Dividing Fractions

Division of fractions = multiplication by the reciprocal.

That sentence is the entire rule. Once you internalize it, fraction division is just fraction multiplication with one extra step.

The reciprocal

The reciprocal of a/b is b/a. You flip it.

The reciprocal of a non-zero number, multiplied by that number, equals 1. For example: 2/3 × 3/2 = 6/6 = 1. That's the whole point of a reciprocal.

The rule

a/b ÷ c/d  =  a/b × d/c

In words: keep the first, flip the second, multiply. Some teachers call it Keep–Change–Flip (KCF).

Worked examples

Example A — basic divide

2/3 ÷ 4/5
= 2/3 × 5/4
= (2 × 5) / (3 × 4)
= 10/12
= 5/6

Example B — divide by a whole number

A whole number n is n/1. Its reciprocal is 1/n.

3/4 ÷ 2
= 3/4 ÷ 2/1
= 3/4 × 1/2
= 3/8

Dividing a fraction by a whole number is the same as putting the whole number into the denominator.

Example C — whole number divided by a fraction

6 ÷ 2/3
= 6/1 × 3/2
= 18/2
= 9

That's why "6 divided by 2/3" feels surprising — the answer is bigger than 6. Dividing by something less than 1 makes the answer larger.

Example D — simplify across the multiplication

4/9 ÷ 8/3
= 4/9 × 3/8

4 and 8 share factor 4 → 1 and 2
3 and 9 share factor 3 → 1 and 3

= (1/3) × (1/2)
= 1/6

Common mistakes

• Flipping the wrong fraction. Always flip the second one (the divisor), never the first (the dividend).
• Forgetting to flip at all. 2/3 ÷ 4/5 ≠ 2/3 × 4/5. The first gives 5/6, the second gives 8/15.
• Trying to find a common denominator. Not needed. Common denominators are for add/subtract. For divide, just flip and multiply.

Try these (answers below)

1. 1/2 ÷ 1/4
2. 3/5 ÷ 6/7
3. 4 ÷ 2/3
4. 7/8 ÷ 7

1. 2 (or 2/1)
2. 7/10
3. 6
4. 1/8