Dividing Fractions by Fractions

Dividing fractions might sound tricky, but there is a simple rule that turns every fraction division problem into a multiplication problem. This rule is often called "Keep, Change, Flip."

The "Keep, Change, Flip" Method

To divide one fraction by another, you multiply the first fraction by the reciprocal of the second fraction. The reciprocal is just the fraction flipped upside down.

Here is how the method works in three easy steps:

1. Keep the first fraction exactly the same.
2. Change the division sign ($\div$) to a multiplication sign ($\times$).
3. Flip the second fraction (the divisor) upside down.

Example 1: Standard Fractions

Let's solve: $\frac{3}{4} \div \frac{2}{5}$

• Keep: $\frac{3}{4}$
• Change: $\times$
• Flip: $\frac{5}{2}$

Now, multiply straight across (numerators together, denominators together): $\frac{3}{4} \times \frac{5}{2} = \frac{3 \times 5}{4 \times 2} = \frac{15}{8}$

You can leave this as an improper fraction or convert it to a mixed number: $1\frac{7}{8}$.

Dividing Mixed Numbers

When dealing with mixed numbers, you must add one extra step at the beginning: convert all mixed numbers into improper fractions first. Once they are improper fractions, you just use "Keep, Change, Flip."

Example 2: Mixed Numbers

Let's solve: $2\frac{1}{3} \div 1\frac{1}{6}$

Step 1: Convert to improper fractions.

• $2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{7}{3}$
• $1\frac{1}{6} = \frac{(1 \times 6) + 1}{6} = \frac{7}{6}$

Now the problem is: $\frac{7}{3} \div \frac{7}{6}$

Step 2: Keep, Change, Flip. $\frac{7}{3} \times \frac{6}{7}$

Step 3: Multiply and simplify. $\frac{7 \times 6}{3 \times 7} = \frac{42}{21} = 2$

Solving Word Problems

Fraction division is incredibly useful in real life, especially when figuring out how many smaller portions fit into a larger total.

Example 3: Recipe Portions

Problem: A recipe uses $\frac{3}{4}$ cup of flour per serving. How many servings can you make from 6 cups of flour?

Step 1: Set up the equation. You are taking a total of 6 cups and dividing it into portions of $\frac{3}{4}$ cup. $6 \div \frac{3}{4}$

Step 2: Turn the whole number into a fraction. $\frac{6}{1} \div \frac{3}{4}$

Step 3: Keep, Change, Flip. $\frac{6}{1} \times \frac{4}{3}$

Step 4: Multiply and simplify. $\frac{6 \times 4}{1 \times 3} = \frac{24}{3} = 8$

You can make 8 servings!