Divide Fractions
Dividing Fractions teaches the foundational 'keep, change, flip' method: to divide by a fraction, multiply by its reciprocal. From OpenStax Prealgebra 2E, the formal rule: (a/b) ÷ (c/d) = (a/b) · (d/c), provided b, c, d ≠ 0. For example, (1/3) ÷ (1/9) = (1/3) · (9/1) = 3. This rule applies equally when dividing whole numbers by fractions, fractions by whole numbers, and variables with fractional coefficients. The reciprocal is formed by flipping numerator and denominator.
Key Concepts
Property If $a$, $b$, $c$, and $d$ are numbers where $b \neq 0$, $c \neq 0$, and $d \neq 0$, then $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$$ 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.
Common Questions
How do you divide fractions?
Keep the first fraction, change the division sign to multiplication, and flip the second fraction (take its reciprocal). Then multiply.
What is (1/3) ÷ (1/9)?
Keep 1/3, flip 1/9 to 9/1, multiply: (1/3)(9/1) = 9/3 = 3.
What is (5/6) ÷ (10/3)?
Keep 5/6, flip 10/3 to 3/10, multiply: (5/6)(3/10) = 15/60 = 1/4.
What is the reciprocal of a fraction?
The reciprocal is formed by swapping numerator and denominator. The reciprocal of 4/7 is 7/4.
How do you divide a whole number by a fraction?
Write the whole number as a fraction over 1, then multiply by the reciprocal. 6 ÷ (2/3) = (6/1)(3/2) = 18/2 = 9.
Why does dividing by a fraction give a larger result?
Dividing by a number less than 1 is like asking 'how many times does a small piece fit?' — it fits more times, giving a larger quotient.