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.

 

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 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.

 

Multiplying Fractions

 

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 :

 

U02_L2_T1_text_image5

 

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

                                    U02_L2_T1_text_image6

Then, choose half of those to get .

                                    U02_L2_T1_text_image7

 

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

 

 

Often, a problem indicates that multiplication by a fraction is needed by using phrases like “half of,” “a third of,” or “ of.”

 

Example

Problem

The cost of a vacation is $4,500 and you are required to pay  of that amount when you reserve the trip. How much will you have to pay when you reserve the trip?

 

You need to find  of 4,500. “Of” tells you to multiply.

 

Change 4,500 to an improper fraction by rewriting it with 1 as the denominator.

 

 

Divide.

 

900

Simplify.

 

Answer

You will need to pay $900 when you reserve the trip.

 

 

Example

Problem

U02_L2_T1_text_image8

The pie chart at left represents the fractional part of daily activities.

Given a 24-hour day, how many hours are spent sleeping? Attending school? Eating? Use the pie chart to determine your answers.

 

Sleeping is  of the pie, so the number of hours spent sleeping is  of 24.

 

Rewrite 24 as an improper fraction with a denominator of 1.

 

8 hours sleeping

Multiply numerators and multiply denominators. Simplify  to 8.

 

Attending school is  of the pie, so the number of hours spent attending school is  of 24.

 

Rewrite 24 as an improper fraction with a denominator of 1.

 

4 hours attending school

Multiply numerators and multiply denominators. Simplify  to 4.

 

Eating is  of the pie, so the number of hours spent eating is  of 24.

 

Rewrite 24 as an improper fraction with a denominator of 1.

 

2 hours spent eating

Multiply numerators and multiply denominators. Simplify  to 2.

Answer

Hours spent:

sleeping: 8 hours

attending school: 4 hours

eating: 2 hours

 

 

Neil bought a dozen (12) eggs. He used  of the eggs for breakfast. How many eggs are left?

 

A)                8

B)                4

C)               9

D)               3

 

Show/Hide Answer

A) 8

Correct.  of 12 is 4 (), so he used 4 of the eggs. Because 12 – 4 = 8, there are 8 eggs left.

 

B) 4

Incorrect.  of 12 is 4, but that gives how many eggs Neil used, not how many he had left. You need to subtract 4 from 12 to find the number of remaining eggs. The correct answer is 8.

 

C) 9

Incorrect. You may have incorrectly found  of 12 to be 3.  of 12 is 4, and then 12 – 4 is 8. The correct answer is 8.

 

D) 3

Incorrect. You need to find  of 12, which is 4. Then subtract 4 from 12 to get 8 remaining eggs.

 

 

Summary

 

You multiply two fractions by multiplying the numerators and multiplying the denominators. Often the resulting product will not be in lowest terms, so you must also simplify. If one or both fractions are whole numbers or mixed numbers, first rewrite each as an improper fraction. Then multiply as usual, and simplify.