Expressing One Quantity as a Percentage

Write one amount as a percentage of another: part over whole times 100, same units first, reading the part and whole from data, and amounts linked by a percentage.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 61 real past-paper questions below

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What Expressing as a Percentage Means

You have part of something and the whole of it. You want to say how big the part is, out of 100.

Put the part over the whole. Then multiply by 100.

Formula

percentage = \frac{part}{whole} \times 100

The part is the amount you are asked about. The whole is the total you compare it with.

Key idea

30 out of 40 is $\frac{30}{40} = \frac{3}{4}$ , which is 75%. The whole is always 100%.

A as a Percentage of B

The 2-mark question: write 36 out of 80 as a percentage. The first number is the part, the second is the whole.

Worked example

1
$\frac{36}{80}$ — Put the part on top and the whole underneath. The fraction says “36 out of 80”.
2
$36 \div 80 = 0.45$ — Divide the part by the whole. This gives the part as a decimal of the whole.
3
$0.45 \times 100 = 45%$ — Multiply by 100 to turn the decimal into “out of 100”. That is what a percentage is.

If the answer is not exact, keep every digit and round only at the end. 53 out of 70, to 1 decimal place:

Worked example

1
$\frac{53}{70} \times 100 = 75.714 \dots$ — Part over whole, times 100. Write down all the digits your calculator shows.
2
$\approx 75.7%$ — Round to 1 decimal place, as asked. The next digit is 1, which is less than 5, so the 7 stays.

Common mistake

Swapping the numbers: $\frac{80}{36} \times 100 = 222.2%$ . The number after “of” or “out of” always goes on the bottom.

Key idea

Write the line $\frac{36}{80} \times 100$ in your answer. It earns a mark for working, even if you press a wrong key after it.

Without a Calculator

Some papers do not allow a calculator. Then make the fraction simpler first, and turn it into a number out of 100.

Write 21 as a percentage of 60.

Worked example

1
$\frac{21}{60}$ — Part over whole, as always. Both 21 and 60 divide by 3.
2
$= \frac{7}{20}$ — Divide the top and the bottom by 3: 21 ÷ 3 = 7 and 60 ÷ 3 = 20. Smaller numbers are easier to work with.
3
$= \frac{7 \times 5}{20 \times 5} = \frac{35}{100}$ — 20 × 5 = 100, so multiply the top and the bottom by 5. Doing the same to both keeps the value the same.
4
$= 35%$ — 35 out of 100 is 35%.

If the bottom cannot become 100, divide instead. 14 out of 16:

Worked example

1
$\frac{14}{16} = \frac{7}{8}$ — Divide the top and the bottom by 2. 8 does not go into 100 exactly, so the bottom cannot become 100.
2
$7 \div 8 = 0.875$ — Short division: 8 into 7.000 gives 0.875.
3
$0.875 \times 100 = 87.5%$ — Multiply by 100: the digits move two places to the left.

Key idea

Know that $\frac{1}{8} = 12.5%$ . Then $\frac{7}{8} = 7 \times 12.5% = 87.5%$ . For more on this, see Introduction to Percentages.

Same Units First

The 2-mark question often mixes units: write 45 minutes as a percentage of 2 hours. Change both to the same unit first.

Worked example

1
$2 hours = 120 minutes$ — 1 hour is 60 minutes, so 2 hours is 2 × 60 = 120 minutes. Now both amounts are in minutes.
2
$\frac{45}{120} \times 100$ — Put the part, 45 minutes, over the whole, 120 minutes. Then multiply by 100.
3
$= 37.5%$ — 45 ÷ 120 = 0.375, and 0.375 × 100 = 37.5.

Common mistake

Working out $\frac{45}{2} \times 100$ . That gives 2250%, far too big. Minutes and hours do not mix.

Rule

Minutes with minutes, grams with grams, cents with cents. Use the smaller unit so you avoid decimals.

Amounts Linked by a Percentage

The 3 to 5-mark question: Ali, Bea and Cal share $570. Bea gets 60% of what Ali gets. Cal gets 25% more than Ali. How much does Ali get?

Call Ali's amount $a$ . Write the others as a multiplier times $a$ .

Worked example

1
$Bea = 0.6 a$ — Bea gets 60% of Ali's amount. 60% = 60 ÷ 100 = 0.6, so Bea is 0.6 times a.
2
$Cal = 1.25 a$ — Cal gets all of Ali's amount (100%) plus 25% more. That is 125%, which is 1.25 as a decimal.
3
$a + 0.6 a + 1.25 a = 570$ — Ali, Bea and Cal share all the money, so their three amounts add up to 570.
4
$2.85 a = 570$ — Collect the a's: 1 + 0.6 + 1.25 = 2.85.
5
$a = 570 \div 2.85 = 200$ — Divide both sides by 2.85 to get a on its own. Ali gets $200.

Common mistake

Writing Cal as $0.25 a$ . That is only the extra. 25% more than Ali means $1.25 a$ .

Key idea

Check: Bea gets $120 and Cal gets $250. Then 200 + 120 + 250 = 570.

