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Multiplication of Fractions and Integers

If you are multiplying one fraction by another fraction, you simply multiply the numerators together and the denominators together.

Rule

Multiplying Fractions

\frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}

Note! When you multiply an integer or fraction by a proper fraction (which is less than $1$ ), you’ll find that the answer you get is actually smaller than the integer or fraction you started with. This is different than normal multiplication with whole numbers where the answer is larger.

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Example 1

Calculate $\frac{2}{3} \times \frac{3}{4}$

Here you don’t have to worry about finding a common denominator:

\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}

At the end, you simplify the factorization: $6 = 2 \times 3$ and $12 = 2 \times 2 \times 3$ , where you cancel a 2 and a 3.

If you want to multiply an integer by a fraction, just multiply the number by the numerator in the fraction. Then, it’s pretty clever to remember that all integers can also be written as fractions: $b = \frac{b}{1}$ .

Rule

Integers and Fractions Multiplied

a \times \frac{b}{c} = \frac{a}{1} \times \frac{b}{c} = \frac{a b}{c}

Example 2

Compute $3 \times \frac{2}{4}$

This can be done in two ways. The first way is to rewrite $3 = \frac{3}{1}$ . Then you get

\frac{3}{1} \times \frac{2}{4} = \frac{3 \times 2}{1 \times 4} = \frac{6}{4} = \frac{3}{2},

which is the same as the alternative way:

3 \times \frac{2}{4} = \frac{3 \times 2}{4} = \frac{6}{4} = \frac{3}{2} .

Both methods work. Just use the one you feel the most comfortable with!

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Activities 4

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