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Multiplying a Fraction by a Whole Number

Suppose we want to multiply the fraction by the number 3. We can think of multiplying the fraction by 3 as taking “3 copies of ”. We can model this multiplication by drawing a rectangle, dividing it into 8 equal pieces, and then shading 3 of them.

The shaded pieces represent of the entire rectangle. Notice that this answer can be found by multiplying the numerator of the fraction by 3 and retaining the denominator.

 

Key Idea

To multiply a fraction by a whole number, multiply the numerator in the fraction by the whole number and place this product over the denominator.

 

Example 1

Example 2

Example 3

 

Why Does This Procedure Work?

The multiplication of two whole numbers can be thought of as repeated addition. For example,

2 × 3 = 3 + 3 3 × 7 = 7 + 7 + 7 5 × 9 = 9 + 9 + 9 + 9 + 9

The multiplication of a fraction by a whole number can be thought of in the same way.

When we add fractions with like denominators, we add them by adding the numerators and retaining the denominator. The repeated additions involve adding fractions with like denominators since the fractions are the same.

The last step of the process in each example above shows that multiplying a fraction by a whole number is accomplished by multiplying the whole number times the numerator and keeping the same denominator.

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