Percentage of an Amount

Break it into benchmarks: 10% of 60 = 6, so 30% = 18. Then 5% of 60 = 3 (half of 10%). So 35% = 18 + 3 = 21. Alternatively, \( \frac{35}{100} \times 60 = \frac{2100}{100} = 21 \).

This question pushes students beyond the “easy” percentages (10%, 25%, 50%) and tests whether they can combine benchmarks fluently. Students who can only find 10% or 50% will struggle — those who can build from known percentages demonstrate real understanding of the multiplicative structure.

Calculate both: 40% of 25 = \( \frac{40}{100} \times 25 = 10 \). And 25% of 40 = \( \frac{25}{100} \times 40 = 10 \). They’re equal. This always works because \( x\% \) of \( y = \frac{xy}{100} = y\% \) of \( x \) — multiplication is commutative, so swapping the percentage and the amount doesn’t change the product.

Think about this: If I have 40% of a pizza, and you have 25% of a different pizza, do we have the same amount of food? No! So why does this work here? It forces us to realise that the base amounts compensate for the percentage sizes, which is the heart of the \( x\% \text{ of } y = y\% \text{ of } x \) proof.

10% means one tenth, so 10% of 45 = 45 ÷ 10 = 4.5 exactly. The answer is not 4 (that would be rounding down) and not 5 (that would be rounding up). Percentages don’t have to give whole number answers — fractions and decimals are perfectly valid results.

Many students expect a “clean” answer and will round 4.5 to either 4 or 5, not realising that the decimal IS the answer. This is the “percentages must give whole numbers” misconception. It often surfaces when students check their work and dismiss a correct decimal answer as a sign they’ve made an error.

20% means 20 out of 100, which simplifies to \( \frac{1}{5} \) (divide numerator and denominator by 20). So 20% of any amount = \( \frac{1}{5} \) of that amount = the amount ÷ 5. For example, 20% of 60 = 60 ÷ 5 = 12. You can verify: \( \frac{20}{100} \times 60 = \frac{1200}{100} = 12 \).

This connects percentages to fractions — a link many students miss entirely. Similar connections include: 10% = \( \frac{1}{10} \), 25% = \( \frac{1}{4} \), 50% = \( \frac{1}{2} \), 75% = \( \frac{3}{4} \). Students who see these links can calculate percentages mentally by using the equivalent fraction division — a crucial skill for building percentage fluency.

Example: 20 (25% of 20 = 5)

Another: 48 (25% of 48 = 12)

Creative: 4 (25% of 4 = 1) — the smallest positive whole number that works. Or 400 (25% of 400 = 100) — a large amount giving a “round” result. Students rarely consider very small or very large values.

Trap: 25 (25% of 25 = 6.25, not a whole number) — a student might think “25% of 25 must work because the numbers match.” But 25% means dividing by 4, and 25 ÷ 4 = 6.25. Only multiples of 4 give whole numbers.

Example: 10% of 120 = 12

Another: 50% of 24 = 12

Creative: 1% of 1200 = 12 — using an unusually small percentage. Or 300% of 4 = 12 — using a percentage greater than 100%. Students rarely think beyond the range 1%–100%.

Trap: 12% of 10 = 1.2, not 12 — a student might see the “12” in the percentage and assume the answer is 12. But 12% of 10 means \( \frac{12}{100} \times 10 \), which is 1.2. The percentage doesn’t directly become the answer unless the amount is 100.

Example: 7 (50% of 7 = 3.5)

Another: 15 (50% of 15 = 7.5)

Creative: 1 (50% of 1 = 0.5) — the simplest possible example. Or 0.6 (50% of 0.6 = 0.3) — using a decimal starting amount, which students rarely consider.

Trap: 30 (50% of 30 = 15, which IS a whole number) — a student might pick 30 because “it’s not a round number like 100” and assume the result won’t be whole. But 50% of any even whole number is always a whole number. It’s the odd whole numbers that produce non-integer results.

Example: 200% of 30 = 60 (double the original)

Another: 150% of 80 = 120

Creative: 101% of 50 = 50.5 — only just exceeds the original. Or 1000% of 3 = 30 — a very large percentage of a small amount.

Trap: 99% of 100 = 99 (still less than 100, not greater) — a student might think “99% is nearly everything, so it must be more.” But any percentage below 100% will always give a result less than the original. You need to cross the 100% threshold.

10% means 10 out of 100, which simplifies to \( \frac{1}{10} \). So 10% of any number = that number × \( \frac{1}{10} \) = that number ÷ 10. This holds universally — for whole numbers, decimals, fractions, and even negative numbers.

