Multiply and Divide Fractions

Simplify fractions

In working with equivalent fractions, we saw that there are many ways to write a fraction that represents the same value. How do we know which one to use? Often we use the fraction that is in simplified form.

For example, $\tfrac{2}{3}$ is simplified because $2$ and $3$ have no common factors, but $\tfrac{10}{15}$ is not simplified because $5$ is a common factor of $10$ and $15$.

Simplifying a fraction is often called reducing it. We use the Equivalent Fractions Property in reverse to simplify: rewriting both forms together,

Since $c$ is a common factor of the numerator and denominator, it can be removed.

Example. Simplify $\tfrac{10}{15}$.

Notice $5$ is a factor of both $10$ and $15$. Factor: $\tfrac{10}{15} = \tfrac{2 \cdot 5}{3 \cdot 5}$. Remove the common factor of $5$:

To simplify a negative fraction, use the same process and keep the negative sign.

Example. Simplify $-\tfrac{18}{24}$.

Both $18$ and $24$ have a factor of $6$: $-\tfrac{18}{24} = -\tfrac{3 \cdot 6}{4 \cdot 6}$. Remove the common factor:

When we simplify an improper fraction, there is no need to convert it to a mixed number first.

Example. Simplify $-\tfrac{56}{32}$.

Both $56$ and $32$ share a factor of $8$: $-\tfrac{56}{32} = -\tfrac{7 \cdot 8}{4 \cdot 8}$. Removing the common factor gives $-\tfrac{7}{4}$.

Sometimes it isn’t easy to spot common factors of the numerator and denominator. A good approach is to factor both into primes first — using the factor tree method — then remove the factors that appear in both.

Example. Simplify $\tfrac{210}{385}$.

Factor tree the numerator and denominator into primes: $210 = 2 \cdot 3 \cdot 5 \cdot 7$ and $385 = 5 \cdot 7 \cdot 11$. Rewrite:

Remove the common factors of $5$ and $7$:

We can also simplify fractions containing variables the same way, removing a common factor from numerator and denominator.

Example. Simplify $\tfrac{5xy}{15x}$.

Rewrite showing the common factors: $\tfrac{5xy}{15x} = \tfrac{5 \cdot x \cdot y}{3 \cdot 5 \cdot x}$. Remove the common factors of $5$ and $x$:

Multiply fractions

A model can help you understand multiplying fractions. Think of $\tfrac{1}{2} \cdot \tfrac{3}{4}$ as “one-half of three-fourths.” If we shade $\tfrac{3}{4}$ of a rectangle, then heavily shade half of that shaded region, we find that $3$ out of $8$ pieces are heavily shaded — so $\tfrac{1}{2} \cdot \tfrac{3}{4} = \tfrac{3}{8}$.

Do you notice we could have gotten the same answer by multiplying the numerators together and the denominators together? $\tfrac{1 \cdot 3}{2 \cdot 4} = \tfrac{3}{8}$.

Example. Multiply, and write the answer in simplified form: $\tfrac{3}{4} \cdot \tfrac{1}{5}$.

Multiply the numerators and denominators: $\tfrac{3 \cdot 1}{4 \cdot 5} = \tfrac{3}{20}$. There are no common factors, so the fraction is already simplified.

When multiplying fractions, the properties of positive and negative numbers still apply — determine the sign of the product first.

Example. Multiply, and write the answer in simplified form: $-\tfrac{5}{8}\left(-\tfrac{2}{3}\right)$.

The signs are the same, so the product is positive. Multiplying: $\tfrac{5 \cdot 2}{8 \cdot 3} = \tfrac{10}{24}$. Rewriting to show the common factor of $2$ and removing it gives $\tfrac{5}{12}$.

It’s often faster to remove common factors before multiplying, rather than after — same result either way.

Example. Multiply, and write the answer in simplified form: $-\tfrac{14}{15} \cdot \tfrac{20}{21}$.

The product is negative (opposite signs). Since $7$ is a common factor of $14$ and $21$, and $5$ is a common factor of $20$ and $15$, remove them before multiplying:

To multiply a fraction by an integer, it helps to write the integer as a fraction — any integer $a$ can be written as $\tfrac{a}{1}$.

Example. Multiply, and write the answer in simplified form: $\tfrac{12}{5}(-20x)$.

Write $-20x$ as $\tfrac{-20x}{1}$. The product is negative. Multiplying and removing common factors of $4$ and $5$:

Find reciprocals

The fractions $\tfrac{2}{3}$ and $\tfrac{3}{2}$ are related in a special way — they look like upside-down (inverted) versions of one another, and $\tfrac{2}{3} \cdot \tfrac{3}{2} = 1$. Such pairs of numbers are called reciprocals.

