Divide Fractions: Keep, Change, Flip Explained (and Why It Works)

What is keep, change, flip?

"Keep, change, flip" is a memory aid for dividing fractions:

1. Keep the first fraction exactly as it is.
2. Change the division sign to a multiplication sign.
3. Flip the second fraction (swap its numerator and denominator).

Then multiply as normal.

It is also called "invert and multiply" or "multiply by the reciprocal". They all mean the same thing.

A quick worked example

Divide 3/4 ÷ 2/5.

• Keep 3/4.
• Change ÷ to ×.
• Flip 2/5 to 5/2.

Now multiply: 3/4 × 5/2 = 15/8 = 1 7/8.

That is the whole procedure. But why does flipping the second fraction turn a division into a multiplication?

Why it works

Division asks: "how many of the second quantity fit into the first?"

3/4 ÷ 2/5 asks: "how many groups of 2/5 fit into 3/4?"

Dividing by a fraction is the same as multiplying by its reciprocal because of a fundamental algebraic identity:

a ÷ b = a × (1/b)

When b is itself a fraction like 2/5, then 1/(2/5) = 5/2. So dividing by 2/5 is the same as multiplying by 5/2.

Another way to see it

Write the division as a complex fraction:

(3/4) / (2/5)

To simplify a complex fraction, multiply numerator and denominator by the reciprocal of the denominator:

(3/4 × 5/2) / (2/5 × 5/2) = (15/8) / (1) = 15/8.

The denominator becomes 1, and you are left with the product of the first fraction and the reciprocal of the second.

More worked examples

Example 1: Simple proper fractions

5/6 ÷ 1/3

Keep 5/6. Change ÷ to ×. Flip 1/3 to 3/1.

5/6 × 3/1 = 15/6 = 5/2 = 2 1/2.

Example 2: Dividing by a whole number

7/8 ÷ 4

Write 4 as 4/1. Flip to 1/4.

7/8 × 1/4 = 7/32. Answer: 7/32.

Notice the result is smaller than 7/8. Dividing by a number greater than 1 makes the result smaller — the opposite of what happens when you divide by a proper fraction.

Example 3: A whole number divided by a fraction

6 ÷ 3/4

Write 6 as 6/1. Flip 3/4 to 4/3.

6/1 × 4/3 = 24/3 = 8.

Six divided by three-quarters is 8, because there are 8 groups of 3/4 in 6.

Example 4: Mixed numbers

2 1/2 ÷ 1 1/4

Convert: 2 1/2 = 5/2, 1 1/4 = 5/4.

Keep 5/2. Change ÷ to ×. Flip 5/4 to 4/5.

5/2 × 4/5 = 20/10 = 2.

For a full walkthrough of dividing fractions, see the How to Divide Fractions guide.

Answer checks

Check 1 — Direction

• Dividing by a number less than 1 → result is bigger than the first fraction.
• Dividing by a number greater than 1 → result is smaller.
• Dividing by 1 → result stays the same.

If your answer violates these rules, re-check your working.

Check 2 — Decimal estimate

Convert each fraction to a decimal, divide, and compare.

5/6 ≈ 0.833, 1/3 ≈ 0.333. Division: 0.833 ÷ 0.333 ≈ 2.5. Our answer was 2 1/2 = 2.5. Confirmed.

Check 3 — Multiply back

If a ÷ b = c, then c × b should equal a.

2 1/2 × 1/3 should equal 5/6. Check: 5/2 × 1/3 = 5/6. Correct.

Common mistakes

Flipping the wrong fraction. You flip the second fraction (the divisor), not the first. This is the most common error.

Forgetting to change the sign. If you flip but still divide, you will get the reciprocal of the correct answer.

Not converting mixed numbers first. You must convert mixed numbers to improper fractions before applying keep-change-flip.

Cancelling before flipping. Always flip first, then cancel. The cancellation rules apply to the multiplication, not the original division. See the worked example for 8/1 ÷ 1/2 for a clear illustration.

When students get confused

Some students memorise the rule but cannot explain why it works. Without understanding, they tend to flip the wrong fraction or apply the rule to multiplication problems by mistake. Spending a few minutes with the "complex fraction" explanation above helps build real understanding rather than rote recall.

FAQ

Does keep-change-flip work for dividing decimals too? Decimals are just fractions in disguise. You can convert to fractions, apply the rule, and convert back. But usually it is simpler to divide decimals directly using long division.

What if the first fraction is zero? 0 ÷ anything = 0 (as long as the divisor is not zero).

What if the second fraction is zero? Division by zero is undefined. You cannot divide by 0/5 or 0.

Can I divide three fractions in a row? Yes. Work left to right: (a ÷ b) ÷ c. Apply keep-change-flip for each step. Be careful — fraction division is not associative, so the grouping matters.

Is "invert and multiply" the only way? It is the standard method. You could also use common denominators: rewrite both fractions with the same denominator, then divide the numerators. The answer is the same. Most people find keep-change-flip faster.