solve their own fraction multiplication problems.
• Understand and use number properties when multiplying fractions.
• Explore and demonstrate the reciprocal relationship between multiplication and division.

Description of Mathematics

This series of lessons builds upon students’ understanding and use of equivalent fractions as they solve fraction addition and subtraction problems.

As in earlier units of work, an emphasis is given to having students model operations with fractions, using a range of materials and record using words and symbols.

Having a sound knowledge of basic multiplication and division facts is fundamental to the students’ success in working with and understanding equivalent fractions, and, in this unit, multiplying fractions.

There are three key understandings that underpin multiplication of fractions. The first is that multiplying two fractions involves finding the fraction of another fraction. For example, 1/2 x 1/4 is interpreted as 1/2 of 1/4. The second is that when two fractions less than one are multiplied, the product is always less than either factor.

In multiplying whole numbers, students expect a product that is larger than either factor. Multiplying fractions requires a conceptual shift for the students who must clearly understand they are finding a part of a part. Thirdly, by understanding the commutative property, students can make problems simpler by changing the order of the factors.

In multiplying fractions students come to recognise that the language of ‘times’ and ‘of’ are interchangeable. The use an array model to visualise and solve problems involving finding a fraction of a fraction, by breaking an area into parts, horizontally and vertically, scaffolds the move from a whole number understanding to a fractional one.

Beginning with problems that involve working with unit parts without subdivision (eg. 1/3 of 3/8), establishes the conceptual understanding of the multiplication operation with fractions. Once this is clearly understood, working with unit parts that involve subdivision (eg. 3/4 of 2/3), focuses the students on recording and calculations as they explore the relationships between the numbers.

Using realistic contexts for finding fractions of fractions is important. Having students respond to these, and create context of their own, will help them recognise the practical application of fraction multiplication.

These ideas are presented in five sessions however, as they include complex concepts that are fundamental to a student’s success with fractions, these sessions can be extended over a longer period of time.

Whilst the games are introduced and used within sessions to consolidate ideas, they can also be added to the class or group independent activities, or be sent home for family challenges and enjoyment.

Required Resource Materials

Session 1

SLOs:

• Review finding a fraction of a whole number.
• Recognise that a fraction of a fraction results in a smaller part.
• Record in words the actions and results of finding a fraction of a fraction.
• Solve problems involving finding a fraction of a fraction, using a regional model.

Activity 1

1. Begin this session by posing these three problems. Have the students discuss in pairs the solutions to each: There are 33 children in Nina’s class. She was asked how many would be in her team if it included 1/3 of the class. She said 12. Is she right? 15 students in Jo’s class were sitting on the mat. This was 3/5 of the class. How many students are in Jo’s class? 18 students in Roly’s class are on the mat. 2/5 are yet to come into the classroom. How many are outside? Have the students pair share their solutions.
2. On the class chart write: Finding fractions of whole numbers. Nina’s class: 1/3 of 33 is: Jo’s class: 3/5 of something is 15 so 5/5 is: Roly’s class: 3/5 of something is 18 so 2/5 is:

Have individual students come and record their results on the class chart and explain their solutions on behalf of their group.

Remind the students that they have been finding fractions of whole numbers.

Activity 2

1. Ask: What if you find a fraction of a fraction. Will the result be larger or smaller than both fractions? Discuss. Record ideas on the class chart.
2. Provide pairs of students with strips of paper and coloured pens. Pose the problem: Can you use the materials to show one half of one half. Have students share the results.

Pose another problem: Use the materials to show three quarters of one half.

Have students share what they did and discuss the results.

3. Distribute Fraction Strips (Material Master 7-7) to the students to supplement the paper strips. Give students time to become familiar with the fraction strips reading the unit fractions shown and talking about the subdivisions that they see. Pose the question: What is one third of one half? Have students find one half on the fraction wall and look for the subdivision of one half into thirds, identifying that one sixth is one third of one half.

Distribute and discuss Attachment 1. Explain that they can refer to the fraction wall or use the paper strips and scissors to complete it. Highlight the importance of their writing realistic stories for each.

4. Have students pair share and check their results. Review the focus of the learning so far: We have been finding fractions of whole numbers and finding fractions of fractions of an area.

Activity 3

1. Conclude the session by having a number of students read aloud (in random order) word problems they have written. The other students work in pairs to image paper strips or the fraction wall, and respond.
2. Pose aloud word problems that build on these examples using repeated addition to reach solutions: One third of one half is one sixth, so what is two thirds of one half? (2/6 or 1/3) One fifth of one half is one tenth, so what is three fifths of one half? (3/10) One third of one third is one ninth, so what is two thirds of one third? (2/9) One quarter of one half is one eighth, so what is three quarters of one half? (3/8).
3. Conclude that a fraction of a fraction results is a smaller part.

Session 2

SLOs:

• Understand that the language ‘of ‘ and ‘times’ is interchangeable.
• Record and respond to written multiplication equations.
• Use arrays to model multiplication equations.
• Use arrays to model and solve multiplication equations that involve subdividing the unit.

Activity 1

Refer to problem 1 from session 1.
There are 33 children in Nina’s class. She was asked how many would be in her team if it was 1/3 of the class. She said 12. Is she right?
Finding fractions of whole numbers.
Nina’s class: 1/3 of 33 is:

Pose: Nina recorded the equation for this problem as 33 ÷ 1/3 = 11. Is she correct? Why? Why not? (33 ÷ 1/3 = 99 because there are 99 thirds in 33 wholes)

Write 1/3 of 33 = 11 and 1/3 x 33 = 11.

Discuss.

Activity 2

Write on the class chart
1/4 x 2 = ?
2/3 x 12 = ?

Have students work in pairs to solve the problems, and pair share their results including any pictures or diagrams used.

Activity 3