To find the reciprocal, keep the same sign and invert the fraction — since a number and its reciprocal multiply to a positive $1$, they must have the same sign. The number $0$ has no reciprocal, since no number $r$ satisfies $0 \cdot r = 1$.

Example. Find the reciprocal of each number, and check that the product with its original is $1$: (a) $\tfrac{4}{9}$; (b) $-\tfrac{1}{6}$; (c) $-\tfrac{14}{5}$; (d) $7$.

(a) The reciprocal of $\tfrac{4}{9}$ is $\tfrac{9}{4}$. Check: $\tfrac{4}{9} \cdot \tfrac{9}{4} = 1$.

(b) The reciprocal of $-\tfrac{1}{6}$ is $-6$. Check: $-\tfrac{1}{6} \cdot (-6) = 1$.

(c) The reciprocal of $-\tfrac{14}{5}$ is $-\tfrac{5}{14}$. Check: $-\tfrac{14}{5} \cdot \left(-\tfrac{5}{14}\right) = 1$.

(d) Write $7$ as $\tfrac{7}{1}$; its reciprocal is $\tfrac{1}{7}$. Check: $7 \cdot \tfrac{1}{7} = 1$.

Recall from the integers chapter that opposite and absolute value behave differently from reciprocal: the opposite of a number flips its sign, the absolute value is never negative, and the reciprocal keeps the same sign but inverts the fraction.

Divide fractions

Why is $12 \div 3 = 4$? Because there are $4$ groups of $3$ in $12$. In the same way, to divide fractions we can ask how many of the divisor fit into the dividend. Using fraction tiles, lining up half- and sixth-tiles shows there are three $\tfrac{1}{6}$ tiles in $\tfrac{1}{2}$, so $\tfrac{1}{2} \div \tfrac{1}{6} = 3$.

Notice that $\tfrac{1}{2} \div \tfrac{1}{6} = 3$ and also $\tfrac{1}{2} \cdot \tfrac{6}{1} = 3$ — dividing by $\tfrac{1}{6}$ gives the same result as multiplying by its reciprocal, $\tfrac{6}{1}$. This leads to the procedure for fraction division.

Example. Divide, and write the answer in simplified form: $\tfrac{2}{5} \div \left(-\tfrac{3}{7}\right)$.

Multiply by the reciprocal of the second fraction: $\tfrac{2}{5} \cdot \left(-\tfrac{7}{3}\right)$. The product is negative:

Example. Divide, and write the answer in simplified form: $-\tfrac{3}{4} \div \left(-\tfrac{7}{8}\right)$.

Multiply by the reciprocal of the second: $-\tfrac{3}{4} \cdot \left(-\tfrac{8}{7}\right)$. Both signs are negative, so the product is positive. Rewrite to show the common factor of $4$ and remove it:

Example. Divide, and write the answer in simplified form: $\tfrac{7}{18} \div \tfrac{14}{27}$.

Multiply by the reciprocal: $\tfrac{7}{18} \cdot \tfrac{27}{14}$. Rewriting to show common factors of $7$, $9$, and $2$ and removing them:

Key terms

simplified fraction — a fraction with no common factors, other than $1$, in its numerator and denominator. fraction multiplication — to multiply two fractions, multiply the numerators and multiply the denominators, $\tfrac{a}{b} \cdot \tfrac{c}{d} = \tfrac{ac}{bd}$. reciprocal — the fraction $\tfrac{b}{a}$, obtained by inverting $\tfrac{a}{b}$ ($a \neq 0$, $b \neq 0$); a number and its reciprocal multiply to $1$. fraction division — to divide by a fraction, multiply by its reciprocal, $\tfrac{a}{b} \div \tfrac{c}{d} = \tfrac{a}{b} \cdot \tfrac{d}{c}$.

Practice

Simplify fractions

Multiply fractions

Find reciprocals

Divide fractions

This section is adapted from Prealgebra 2e, Section 4.2: Multiply and Divide Fractions by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the fraction-tile and shaded-rectangle multiplication models as prose walkthroughs and the factor-tree simplification as typeset math; omitted the Be Prepared quiz, Manipulative Mathematics callouts, the opposite/absolute value/reciprocal comparison chart’s fill-in-the-blank format, and media links; converted the practice problems (“Try Its”) into interactive exercises with instant feedback; and adapted selected end-of-section exercises into the interactive Practice block.