Dividing fractions
To divide by a fraction, multiply by its reciprocal: keep the first fraction, change ÷ to ×, flip the second. Below is the calculator, the method, and the reason it works.
Working
- Keep the first fraction, change ÷ to ×, and flip the second fraction (its reciprocal is 8).
3/4 × 8/1 - Multiply straight across.
(3 × 8) / (4 × 1) = 24/4 - Simplify: divide top and bottom by their greatest common factor, 4.
24/4 = 6
How to divide fractions, step by step
- Convert mixed and whole numbers. 2 1/4 becomes 9/4; 6 becomes 6/1.
- Keep the first fraction. It does not change.
- Change ÷ to ×. Division by a number is multiplication by its reciprocal.
- Flip the second fraction. The reciprocal of 1/8 is 8/1. The reciprocal of a/b is b/a.
- Multiply and simplify. 3/4 × 8/1 = 24/4 = 6.
What dividing by a fraction really asks
3/4 ÷ 1/8 asks: how many 1/8-sized pieces fit into 3/4? Put both on the same eighths ruler and count.
3/4 is 6 eighths; 1/8 is 1 eighths. 6 ÷ 1 = 6 pieces fit. Flipping and multiplying (3/4 × 8/1) is a shortcut to this same count.
What is the reciprocal of a fraction?
The reciprocal is the fraction flipped upside down: the reciprocal of 2/5 is 5/2, of 7 (7/1) is 1/7, and of 1 3/4 (7/4) is 4/7. A number times its reciprocal is always 1. Zero has no reciprocal, which is why you cannot divide by zero.
Dividing a whole number by a fraction
4 ÷ 2/5 = 4/1 × 5/2 = 20/2 = 10. Ten pieces of size 2/5 fit into 4.
Dividing a fraction by a whole number
3/4 ÷ 3 = 3/4 × 1/3 = 3/12 = 1/4. Sharing three-quarters between three people gives each a quarter.
Why the answer can be bigger
Dividing by a number less than 1 asks how many small pieces fit, so the answer grows: 3/4 ÷ 1/8 = 6. That surprises people who learned that "division makes smaller" with whole numbers.
Common mistakes
- Flipping the first fraction instead of the second.
- Flipping before converting a mixed number (flip 9/4, not 2 1/4 → 2 4/1).
- Forgetting to change ÷ to × after flipping.
Questions people ask
Why do you flip the second fraction when dividing?
Dividing by a number undoes multiplying by it, and multiplying by the reciprocal does exactly that. Put another way: a ÷ (c/d) asks how many c/d fit into a; there are d/c of them in every 1, so the answer is a × d/c.
What is 1/2 ÷ 1/4?
1/2 × 4/1 = 4/2 = 2. Two quarters fit in a half.
What is the reciprocal of 3?
1/3, because 3 × 1/3 = 1.
How do you divide mixed numbers?
Convert both to improper fractions, then keep–change–flip. 2 1/2 ÷ 1 1/4 = 5/2 × 4/5 = 20/10 = 2.