To multiply or divide mixed numbers, first convert each one into an improper fraction. Then multiply numerators together and denominators together; for division, keep the first fraction, change ÷ to ×, and flip the second. Converting to improper fractions avoids the errors that arise from trying to handle the whole-number and fraction parts separately.
Why do you need to convert to improper fractions first?
Mixed numbers like 2¾ have a whole part and a fractional part sitting side by side. If you try to multiply or divide them directly, you end up treating the whole part separately from the fraction, which introduces errors. Converting to an improper fraction first merges everything into a single number, so the standard "multiply across" or "keep-change-flip" rules work cleanly.
How do you convert a mixed number to an improper fraction?
The method takes two steps:
- Multiply the whole number by the denominator.
- Add the numerator, then write that total over the original denominator.
Example: convert 3¼ to an improper fraction.
- Multiply the whole by the denominator: 3 × 4 = 12.
- Add the numerator: 12 + 1 = 13.
- Write over the denominator: 13/4.
To convert back from an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole part, and the remainder is the new numerator.
How do you multiply two mixed numbers?
Follow these four steps:
- Convert each mixed number to an improper fraction.
- Multiply the numerators together.
- Multiply the denominators together.
- Simplify the answer, converting back to a mixed number if needed.
Worked example: calculate 1½ × 2⅔.
| Step | Working |
|---|---|
| Convert 1½ | 1 × 2 + 1 = 3, so 3/2 |
| Convert 2⅔ | 2 × 3 + 2 = 8, so 8/3 |
| Multiply numerators | 3 × 8 = 24 |
| Multiply denominators | 2 × 3 = 6 |
| Simplify 24/6 | 24 ÷ 6 = 4 |
Answer: 4
A useful shortcut: cancel common factors between any numerator and any denominator before you multiply, rather than simplifying a large fraction afterwards. In the example above, 3 and 3 share a factor of 3, and 2 and 8 share a factor of 2, giving (1/1) × (4/1) = 4 directly.
How do you divide one mixed number by another?
Division of fractions uses the keep-change-flip rule (also called "multiply by the reciprocal"):
- Keep the first fraction unchanged.
- Change ÷ to ×.
- Flip the second fraction (write its reciprocal).
- Then multiply as above.
Worked example: calculate 3½ ÷ 1¾.
| Step | Working |
|---|---|
| Convert 3½ | 3 × 2 + 1 = 7, so 7/2 |
| Convert 1¾ | 1 × 4 + 3 = 7, so 7/4 |
| Keep-change-flip | 7/2 × 4/7 |
| Cancel 7s | 1/2 × 4/1 |
| Multiply | 2 |
Answer: 2
Always check this makes sense: 3½ divided by something close to 2 should give roughly 1¾ — and it does.
How do mixed number multiplication and division compare?
| Operation | Key step | Memory cue |
|---|---|---|
| Multiplication | Multiply numerators; multiply denominators | "Straight across" |
| Division | Flip the second fraction, then multiply | "Keep-change-flip" |
| Both | Convert to improper fractions first | "Whole numbers make errors" |
| Both | Simplify by cancelling before multiplying | "Cancel early, fewer big numbers" |
What mistakes do students commonly make?
- Forgetting to convert first. Multiplying 2¾ × 1½ as "2 × 1 = 2, then ¾ × ½ = 3/8, so 2 3/8" gives the wrong answer (the correct answer is 4⅛). Always convert before multiplying.
- Flipping the wrong fraction. In a ÷ b, only b (the divisor) gets flipped. Flipping both fractions produces the original problem again.
- Not simplifying fully. After multiplying, check whether the resulting fraction can be reduced. An answer of 18/12 should become 3/2 = 1½.
- Forgetting to convert back. Unless the question asks for an improper fraction, convert 11/4 to 2¾ in your final answer.
Frequently asked questions
Can I multiply the whole-number parts and fraction parts separately?
No, not directly — this is the most common source of errors. The cross-product terms are missing: (a + b/c) × (d + e/f) is not the same as (a × d) + (b/c × e/f). Converting to improper fractions first means you capture all the cross terms automatically and the "straight across" rule applies safely.
What if one of the numbers is a whole number, not a mixed number?
Write the whole number as a fraction over 1. So 5 becomes 5/1, and the multiplication or division proceeds exactly as above. For example, 2¼ × 6 becomes (9/4) × (6/1) = 54/4 = 13½.
How do I know whether to convert my answer back to a mixed number?
Follow the form of the question. If the question uses mixed numbers throughout, give your answer as a mixed number. If it uses improper fractions, an improper fraction is fine. In a real-life or word-problem context (e.g. "how many 1½-metre lengths fit in 9 metres?"), a whole number or mixed number is almost always expected.
Why does "keep-change-flip" work for division?
Dividing by a number gives the same result as multiplying by its reciprocal. The reciprocal of 7/4 is 4/7 because 7/4 × 4/7 = 1 (they cancel to give 1, the identity for multiplication). So a ÷ 7/4 = a × 4/7. Keep-change-flip is just a memory device for this algebraic identity.
For step-by-step KS3 fraction practice with Professor Pi, visit aitutors.me.