The conceptual reason behind the reciprocal rule for dividing fractions
Reserve & Extensions • K-12
When you were first taught to divide fractions, you learned a rule: "keep, change, flip" -- keep the first fraction, change division to multiplication, and flip the second fraction. But why does this work? This lesson reveals the mathematical reasoning behind the rule.
The expression 6 ÷ 2 asks: "How many groups of 2 fit into 6?" The answer is 3. Fraction division asks the same kind of question:
Since 3/4 = 6/8, the answer is 6. Let us see how "flip and multiply" gets us there.
Calculate 3/4 ÷ 1/8 using "flip and multiply."
Indeed, six copies of 1/8 make 6/8 = 3/4. The answer checks out.
Here is why the rule works, step by step:
Any division can be written as a fraction (a complex fraction):
To simplify a complex fraction, multiply the top and bottom by the same thing -- specifically, by the reciprocal of the denominator:
Simplify (2/3) ÷ (4/5).
Multiplying by the reciprocal always turns the denominator into 1, leaving just the numerator -- which is the first fraction times the flipped second fraction.
The reciprocal of a number is what you multiply it by to get 1:
| Number | Reciprocal | Product |
|---|---|---|
| 3/4 | 4/3 | (3/4) × (4/3) = 12/12 = 1 |
| 5 | 1/5 | 5 × (1/5) = 1 |
| 2/7 | 7/2 | (2/7) × (7/2) = 14/14 = 1 |
Division by a number is the same as multiplication by its reciprocal. This is true for all numbers, not just fractions: 10 ÷ 5 = 10 × (1/5) = 2.
Simplify: (5/6) / (10/3)
Before multiplying, you can simplify across the fractions. In (5/6) × (3/10), notice that 5 and 10 share a factor of 5, and 3 and 6 share a factor of 3. Cancel first: (1/2) × (1/2) = 1/4. Same answer, simpler arithmetic.
Always flip the second fraction (the divisor), never the first. In (2/3) ÷ (4/5), flip 4/5 to get 5/4. Do not flip 2/3.
1. Calculate: (3/5) ÷ (2/3)
(3/5) × (3/2) = 9/10
2. Calculate: (7/8) ÷ (7/4)
(7/8) × (4/7) = 28/56 = 1/2. (Cross-cancel the 7s and simplify 4/8 to get 1/2 directly.)
3. Simplify the complex fraction: (1/2) / (3/4)
(1/2) × (4/3) = 4/6 = 2/3
4. A recipe calls for 3/4 cup of sugar. Each scoop holds 1/8 cup. How many scoops do you need?
(3/4) ÷ (1/8) = (3/4) × (8/1) = 24/4 = 6 scoops.
5. Use the "multiply both sides by the reciprocal" method to show that (a/b) ÷ (c/d) = (ad)/(bc).
Write as a complex fraction: (a/b) / (c/d). Multiply numerator and denominator by d/c. Numerator becomes (a/b)(d/c) = ad/(bc). Denominator becomes (c/d)(d/c) = 1. Result: ad/(bc).