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Level 7Grade 7skill: fraction_multiply· 8 min read

How to Multiply Fractions

In short: To multiply fractions, multiply the numerators and multiply the denominators — no common denominator needed — then simplify. Cancel shared factors before multiplying to keep numbers small, turn percentages into fractions first, and remember that multiplying by a fraction below 1 makes the answer smaller, which catches the most common error.

The method: multiply straight across

Unlike adding and subtracting, multiplying fractions needs no common denominator — that requirement belongs to the other operations. To multiply, you go straight across: multiply along the top, multiply along the bottom. That single fact makes it the most forgiving fraction operation, and it still trips people up because they reach for the addition habit out of reflex.

The word to watch for in a question is of. "Three-quarters of 80", "two-thirds of the water", "40% of the price" all mean multiply. A percentage is just a fraction with a denominator of 100, so 40% becomes 40/100 = 2/5 before you do anything else.

The habit that separates a fast, accurate solver from a slow, error-prone one is cancelling before you multiply. If a number on any top shares a factor with a number on any bottom, divide both by it first. You get smaller numbers, no giant products to simplify, and far fewer arithmetic slips.

  1. Rewrite everything as a fraction: whole numbers go over 1, percentages become the percent over 100, mixed numbers become improper fractions.
  2. Cancel any common factor between a numerator and a denominator (they do not have to be in the same fraction).
  3. Multiply the numerators together, then multiply the denominators together.
  4. Simplify to lowest terms, then convert back to a mixed number, decimal, or amount if the question asks for one.
  5. Sanity check: multiplying by a fraction below 1 must make the quantity smaller. If your answer grew, you added by mistake.

Why it works, and the one check that saves you Why it works

"of" = ×
The word "of" between a fraction and a quantity is a multiplication sign in disguise.

1/2 of 8 = 1/2 × 8/1 = 8/2 = 4.

Straight across
Multiply the tops, multiply the bottoms. No common denominator, ever.

2/3 × 4/5 = (2×4)/(3×5) = 8/15.

Cancel first
Divide a top and a bottom by a shared factor before multiplying to keep numbers tiny.

4/9 × 3/8 → cancel 4 & 8, and 3 & 9 → 1/3 × 1/2 = 1/6.

Percent to fraction
A percentage is a fraction over 100; simplify it, then multiply as normal.

25% of 60 = 1/4 × 60 = 15.

Shrink check
Times a proper fraction, any positive number shrinks.

5/8 × 4/5 = 1/2 — smaller than both.

Here is the insight worth keeping: because multiplying by a proper fraction always shrinks a quantity, while adding fractions grows it, the two operations can never give similar answers. So the classic blunder — hunting for a common denominator and adding when the question said multiply — is not just wrong, it points the wrong way. One glance at whether your answer got bigger or smaller than the numbers you started with will catch it every time.

Worked examples

In a year group of 40 students, 3/5 play a sport. Of those, 5/6 play a team sport. How many play a team sport? = 20 students
  1. "3/5 of 40": 40 × 3/5 = 120/5 = 24 students play a sport.
  2. "5/6 of those": 24 × 5/6. Cancel 24 and 6 → 4 and 1: 4 × 5 = 20.
  3. 20 is the final count — do not stop at the 24 from step one.
A jacket costs £80. It is reduced by 1/4, then a further 1/5 is taken off the reduced price. What is the final price? = £48
  1. First reduction: keep 3/4 of the price. 80 × 3/4 = 60, so the jacket is now £60.
  2. Second reduction is 1/5 off the NEW price, so keep 4/5 of £60: 60 × 4/5 = 48.
  3. The 1/5 must come off £60, not the original £80 — the base changes after the first cut.
A recipe uses 0.75 kg of flour. You scale it to make 0.4 of the recipe. How much flour do you need? = 0.3 kg
  1. Rewrite as fractions: 0.75 = 3/4 and 0.4 = 2/5.
  2. 3/4 × 2/5 = 6/20 = 3/10.
  3. 3/10 = 0.30, so you need 0.3 kg. Check: 0.75 × 0.4, two numbers below 1, must give an answer below 0.75 — and 0.3 is.
A tank is 3/4 full. You use 2/3 of the water in it. What fraction of the full tank is left? = 1/4 of the full tank
  1. The 2/3 is 2/3 of the WATER, not of the tank. Water used = 2/3 × 3/4 = 6/12 = 1/2 of the full tank.
  2. Water started at 3/4 of the tank, so water left = 3/4 − 1/2 = 1/4 of the tank.
  3. The tempting 1 − 2/3 = 1/3 answers a different question — it ignores that the tank was only 3/4 full to begin with.

