Fractions of an Amount

To find \( \frac{3}{4} \) of 20, first find \( \frac{1}{4} \) of 20 by dividing by 4: \( 20 \div 4 = 5 \). Then multiply by 3: \( 5 \times 3 = 15 \). You can visualise this as splitting 20 into 4 equal groups of 5 — three of those groups give \( 5 + 5 + 5 = 15 \).

You can also think of it as multiplication: \( \frac{3}{4} \times 20 = \frac{3 \times 20}{4} = \frac{60}{4} = 15 \). Or use a bar model: draw a bar worth 20, split it into 4 equal parts, shade 3 of them. Each part is worth 5, so the shaded region is 15.

\( \frac{2}{5} \) of 30 = \( 30 \div 5 \times 2 = 6 \times 2 = 12 \). And \( \frac{4}{10} \) of 30 = \( 30 \div 10 \times 4 = 3 \times 4 = 12 \). Both give 12 because \( \frac{2}{5} \) and \( \frac{4}{10} \) are equivalent fractions (multiply numerator and denominator of \( \frac{2}{5} \) by 2 to get \( \frac{4}{10} \)). Since the fractions represent the same proportion, the same proportion of any amount must be the same.

This matters because students sometimes treat equivalent fractions as a “number fact” that has nothing to do with finding fractions of amounts. But equivalent fractions are equal — they describe the same part of any whole. So \( \frac{2}{5} \) of anything will always give the same result as \( \frac{4}{10} \) of the same thing.

If you take \( \frac{1}{4} \) away from the whole, you’re left with \( \frac{3}{4} \) of the whole. Using 20 as an example: \( \frac{1}{4} \) of 20 = 5, so \( 20 – 5 = 15 \). And \( \frac{3}{4} \) of 20 = 15. They match because \( \frac{1}{4} + \frac{3}{4} = \frac{4}{4} = 1 \), so the fraction you keep and the fraction you remove always add up to the whole.

This relationship works for any fraction: finding \( \frac{2}{3} \) of a number is the same as subtracting \( \frac{1}{3} \) from the whole. Finding \( \frac{7}{10} \) is the same as subtracting \( \frac{3}{10} \). This “complementary” approach is often an easier mental method — for example, to find \( \frac{9}{10} \) of 60, it’s quicker to calculate \( \frac{1}{10} \) of 60 = 6 and subtract: \( 60 – 6 = 54 \).

If \( \frac{1}{4} \) of the number is 5, that means one equal part is worth 5. Since the denominator is 4, the whole number is made up of 4 of these parts. So the whole number is \( 5 \times 4 = 20 \).

This is the reverse of the usual process. Normally we divide the whole by 4 to find the part. Here, we multiply the part by 4 to find the whole. You can verify it by working forwards: does \( \frac{1}{4} \) of 20 equal 5? Yes, \( 20 \div 4 = 5 \).

Example: 8 — \( \frac{3}{4} \) of 8 = 6

Another: 20 — \( \frac{3}{4} \) of 20 = 15

Creative: 4 — \( \frac{3}{4} \) of 4 = 3, the smallest positive whole number that works. Or −12 — \( \frac{3}{4} \) of −12 = −9, using a negative number.

Trap: 6 — a student might pick this because \( 6 \div 3 = 2 \) (a whole number), thinking the numerator is what matters. But \( \frac{3}{4} \) of 6 = 4.5, which is not a whole number. For \( \frac{3}{4} \) of a number to be whole, the number must be divisible by the denominator (4), not the numerator (3). Numbers that work are multiples of 4: 4, 8, 12, 16, 20…

Example: \( \frac{1}{2} \) of £5 = £2.50

Another: \( \frac{1}{4} \) of £10 = £2.50

Creative: \( \frac{5}{2} \) of £1 — using an improper fraction of a smaller amount. Or \( \frac{250}{100} \) of £1.

Trap: \( \frac{1}{2} \) of 500 — a student might write this thinking of 500 pennies, but if the unit is pounds, this is £250, not £2.50. Units matter!

Example: \( \frac{1}{3} \) of 10 = \( \frac{10}{3} \) or \( 3\frac{1}{3} \). (Avoid using decimals like 3.33… in fraction work if possible!)

Another: \( \frac{1}{4} \) of 7 = \( \frac{7}{4} \) or \( 1\frac{3}{4} \).

Creative: \( \frac{1}{2} \) of 5 = \( 2.5 \) — even halving, the simplest unit fraction, can give a non-whole answer. Or \( \frac{1}{3} \) of 1 = \( \frac{1}{3} \) — using 1 as the amount.

Trap: \( \frac{1}{5} \) of 15 — a student might offer this thinking “15 isn’t a round number so it won’t divide by 5 evenly.” But \( 15 \div 5 = 3 \), which is a whole number. 15 is indeed a multiple of 5 (5, 10, 15, 20…). The test is whether the number is a multiple of the denominator — not whether it “looks” divisible.

