4.2 Multiply and Divide Fractions

Multiplying and Dividing Fractions

Multiplying and dividing fractions comes up constantly in math, so getting comfortable with these operations now will pay off in algebra and beyond. The core idea is simple: multiplying fractions means multiplying straight across, and dividing fractions means flipping the second fraction and then multiplying.

Simplifying Fractions

Before you multiply or divide, you should know how to simplify (or "reduce") a fraction to its lowest terms. A simplified fraction is easier to work with and easier to compare to other fractions.

To simplify a fraction:

1. Find the greatest common factor (GCF) of the numerator and denominator. The GCF is the largest number that divides evenly into both.
2. Divide the numerator and denominator by that GCF.

For example, to simplify $\frac{12}{18}$:

• The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The GCF is 6.
• $\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$

You know a fraction is fully simplified when the numerator and denominator share no common factor other than 1. So $\frac{2}{3}$ is in simplest form because 2 and 3 share no factors.

One more thing: the fraction bar itself represents division. Writing $\frac{12}{18}$ is the same as writing $12 \div 18$.

Multiplying Fractions

Multiplying fractions is the most straightforward fraction operation. Here's the formula:

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

Multiply the numerators together, multiply the denominators together, then simplify.

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

Shortcut: Cross cancellation. Before you multiply, check if any numerator shares a common factor with any denominator (even diagonally). Cancel those common factors first, and you'll have less simplifying to do at the end. In the example above, the 3 in $\frac{2}{3}$ and the 3 in $\frac{3}{4}$ cancel to 1, giving you $\frac{2}{1} \times \frac{1}{4} = \frac{2}{4} = \frac{1}{2}$ right away.

Multiplying with mixed numbers. You can't multiply mixed numbers directly. Convert them to improper fractions first:

1. Multiply the whole number by the denominator.
2. Add that result to the numerator.
3. Place the sum over the original denominator.

Example: Convert $2\frac{1}{3}$ to an improper fraction:

$2 \times 3 = 6$, then $6 + 1 = 7$, so $2\frac{1}{3} = \frac{7}{3}$

Now you can multiply normally: $\frac{7}{3} \times \frac{4}{5} = \frac{28}{15}$

Since 28 and 15 share no common factors, $\frac{28}{15}$ is already simplified. You could convert it back to a mixed number: $1\frac{13}{15}$.

Reciprocals

A reciprocal (also called the multiplicative inverse) is what you get when you flip a fraction's numerator and denominator.

• The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$.
• The reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$.
• The reciprocal of a whole number like 7 (which is $\frac{7}{1}$) is $\frac{1}{7}$.

The key property: any fraction multiplied by its reciprocal equals 1.

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

This property is exactly why reciprocals let you turn division into multiplication.

Dividing Fractions

To divide fractions, multiply the first fraction by the reciprocal of the second. This is sometimes called "keep, change, flip": keep the first fraction, change division to multiplication, flip the second fraction.

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

Example: $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2}$

Dividing mixed numbers follows the same process, with one extra step at the start:

1. Convert both mixed numbers to improper fractions.
2. Multiply the first fraction by the reciprocal of the second.
3. Simplify the result.

Example: $1\frac{1}{3} \div 2\frac{1}{4}$

• Convert: $1\frac{1}{3} = \frac{4}{3}$ and $2\frac{1}{4} = \frac{9}{4}$
• Flip and multiply: $\frac{4}{3} \times \frac{4}{9} = \frac{16}{27}$
• 16 and 27 share no common factors, so $\frac{16}{27}$ is the final answer.

Special Types of Fractions

Two types worth knowing:

• A unit fraction has a numerator of 1, like $\frac{1}{2}$, $\frac{1}{3}$, or $\frac{1}{4}$. Every unit fraction is the reciprocal of a whole number ($\frac{1}{4}$ is the reciprocal of 4).
• A complex fraction has a fraction in its numerator, its denominator, or both. Something like $\frac{\frac{1}{2}}{\frac{3}{4}}$. To simplify a complex fraction, treat the fraction bar as division: $\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}$.