O-Level E-Math notes · Chapter 3

Percentage: O-Level E-Math Notes

Percentages are in almost every real-life question: discounts, GST, profit, growth and rates. This chapter shows how to find, compare, increase, decrease and reverse them, and how to avoid the one mistake that causes most lost marks: choosing the wrong base.

Your outcome tracker0 / 9

Let's go! Tick an outcome once you can pass its exam check. Saved on this device.

Chapter
3

Figure 1 · Number of students choosing each CCA type (data for Example 3.7)

5 outcomes · 35 worked examples · every step shown ✦

Percentages and fractions

A percentage is a fraction whose denominator is one hundred.

One quantity as a percentage

percentage = (part ÷ whole) × 100%

Comparing by percentage

Change the base and the percentage changes.

By the end of this chapter

Express one quantity as a percentage of another, and compare two quantities by percentage.
Work with percentages greater than one hundred.
Increase or decrease a quantity by a given percentage, find a percentage change, and use percentage points correctly.
Find the original value from the final value and the rate (reverse percentages), including GST and discount problems.

Start here

Maths In Real Life

Three real-looking situations, each with a different percentage story. All the prices, rates and counts are made-up practice data.

$A shop window with a rack of trainers, a price tag on one pair and a till at the counter; bars drawn over the picture show the marked price, the price after discount and the price after GST$

Maths In Real Life 1

The sneaker sale

A till receipt applies several percentage changes in a row. Each one has its own base.

A pair of trainers has a marked price of $\text{\textdollar}$ 140. The shop gives a 30% discount, and then GST at 9% is added at the till. Give money answers to the nearest cent.

(a)Find the amount paid at the till.
(b)Another pair costs $\text{\textdollar}$ 152.60 at the till after the same discount and GST. Find its marked price.
(c)The till price is what percentage of the marked price?

Show full solutionHide solution

(a)

after discount: 140 × 0.7 =

\text{\textdollar}

98.00

100% − 30% = 70%

after GST: 98.00 × 1.09 =

\text{\textdollar}

106.82

(b)

till price = marked × 0.7 × 1.09 = marked × 0.763

marked = 152.6 ÷ 0.763 =

\text{\textdollar}

200.00

(c)

0.763 × 100% = 76.3%

Final answer

(a)

\text{\textdollar}

106.82

(b)

\text{\textdollar}

200.00

$A busy hawker stall with plates of noodles on the counter and a cost and a selling price drawn over the picture as two bars$

Maths In Real Life 2

The hawker stall margin

A stall owner watches two numbers: what a plate costs to make, and what it sells for. A percentage profit compares them.

A hawker stall spends $\text{\textdollar}$ 2.40 on ingredients for each plate of noodles and sells each plate for $\text{\textdollar}$ 3.60. Give percentages to three significant figures if they are not exact.

(a)Find the percentage profit on cost.
(b)The cost of ingredients rises by 15% but the selling price stays the same. Find the new percentage profit.
(c)What percentage increase in the selling price would give back the original percentage profit?

Show full solutionHide solution

(a)

(3.6 − 2.4)/2.4 × 100 = 50%

profit uses the cost as the base

(b)

new cost = 2.4 × 1.15 =

\text{\textdollar}

2.76

(3.6 − 2.76)/2.76 × 100 ≈ 30.4%

(c)

needed price = 2.76 × 1.5 =

\text{\textdollar}

4.14

(4.14 − 3.6)/3.6 × 100 = 15%

Final answer

(a)50%

(b)30.4%

$A bright community library with readers at tables, and three bars drawn over the picture for the number of visitors in three months$

Maths In Real Life 3

Visitors at the community library

Bar charts of visitors let you compare months by percentage change instead of by raw counts.

A community library counted its visitors: 1200 in March, 1500 in April and 1350 in May.

(a)Find the percentage increase from March to April.
(b)Find the percentage decrease from April to May.
(c)Express May's visitors as a percentage of March's. Is this over one hundred?

Show full solutionHide solution

(a)

(1500 − 1200)/1200 × 100 = 25%

(b)

(1500 − 1350)/1500 × 100 = 10%

the base is April, the old value

(c)

1350/1200 × 100 = 112.5%

Final answer

(a)25%

(b)10%

Supports K210-N3.1

Percentages as fractions, decimals and parts of a quantity

Quick refresher (supports K310-N3). The word percent means out of 100. So 25% means 25/100, which simplifies to 1/4, and as a decimal it is 0.25. Everything in this chapter is built on that one idea.

The Core Idea

A percentage is a fraction whose denominator is one hundred.