Common mistakes

Finding a common denominator and adding

✗ 2/3 × 3/4 = 8/12 + 9/12 = 17/12

Common denominators belong to addition and subtraction. Multiplication never needs them, and adding makes the answer bigger — the opposite of what multiplying by a proper fraction should do.

Fix: Go straight across: 2/3 × 3/4 = 6/12 = 1/2. Notice 1/2 is smaller than both fractions, exactly as multiplication should be.

wrong_operation

Stopping after the first step

✗ 2/3 of 45, then 3/5 of that → answer 30

45 × 2/3 = 30 is only the halfway value. The question asked for 3/5 of that, so 30 is a correct intermediate number reported as if it were the final answer.

Fix: Finish the chain: 30 × 3/5 = 18. Always re-read what the final quantity is before you write your answer.

partial_computation

Taking the second fraction off the wrong price

✗ £50 reduced by 1/5 then 1/4: 50 − 10 − 12.50 = £27.50

The 1/4 was taken off the original £50 (giving 12.50) instead of the reduced £40. The base changed after the first discount, so the second fraction must act on the new amount.

Fix: After the first cut the price is £40, so the 1/4 off is 40 × 1/4 = 10, giving 40 − 10 = £30.

base_confusion

Using the percent number as the answer

✗ A 40% deposit on a £90 booking → £40

The 40 was lifted straight from "40%" and used as pounds. A percent is a fraction over 100, not a finished amount.

Fix: 40% = 2/5, so the deposit is 2/5 × 90 = 36. The answer is £36, comfortably less than the full £90.

percent_as_whole

Misplacing the decimal point

✗ 0.6 × 0.5 = 0.03

The digit 3 is right, but the point landed one place too far left. Multiplying two numbers each near a half should give roughly a quarter to a third, not three-hundredths.

Fix: Count decimal places: 0.6 (one) × 0.5 (one) = two places, so 3 becomes 0.30 = 0.3. As fractions, 3/5 × 1/2 = 3/10 confirms it.

decimal_point_shift

Practice

Work through these adaptive questions. Several are built to punish the naive move — someone who adds instead of multiplies, stops a step early, or takes the fraction off the wrong base will find their tempting answer waiting as a wrong option. Cancel before you multiply, and use the shrink check on every answer.

Related skills

First, make sure of

Frequently asked questions

Do I need a common denominator to multiply fractions?

No. Common denominators are only for adding and subtracting. To multiply, go straight across: multiply the numerators together and the denominators together, then simplify.

Does the word "of" always mean multiply?

With fractions and percentages, yes. "3/4 of 20" means 3/4 × 20 = 15, and "30% of 50" means 30/100 × 50 = 15. Spotting "of" tells you the operation.

Why does multiplying sometimes make the number smaller?

Multiplying by a proper fraction (one between 0 and 1) takes a part of the amount, so the result shrinks. 8 × 1/2 = 4. This is a handy check: if your answer grew, you probably added by accident.

Should I cancel before or after multiplying?

Before, whenever you can. Divide any numerator and any denominator by a shared factor first — the numbers stay small and simplifying at the end is easier or unnecessary. 4/9 × 3/8 becomes 1/3 × 1/2 = 1/6.

How do I multiply a fraction by a whole number?

Write the whole number over 1. So 6 × 2/3 = 6/1 × 2/3 = 12/3 = 4. It is the same straight-across rule.