Read the Part and Whole From Data

The 2-mark question gives a table: 18 students walk, 27 take the bus, 15 come by car. What percentage walk?

Worked example

1
$18 + 27 + 15 = 60$ — Add every group in the table. Together they are the whole: everyone who was asked.
2
$\frac{18}{60} \times 100 = 30%$ — The walkers are the part. 18 ÷ 60 = 0.3, and 0.3 × 100 = 30.

Sometimes the part is several groups. Journey times of some students: 0–10 min: 9, 10–20 min: 17, 20–30 min: 10, more than 30 min: 4. What percentage take more than 20 minutes?

Worked example

1
$10 + 4 = 14$ — Only the groups above 20 minutes fit: 20–30 min and more than 30 min. Together they are the part.
2
$9 + 17 + 10 + 4 = 40$ — The whole is every student in the table, not just the ones that fit.
3
$\frac{14}{40} \times 100 = 35%$ — Part over whole: 14 ÷ 40 = 0.35, and 0.35 × 100 = 35.

Common mistake

Dividing by one other group, such as 18 over 27. The whole is everyone in the table.

Key idea

In a two-way table, the part is one cell or one row total. The whole is the grand total in the corner, unless the question names a smaller group, such as “what percentage of the girls…”.

From a Ratio

The 1 or 2-mark question gives a ratio and asks what percentage one share is. The whole is all the ratio parts added together.

Paint is mixed red : white : blue = 3 : 9 : 4. What percentage of the paint is white?

Worked example

1
$3 + 9 + 4 = 16$ — Add all the ratio numbers. The paint is 16 equal parts, so 16 is the whole.
2
$\frac{9}{16}$ — White is 9 of those 16 parts, so 9 is the part.
3
$\frac{9}{16} \times 100 = 56.25%$ — 9 ÷ 16 = 0.5625, then multiply by 100.

One share as a percentage of another share: the whole is the share after “of”. Ann : Ben : Cy = 3 : 4 : 6. Write Ben's share as a percentage of Cy's.

Worked example

1
$\frac{4}{6} \times 100$ — “Of Cy's” makes Cy's 6 parts the whole. Ben's 4 parts are the part. Ann is not needed.
2
$= 66.666 \dots \approx 66.7%$ — 4 ÷ 6 × 100 = 66.666…, which is 66.7% to 3 significant figures.

Common mistake

Dividing by one other part, like 9 over 4 for the paint. For a percentage of the whole mix, the bottom is all the parts added: 16.

How Much More, as a Percentage

The 4-mark question: Ali paid $56 and Ben paid $50. How much more did Ali pay, as a percentage of what Ben paid?

Worked example

1
$56 - 50 = 6$ — Ali paid $6 more. This difference is the part the question asks about.
2
$\frac{6}{50} \times 100 = 12%$ — “As a percentage of what Ben paid” makes Ben's $50 the whole. 6 ÷ 50 = 0.12, and 0.12 × 100 = 12.

Common mistake

Dividing by 56. The words “as a percentage of what Ben paid” tell you the whole is 50.

Key idea

Another way: 56 ÷ 50 = 1.12, so Ali paid 112% of Ben's amount. Take away the 100% to get 12% more. This is the same sum as a percentage increase. See Percentage Increases and Decreases.

Find the Amounts First

The 4-mark question hides the part or the whole. Work out the missing amounts first. Then use part over whole.

A club has 120 members. Each plays one of tennis, golf, squash or chess. 18 play tennis and 40% play golf. Write the number who play squash or chess as a percentage of the number who play tennis or golf.

Worked example

1
$0.4 \times 120 = 48$ — Golf is given as a percentage, so change it to a number of members first. 40% = 0.4, so 48 play golf.
2
$18 + 48 = 66$ — Tennis or golf: 18 + 48 = 66. This is the whole, because the question says “of the number who play tennis or golf”.
3
$120 - 66 = 54$ — Everyone else plays squash or chess, so take 66 from all 120 members. This is the part.
4
$\frac{54}{66} \times 100 = 81.818 \dots$ — Part over whole, times 100. Keep all the digits for now.
5
$\approx 81.8%$ — No accuracy was asked for, so give 3 significant figures.

Key idea

Write each amount with a label, like “golf = 48”. Each correct amount earns a mark for working, even if the last line goes wrong.

Which Method? Read the Question

Find the key words. They tell you which number is the part and which is the whole.

You see	You do
A as a percentage of B	A over B, times 100
no calculator	simplify, then make the bottom 100
minutes and hours, grams and kg	same units first
a table of groups	add them all for the whole
a ratio, like 3 : 9 : 4	share over all the parts added
how much more, as a percentage of	difference over that amount
a percentage or total hidden in the question	find the amounts first
gets 60% of, gets 25% more than	write each share with a multiplier
find 15% of an amount	Calculating a Percentage of a Quantity
percentage profit or loss	Profit and Loss
the price before the change	Reverse Percentages

Key idea

The whole comes after the word "of". If the part is smaller than the whole, your answer must be under 100%. New to per cent? See Introduction to Percentages.

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