To turn a percentage into a decimal, divide by 100. To turn a decimal or fraction into a percentage, multiply by 100. To find a percentage of a quantity, multiply the quantity by the percentage written over 100.

Example 3.1

Percentage, fraction, decimal

Convert between percentages, fractions and decimals.

(i)Write 35% as a fraction in its simplest form and as a decimal.
(ii)Write 3/8 as a decimal and as a percentage.

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplify

(i)

35% = 35/100

per cent means out of one hundred

= 7/20

divide top and bottom by their highest common factor

= 0.35

as a decimal: divide by one hundred, so the point moves two places left

(ii)

3 ÷ 8 = 0.375

divide the top by the bottom

0.375 × 100 = 37.5%

multiply by one hundred to turn a decimal into a percentage

Final answer

(i)35% = 7/20 = 0.35

(ii)0.375 = 37.5%

Example 3.2

Finding a percentage of a quantity

Find the quantities asked for.

(i)Find 12.5% of 64 m.
(ii)In a class of 40 students, 35% walk to school. How many students do not walk to school?

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplify

(i)

12.5% of 64 = 12.5/100 × 64

of means multiply; write the percentage over one hundred

= 8

(ii)

Walk: 35/100 × 40 = 14

find the walkers first

Not walking: 40 − 14 = 26

subtract from the whole class

Check: 65% of 40 = 26

the other percentage gives the same answer

Final answer

(i)8 m

(ii)26 students

SGSchoolKaki Tip

A quick check: 10% of a number is that number divided by 10, and 50% is half of it. Use these to see whether an answer is sensible.

Example 3.3

Ordering mixed forms

Arrange these four numbers in ascending order (smallest first), by changing them all into decimals. 0.45, 43%, 3/8, 2/5

Show solutionHide solution

Colour keyexpand, rewrite, substitute or solve

43% = 0.43, 3/8 = 0.375, 2/5 = 0.4

the first number is already a decimal; write each of the others as a decimal so they can be compared fairly

0.375 < 0.4 < 0.43 < 0.45

order the decimals

Final answer

3/8 < 2/5 < 43% < 0.45

Think About It

A shop says "50% off" on a price, and later "50% off the new price". Is that the same as 100% off?

Show the worked answerHide the worked answer

Colour keyadd, subtract, multiply, divide or simplify

start with 100: the first 50% off gives 100 × 0.5 = 50

multiplier for 50% off is 0.5

the second 50% off is taken from 50: 50 × 0.5 = 25

a percentage of what? of the new, smaller price

the item still costs 25, so the total reduction is 75%, not 100%

100 − 25 = 75

Final answerNo. The second 50% is taken from the smaller, already reduced price, so the item is not free. Always ask: a percentage of what?

Links To

Want a free overview of number skills first? See the matching part of the free O-Level maths guide

Supports N3.1

Expressing one quantity as a percentage of another

To say how big a part is compared with a whole, write it as a fraction of the whole and then change that fraction into a percentage.

The Core Idea

percentage = (part ÷ whole) × 100%

Both quantities must be in the same unit. The whole is the number after the word of.

Example 3.4

Same units

Express 14 as a percentage of 40.

Show solutionHide solution

14/40 × 100%

part ÷ whole, then × one hundred

= 35%

Final answer

35%

Example 3.5

Different units

The two quantities are in different units, so change them to the same unit first. A piece of ribbon of 45 cm is taken from a roll of ribbon of 3 m. Express the first as a percentage of the second.

Show solutionHide solution

3 m = 300 cm

use one unit for both lengths

45/300 × 100% = 15%

part ÷ whole × one hundred

Final answer

15%

SGSchoolKaki Tip

On a calculator key the division first, then multiply: (part ÷ whole) × 100. Do not key part ÷ (whole × 100). If your answer is less than 1%, check you did not forget the × 100.

Example 3.6

Answer needs rounding

Express 17 as a percentage of 60. Give your answer correct to three significant figures.

Show solutionHide solution

Colour keyround, estimate or compare

17/60 × 100%

part ÷ whole, then × one hundred

≈ 28.333%

carry more figures than the answer needs

≈ 28.3%

round once, at the end

Final answer

28.3%

Think About It

In Example 3.6 the exact answer goes on for ever. Why should we not round the working line to 28 or 28.3 before the end?

Show the worked answerHide the worked answer

Colour keyround, estimate or compare

17/60 × 100 ≈ 28.333%

the exact working value

using 28%: 28% of 60 = 16.8

rounded too early, the check does not give 17

using 28.3%: 28.3% of 60 = 16.98

closer, but still not 17

so keep the extra figures and round once at the end

to three significant figures

Final answerRounding early can make the last digit wrong. Keep extra figures in the working and round once, at the end, to the accuracy the question gives (three significant figures unless stated otherwise).

