Multiplying a fraction by an integer means taking that many equal copies of the fraction. Three-quarters multiplied by 4 is four lots of three-quarters, giving twelve-quarters or 3. The rule is simply to multiply the numerator by the integer and keep the denominator, then simplify.
What does it mean to multiply a fraction by an integer?
Multiplication is repeated addition. So ¾ × 4 means ¾ + ¾ + ¾ + ¾:
¾ + ¾ + ¾ + ¾ = 12/4 = 3
The numerators add up (3 + 3 + 3 + 3 = 12) while the denominator stays the same. This is why the shortcut works: multiply the numerator by the integer, keep the denominator.
In symbols: a/b × n = (a × n) / b
How do you multiply a fraction by an integer?
- Multiply the numerator by the integer.
- Keep the denominator unchanged.
- Simplify (reduce) the resulting fraction if possible.
Worked examples:
| Calculation | Step 1: multiply numerator | Step 2: keep denominator | Step 3: simplify |
|---|---|---|---|
| ²⁄₅ × 3 | 2 × 3 = 6 | 6/5 | 1⅕ |
| ⅓ × 9 | 1 × 9 = 9 | 9/3 | 3 |
| ⁵⁄₈ × 4 | 5 × 4 = 20 | 20/8 | 2½ |
Notice that ⅓ × 9 gives a whole number because 9 is a multiple of 3.
How do you cancel before multiplying to keep numbers small?
Instead of multiplying first and simplifying at the end, look for a common factor between the integer and the denominator before you begin. Dividing both by that factor first makes the numbers smaller and reduces the chance of errors.
Example: Calculate ⁵⁄₁₂ × 8.
Method A (multiply then simplify): 5 × 8 = 40; 40/12 = 10/3 = 3⅓
Method B (cancel first): 8 and 12 share a factor of 4. Divide 8 by 4 to get 2; divide 12 by 4 to get 3. ⁵⁄₁₂ × 8 → ⁵⁄₃ × 2 = 10/3 = 3⅓
Both methods give the same answer, but Method B uses smaller numbers throughout.
When to cancel: look at the denominator and the integer — if they share a common factor, cancel it before multiplying.
What if the answer is an improper fraction?
If the result has a numerator larger than the denominator, convert it to a mixed number by dividing the numerator by the denominator.
Example: ⁷⁄₄ × 6
- 7 × 6 = 42; 42/4
- Simplify: HCF(42, 4) = 2; 42/4 = 21/2
- Convert: 21 ÷ 2 = 10 remainder 1 → 10½
How do you apply this in word problems?
Worked problem: A recipe needs ¾ of a cup of flour for one batch of biscuits. How much flour is needed for 5 batches?
- Quantity per batch: ¾ cup
- Number of batches: 5
- Total: ¾ × 5 = 15/4 = 3¾ cups
Another example: A piece of ribbon is ⅝ m long. How long are 12 identical pieces?
- ⅝ × 12 — cancel: 12 and 8 share factor 4; 12/4 = 3 and 8/4 = 2.
- ⅝ × 12 = ⁵⁄₂ × 3 = 15/2 = 7½ m
What mistakes should you avoid?
Mistake 1 — Multiplying the denominator as well as the numerator. The rule is numerator × integer; the denominator stays the same. Calculating ²⁄₅ × 3 as 6/15 (multiplying both) gives the wrong answer.
Mistake 2 — Forgetting to simplify. 20/8 is a valid fraction, but the expected answer is 2½. Always check whether the fraction simplifies.
Mistake 3 — Cancelling with the numerator instead of the denominator. Cancelling is between the integer and the denominator, not the integer and the numerator.
Frequently asked questions
Does the order of multiplication matter for fractions and integers?
No — multiplication is commutative: ¾ × 4 = 4 × ¾ = 3. You can write the integer first or the fraction first; the result is the same. This mirrors the familiar rule that 3 × 4 = 4 × 3.
Why does multiplying by a fraction give a smaller answer than the original integer?
When you multiply by a proper fraction (numerator < denominator), you are taking a part of the number, so the result is smaller. For example, ½ × 8 = 4. Multiplying by an improper fraction (e.g., ⁵⁄₃) gives a result larger than the original integer.
How does multiplying fractions by integers connect to multiplying two fractions?
Any integer n can be written as n/1. So ²⁄₃ × 5 = ²⁄₃ × ⁵⁄₁ = (2 × 5)/(3 × 1) = 10/3. This shows that the standard rule for multiplying two fractions (multiply numerators, multiply denominators) includes multiplying a fraction by an integer as a special case.
What is the quickest mental method for ²⁄₃ × 12?
Notice that 12 is a multiple of 3. So ²⁄₃ × 12 = 2 × (12 ÷ 3) = 2 × 4 = 8. Whenever the integer is a multiple of the denominator, divide the integer by the denominator and multiply by the numerator — the answer is a whole number and no simplifying is needed.
For Socratic KS3 fraction practice with a tutor who explains the reasoning, visit aitutors.me.