Students often confuse this with dividing by 100 (which gives 1%, not 10%). Confirming this rule is essential because 10% is the most common “building block” percentage — once students can reliably find 10%, they can build 5%, 20%, 30%, and so on.

True when the percentage is less than 100%: e.g. 30% of 80 = 24 < 80. Equal when the percentage is exactly 100%: 100% of 80 = 80. False when the percentage exceeds 100%: 150% of 80 = 120 > 80.

This targets the common belief that “finding a percentage” always makes things smaller — possibly because most textbook questions use percentages under 100%. Students need to understand that a percentage simply scales an amount: below 100% shrinks it, above 100% enlarges it.

Dividing by 20 gives \( \frac{1}{20} \) = 5%, not 20%. The correct method for 20% is dividing by 5 (since 20% = \( \frac{1}{5} \)). For example, 20% of 100: dividing by 20 gives 5 (that’s 5%, wrong), whereas dividing by 5 gives 20 (that’s 20%, correct).

The “divide by the percentage number” misconception is extremely common and feels logical to students. It happens to work for 10% (divide by 10) because 10% = \( \frac{1}{10} \), but that’s a coincidence — 10 is both the percentage number and the denominator. The general rule is: \( x\% = \frac{x}{100} \), and you multiply by that fraction.

True for even numbers: 50% of 8 = 4. False for odd numbers: 50% of 7 = 3.5. Also false for many decimals: 50% of 3.3 = 1.65.

Students often assume percentages of whole numbers must produce whole numbers. This is especially common with 50% because “half” feels clean. But half of an odd number is always a decimal — and recognising this is important for building percentage fluency.

This is mathematically sound and tests the distributive property of multiplication. Since 15% is made up of 10% + 5%, and 5% is exactly half of 10%, finding 10%, halving it, and adding the two parts will always yield 15%.

Students who understand this have moved beyond relying purely on rigid formulas and are showing true fluency with compound percentages.

The student has confused 10% with 1%. Dividing by 100 finds 1% of a number, not 10%. This is the “divide by 100” misconception — students associate any percentage calculation with dividing by 100 because they’ve learned that “percent means out of 100.”

The correct method: 10% = \( \frac{10}{100} = \frac{1}{10} \), so you divide by 10. 250 ÷ 10 = 25. The student’s answer of 2.5 is actually 1% of 250, which is ten times too small. A quick sense check helps: 10% should be a noticeable chunk of the original — 2.5 out of 250 is tiny, while 25 out of 250 feels right (it’s a tenth).

The answer is correct, but the reasoning is dangerously wrong. The student’s rule — “remove the zero” — only works when the number happens to end in zero. It fails completely for numbers like 45, 73, or 128, where there is no zero to remove.

The correct reasoning is: 10% means dividing by 10. For 30 ÷ 10 = 3, this happens to look like “removing the zero.” But 10% of 45 = 4.5 (no zero to remove), and 10% of 7 = 0.7 (again, no zero). Students who rely on “remove the zero” will be completely stuck when the number doesn’t cooperate, because they haven’t understood that 10% means ÷10.

The student is using the “divide by the percentage number” misconception — believing that to find \( x\% \), you divide by \( x \). This rule is simply wrong. Dividing 50 by 30 gives \( \frac{50}{30} \approx 1.67 \), which bears no relation to 30% of 50.

The correct method: 30% of 50 = \( \frac{30}{100} \times 50 = 0.3 \times 50 = 15 \). Alternatively, 10% of 50 = 5, so 30% = 3 × 5 = 15. The student’s answer of 1.67 is far too small — a quick sense check shows that 30% should be nearly a third of 50, which is about 16 or 17, not 1.67.

The student has correctly recalled that “of means multiply” but has forgotten to convert the percentage to a fraction or decimal first. This is the “missing conversion” misconception — the student multiplies by 40 instead of by 0.4 (or \( \frac{40}{100} \)).

The correct calculation: 40% of 70 = \( \frac{40}{100} \times 70 = 0.4 \times 70 = 28 \). The student’s answer of 2800 is 100 times too large. A quick sense check makes the error obvious: 40% is less than half, so the answer must be less than 35. An answer of 2800 — many times larger than the original 70 — is clearly impossible.

The student has shifted from multiplicative reasoning to additive reasoning. This is the “additive misconception”. You cannot add raw numbers to percentages; you must find 5% of 40 (which is 2) and add that to the 10% (which is 4) to get 6.

This reveals a fragile understanding of what percentages represent. The student knows 15 is 5 more than 10, but forgets that “5%” means “5 parts per 100 of the original amount,” not simply the number 5.