Lesson 4.2: Multiply and Divide Fractions

New Concept

Ready to master fraction operations? You'll learn how to multiply fractions by combining numerators and denominators, and divide them using the clever trick of reciprocals. Simplifying will be the key to getting the final, correct answer.

What’s next

Next, you'll put this into practice with interactive examples of multiplication and division, followed by a series of challenge problems to master the concepts.

Property

A fraction is considered simplified if there are no common factors, other than 1, in the numerator and denominator. We can use the Equivalent Fractions Property in reverse to simplify fractions. If $a$, $b$, $c$ are numbers where $b \neq 0$, $c \neq 0$, then $\frac{a}{b} = \frac{a \cdot c}{b \cdot c}$ and $\frac{a \cdot c}{b \cdot c} = \frac{a}{b}$. Anytime we have a common factor in the numerator and denominator, it can be removed.

Examples

• To simplify $\frac{9}{21}$, we find the common factor of 3. Factoring gives $\frac{3 \cdot 3}{7 \cdot 3}$. Removing the common factor leaves $\frac{3}{7}$.
• Simplify $-\frac{16}{40}$. The greatest common factor of 16 and 40 is 8. So, we rewrite the fraction as $-\frac{2 \cdot 8}{5 \cdot 8}$. After removing the 8s, we get $-\frac{2}{5}$.
• Simplify $\frac{4a}{12ab}$. We factor the top and bottom to get $\frac{4 \cdot a}{3 \cdot 4 \cdot a \cdot b}$. Removing the common factors $4$ and $a$ gives $\frac{1}{3b}$.

Explanation

Simplifying a fraction means to reduce it to its 'simplest' form. You do this by finding any factors that are common to both the numerator and the denominator and then removing them. It's like canceling out matching pairs from the top and bottom.

Property

If $a$, $b$, $c$, and $d$ are numbers where $b \neq 0$ and $d \neq 0$, then

To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.

Examples

• To multiply $\frac{2}{3} \cdot \frac{4}{5}$, multiply the numerators ($2 \cdot 4 = 8$) and the denominators ($3 \cdot 5 = 15$). The result is $\frac{8}{15}$.
• Multiply $-\frac{4}{7} \cdot \frac{14}{16}$. The product will be negative. We can simplify before multiplying: $-\frac{4}{7} \cdot \frac{2 \cdot 7}{4 \cdot 4} = -\frac{1}{1} \cdot \frac{2}{4} = -\frac{2}{4}$, which simplifies to $-\frac{1}{2}$.
• To multiply $8 \cdot \frac{3}{4}$, first write 8 as $\frac{8}{1}$. Then multiply: $\frac{8}{1} \cdot \frac{3}{4} = \frac{24}{4}$. Simplifying this fraction gives 6.

Explanation

Multiplying fractions is straightforward: multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. Remember to simplify the resulting fraction by canceling any common factors for the final answer.

Property

The reciprocal of the fraction $\frac{a}{b}$ is $\frac{b}{a}$, where $a \neq 0$ and $b \neq 0$. A number and its reciprocal have a product of 1. To find the reciprocal of a fraction, we invert the fraction. Reciprocals must have the same sign. The number zero does not have a reciprocal.

Examples

• The reciprocal of $\frac{5}{8}$ is $\frac{8}{5}$. Check: $\frac{5}{8} \cdot \frac{8}{5} = \frac{40}{40} = 1$.
• The reciprocal of $-9$ is $-\frac{1}{9}$. First, write $-9$ as $-\frac{9}{1}$, then invert it. Check: $-9 \cdot (-\frac{1}{9}) = 1$.
• The reciprocal of $-\frac{1}{4}$ is $-4$. Check: $-\frac{1}{4} \cdot (-4) = 1$.

Explanation

Finding a reciprocal is like flipping a fraction upside down. The numerator becomes the denominator and the denominator becomes the numerator. When you multiply any number by its reciprocal, the result is always 1. Keep the sign the same!

Property

If $a$, $b$, $c$, and $d$ are numbers where $b \neq 0$, $c \neq 0$, and $d \neq 0$, then

To divide fractions, multiply the first fraction by the reciprocal of the second.

Examples

• To divide $\frac{1}{3} \div \frac{1}{9}$, we multiply $\frac{1}{3}$ by the reciprocal of $\frac{1}{9}$, which is $\frac{9}{1}$. So, $\frac{1}{3} \cdot \frac{9}{1} = \frac{9}{3} = 3$.
• Let's divide $\frac{5}{6} \div \frac{10}{3}$. We keep $\frac{5}{6}$, change to multiplication, and flip $\frac{10}{3}$ to $\frac{3}{10}$. This gives $\frac{5}{6} \cdot \frac{3}{10} = \frac{15}{60}$, which simplifies to $\frac{1}{4}$.
• For $-\frac{2}{5} \div \frac{4}{15}$, we calculate $-\frac{2}{5} \cdot \frac{15}{4}$. The result is negative. $\frac{2 \cdot 15}{5 \cdot 4} = \frac{30}{20}$. This simplifies to $-\frac{3}{2}$.

Explanation

To divide fractions, use the 'Keep, Change, Flip' method. Keep the first fraction the same, change the division sign to multiplication, and flip the second fraction to its reciprocal. Then, simply multiply the fractions as usual.