Number of students choosing each CCA type (data for Example 3.7)

Example 3.7

Shares of a total

Each of the students in a survey chose the type of CCA they like best. The graph shows how many chose each type.

(i)Find the percentage for each category.
(ii)What percentage chose sports or clubs?

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplify

(i)

36/80 × 100% = 45%

sports

20/80 × 100% = 25%

music

16/80 × 100% = 20%

clubs

8/80 × 100% = 10%

others

Total: 45 + 25 + 20 + 10 = 100%

a good check: all the shares must add to one hundred percent

(ii)

45% + 20% = 65%

add the two shares

Final answer

(i)sports 45%, music 25%, clubs 20%, others 10%

(ii)65%

Supports N3.2

Comparing two quantities by percentage

Comparing is where most percentage mistakes happen, because the answer depends on which quantity is the base. The base is the quantity you compare with. It is always treated as one hundred percent.

The Core Idea

Change the base and the percentage changes.

In the case table, A and B are the same two quantities, but "A as a percentage of B" and "B as a percentage of A" are different answers because the base is different.

Case Table 1: the base is the quantity after of or *than* (answers that are not exact are given to three significant figures)
Statement	Base (100%)	Calculation	Result
A as a percentage of B	B	60/80 × 100	75%
B as a percentage of A	A	80/60 × 100	133%
A is what percentage less than B	B	(80 − 60)/80 × 100	25% less
B is what percentage more than A	A	(80 − 60)/60 × 100	33.3% more

Example 3.8

Two-way comparison

Wei Ling saves $\text{\textdollar}$ 45 a month and Kai Xiang saves $\text{\textdollar}$ 60 a month. Give each answer correct to three significant figures if it is not exact.

(i)Express Wei Ling's saving as a percentage of Kai Xiang's.
(ii)Express Kai Xiang's saving as a percentage of Wei Ling's.

Show solutionHide solution

(i)

45/60 × 100

the base is Kai Xiang's saving, because it follows of

= 75%

(ii)

60/45 × 100

now the base is Wei Ling's saving

≈ 133.33

≈ 133%

Final answer

(i)75%

(ii)133%

Example 3.9

How much more, how much less

Phone P costs $\text{\textdollar}$ 480 and Phone Q costs $\text{\textdollar}$ 420. Give answers to three significant figures where they are not exact.

(i)How much percent cheaper is Phone Q than Phone P?
(ii)How much percent dearer is Phone P than Phone Q?

Show solutionHide solution

(i)

(

\text{\textdollar}

480 −

\text{\textdollar}

420) /

\text{\textdollar}

480 × 100

difference ÷ base × one hundred; the base is

\text{\textdollar}

480

= 12.5%

(ii)

(

\text{\textdollar}

480 −

\text{\textdollar}

420) /

\text{\textdollar}

420 × 100

difference ÷ base × one hundred; the base is

\text{\textdollar}

420

≈ 14.286

≈ 14.3%

Final answer

(i)12.5% cheaper

(ii)14.3% dearer

SGSchoolKaki Tip

"Q is 12.5% cheaper than P" does not mean "P is 12.5% dearer than Q". The first uses P as the base, the second uses Q.

Example 3.10

Which is better?

Team A won 21 of their 28 matches. Team B won 27 of their 40 matches. Which team has the higher winning percentage? Show your working.

Show solutionHide solution

Colour keyround, estimate or compare

Team A: 21/28 × 100 = 75%

different totals, so compare percentages, not counts

Team B: 27/40 × 100 = 67.5%

75% > 67.5%

Final answer

Team A (the higher percentage, 75%)

Example 3.11

A percentage of a percentage

A is 20% of B, and B is 50% of C. Express A as a percentage of C.

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplifyexpand, rewrite, substitute or solve

A = 20/100 × B

write the first statement as a multiplication

B = 50/100 × C

A = 20/100 × 50/100 × C = 0.1 × C

substitute: the base is now the last quantity

0.1 × 100% = 10%

Final answer

10%

Locked · Chapter 3

You've read the free 29%. Unlock the rest of Chapter 3.

Still to read in this chapter:

Percentages above one hundred
Increase and decrease
Percentage change and points
Reverse percentages
Exam-style problems
Quick answers
Formula panel
Exercise

$9 one-time · all 19 chapters · no subscription

Worked examples, formulas and exam checks.
The whole E-Math syllabus.

ONE PRICE19chapters

Premium members get all 19 chapters included.

Notes version 1.0 · Last updated 3 Nov 2026 · Chapter 3 of 19