Multiplying and Dividing Fractions
Multiplying and Dividing Fractions
Multiplying and dividing fractions follow rules that are, in some ways, simpler than addition and subtraction. To multiply, you simply multiply the numerators together and the denominators together — no common denominator needed.
Division of fractions introduces the powerful "multiply by the reciprocal" technique: to divide by a fraction, flip it and multiply. This rule is not a trick — it follows directly from the definition of division as the inverse of multiplication.
In this lesson you will practice both operations with proper fractions, improper fractions, and mixed numbers, and you will see how cross-cancellation can simplify your work before you even start multiplying.
Multiplication
$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$
Simplify by cross-canceling before multiplying.
$\dfrac{3}{8} \times \dfrac{4}{9}$
Cross-cancel: $3$ and $9$ (÷3); $4$ and $8$ (÷4).
$$\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$$
$2\dfrac{1}{3} \times 1\dfrac{1}{2}$
Convert: $\dfrac{7}{3} \times \dfrac{3}{2}$. Cancel the 3s: $\dfrac{7}{2} = 3\dfrac{1}{2}$.
Division — "Keep, Change, Flip"
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$
$\dfrac{5}{6} \div \dfrac{2}{3}$
$$\frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4} = 1\frac{1}{4}$$
$3\dfrac{3}{4} \div 1\dfrac{1}{2}$
$\dfrac{15}{4} \div \dfrac{3}{2} = \dfrac{15}{4} \times \dfrac{2}{3} = \dfrac{5}{2} = 2\dfrac{1}{2}$
Practice Problems
Show Answer Key
1. $\dfrac{10}{21}$
2. $\dfrac{1}{6}$
3. $\dfrac{2}{3}$
4. $4$
5. $\dfrac{1}{2}$
6. $3\dfrac{1}{2}$
7. $\dfrac{3}{4}$
8. $4$
9. $\dfrac{1}{2}$
10. $3\dfrac{3}{5}$
11. $\dfrac{2}{3}$
12. $6$
13. $\dfrac{2}{5}$
14. $7\dfrac{1}{2}$
15. $3\dfrac{2}{3}$