Example: \( \frac{5}{4} \) of 20 = 25

Another: \( \frac{3}{2} \) of 10 = 15

Creative: \( \frac{7}{3} \) of 6 = 14 — using a fraction greater than 2 to get an answer more than double the original. Or \( \frac{101}{100} \) of 200 = 202 — only just exceeding the original amount.

Trap: \( \frac{4}{5} \) of 30 = 24 — a student might think \( \frac{4}{5} \) is “nearly all of it and a bit more” because 4 is close to 5. But for positive amounts, a proper fraction (less than 1) always gives a result smaller than the original. You need an improper fraction (numerator > denominator) to increase a positive amount.

It depends on whether the number is divisible by the denominator. For example, \( \frac{1}{4} \) of 20 = 5 (whole number) because 20 is a multiple of 4. But \( \frac{1}{3} \) of 10 is not a whole number because 10 is not a multiple of 3.

The key is divisibility: \( \frac{1}{n} \) of a number gives a whole number only when that number is a multiple of \( n \). Students who only practise with “friendly” numbers may develop the misconception that fractions of amounts always work out neatly.

True for proper fractions (where the numerator is less than the denominator): \( \frac{3}{4} \) of 20 = 15, and \( 15 < 20 \). But false for improper fractions: \( \frac{5}{4} \) of 20 = 25, and \( 25 > 20 \). And exactly equal when the fraction is 1: \( \frac{4}{4} \) of 20 = 20.

Many students believe fractions always represent “a part” of something, so the result must be smaller. But fractions like \( \frac{5}{4} \) or \( \frac{3}{2} \) represent more than one whole, so the result exceeds the original. The word “fraction” can reinforce this misconception — it comes from the Latin for “broken,” suggesting something smaller.

“Of” means “multiply” in mathematics. So \( \frac{3}{4} \) of 20 = \( \frac{3}{4} \times 20 = 15 \). This works for every fraction and every number: \( \frac{a}{b} \) of \( n \) = \( \frac{a}{b} \times n = \frac{a \times n}{b} \). The “Divide by the bottom, times by the top” method is just one way of carrying out this multiplication.

Students often see “finding a fraction of” as a separate skill from “multiplying by a fraction.” Recognising they are the same operation is a key conceptual shift that connects primary fraction work to secondary-level proportional reasoning.

\( \frac{1}{3} \) is a larger fraction than \( \frac{1}{4} \) (thirds are bigger pieces than quarters), so \( \frac{1}{3} \) of any positive amount is always larger, not smaller. For example, \( \frac{1}{3} \) of 12 = 4 and \( \frac{1}{4} \) of 12 = 3, and \( 4 > 3 \).

This targets the common “bigger denominator means bigger result” misconception — students see \( 4 > 3 \) in the denominators and assume \( \frac{1}{4} > \frac{1}{3} \). In fact, the opposite is true: dividing into more pieces makes each piece smaller. Picturing a pizza cut into 3 slices vs 4 slices makes this clear — each third is bigger than each quarter.

The student has only completed the first step — finding \( \frac{1}{5} \) of 40 = 8 — and stopped there. This is the “unit fraction only” misconception: the student treats every fraction-of-an-amount as if it were a unit fraction.

The correct method requires a second step: \( 40 \div 5 = 8 \), and then \( 8 \times 3 = 24 \).

The answer is correct, but the method is dangerously wrong. The student’s rule — “divide by the numerator, then divide by the denominator” — only works for unit fractions, where dividing by 1 in the first step has no effect. But applying this rule to a non-unit fraction fails catastrophically.

For example, applying the same method to \( \frac{3}{4} \) of 24: divide by 3 to get 8, then divide by 4 to get 2. The student would answer 2, but the correct answer is 18.

The student has swapped the roles of the numerator and denominator. They divided by the numerator (2) and multiplied by the denominator (3), when it should be the other way around: divide by the denominator, multiply by the numerator.

A quick sense check reveals the error: \( \frac{2}{3} \) is less than 1, so the result must be less than 18. Getting 27 (which is bigger than 18) should be an immediate red flag.

The student correctly identifies that “of” means multiply, but then multiplies by the numerator only, ignoring the denominator entirely. They calculated \( 3 \times 32 \) instead of \( \frac{3}{4} \times 32 \). The denominator is a critical part of the fraction — it tells you how many equal parts the whole is split into.

The correct answer is: \( \frac{3}{4} \times 32 = \frac{3 \times 32}{4} = \frac{96}{4} = 24 \). Or equivalently, \( 32 \div 4 = 8 \), then \( 8 \times 3 = 24 \).