Overview
What you’ll learn
- Convert between fractions, decimals and percentages.
- Express one quantity as a percentage of another, with matching units.
- Find a percentage of a quantity, and increase or decrease a quantity by a given percentage.
- Calculate percentage change, stating whether it is an increase or a decrease and what it is compared with.
- Solve reverse percentage problems to find the original amount.
- Solve real-world problems involving discount, GST and simple interest.
In 30 seconds
- 1Per cent means out of 100: 45% = 45/100 = 0.45.
- 2Percentage of a quantity: 24% of 75 = 75 × 24/100 — multiply by the fraction.
- 3Percentage change compares with the original: change ÷ original × 100%.
- 4An increase of 20% makes the new value 120% of the old; a 20% discount leaves 80%.
- 5Reverse percentages divide: sale price ÷ 0.8 undoes a 20% discount — never add 20% back onto the sale price.
- 6Simple interest each year = principal × rate; total interest = P × r% × n years.
Notes · 1 of 5
Fractions, decimals and percentages
Three costumes for the same quantity — pick whichever makes the question easiest.
| Direction | How | Example |
|---|---|---|
| Fraction → percentage | Multiply by 100% | 3/8 = 37.5% |
| Decimal → percentage | Multiply by 100 (shift two places right) | 0.375 → 37.5% |
| Percentage → decimal | Divide by 100 (shift two places left) | 45% → 0.45 |
| Percentage → fraction | Write over 100 and simplify | 45% = 9/20 |
Comparing quantities is easiest when everything wears the same costume. To order 0.62, 3/5 and 58%, convert all to decimals: 0.62, 0.6 and 0.58 — now the order is obvious.
Notes · 2 of 5
Finding a percentage of a quantity
Turn the percentage into a fraction or decimal, then multiply.
- 1Write the multiplier
24% becomes 24/100 or 0.24.
- 2Multiply
0.24 × 75 = 18.
- 3Sanity-check
The answer (18) is less than 75, as it should be for a part under half.
Friendly percentages have shortcuts worth knowing: 10% is a tenth (divide by 10), 5% is half of 10%, and 1% is a hundredth. So 35% of 64 = 3 × 10% (19.2) + ½ × 10% (3.2) = 22.4 — often faster than one long multiplication, and it catches typos.
Notes · 3 of 5
Percentage increase and decrease
Add or subtract the percentage, but always compare the change with the original value.
A percentage change is always measured against the original value: change ÷ original × 100%. If a price rises from $50 to $62, the change is $12 and the original is $50, so the rise is 12/50 × 100% = 24%. Comparing with the new price (12/62 ≈ 19.4%) is the classic slip.
| Change | New value is | Multiplier |
|---|---|---|
| increase by 15% | 115% of the original | × 1.15 |
| decrease by 30% | 70% of the original | × 0.70 |
| discount 20% | you pay 80% | × 0.80 |
Notes · 4 of 5
Reverse percentages: finding the original
When you know the after value, divide by the multiplier — never “add the percentage back”.
A bag sells at $96 after a 20% discount. The $96 is 80% of the original, so the original is 96 ÷ 0.8 = $120. Adding 20% of $96 ($19.20) to get $115.20 is wrong — that 20% would be measured from the sale price, not the original.
| Given | The given value is | Original = |
|---|---|---|
| $96 after a 20% discount | 80% of the original | 96 ÷ 0.8 |
| $75 after a 25% increase | 125% of the original | 75 ÷ 1.25 |
| $109 including 9% GST | 109% of the pre-GST price | 109 ÷ 1.09 |
Notes · 5 of 5
Discount, GST and simple interest
Three real-world appearances of percentages, each with its own vocabulary.
Shopping
Discount
Sale price = marked price − discount. A 20% discount means you pay 80%.
Tax
GST
GST is added on top of the pre-GST price: total = price × 1.09 at 9% GST.
Banking
Simple interest
Interest = principal × rate × years. The total amount = principal + interest.
With simple interest, the rate applies to the original principal every year: $800 at 3% per year earns $24 each year, so 2 years earn $48 and the account holds $848. Keeping interest (the earnings) separate from amount (principal + interest) is a marked distinction in exams.
Revise
Key terms
The words examiners expect you to use precisely.
- Percentage
- A fraction out of 100; 45% means 45 out of every 100.
- Multiplier
- The decimal form of a percentage change: ×1.15 increases by 15%, ×0.8 decreases by 20%.
- Percentage change
- The change divided by the original value, written as a percentage.
- Original value
- The value before a percentage change is applied; the base the percentage refers to.
- Reverse percentage
- Finding the original value from the after-the-change value, by dividing by the multiplier.
- Discount
- A reduction from the marked price, usually given as a percentage.
Show 4 more key termsShow fewer key terms
- GST
- Goods and Services Tax, added on top of a pre-tax price (9% in Singapore at the time of writing).
- Principal
- The original sum of money deposited or borrowed.
- Simple interest
- Interest computed each year on the original principal only: P × r% × number of years.
- Amount
- The principal plus the interest earned; the total in the account.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Comparing a change with the new value instead of the original, e.g. 12/62 for a rise from 50 to 62.
Percentage change is defined against the value before the change.
Instead: Percentage change = change ÷ original × 100% = 12/50 × 100% = 24%.
Mistake: Undoing a 20% discount by adding 20% back: 96 × 1.2 = $115.20.
After the discount, the price is 80% of the original; 20% of 96 is measured from the wrong base.
Instead: Original = 96 ÷ 0.8 = $120.
Show 8 more mistakesShow fewer mistakes
Mistake: Reading “increase by 200%” as “new value is twice the old”.
A 200% increase adds twice the original on top, tripling it.
Instead: Increase by 200% → new value is 300% of the old (×3).
Mistake: Treating two successive changes as one: “−20% then +20% returns to the start”.
The second multiplier acts on the already-changed value: 90 × 0.8 × 1.2 = 86.4.
Instead: Multiply the multipliers: 0.8 × 1.2 = 0.96 — a 4% net decrease.
Mistake: Reporting the interest as the “amount” in the account.
The amount includes the principal as well as the interest earned.
Instead: Amount = principal + interest; $800 + $48 = $848.
Mistake: Adding GST of 9% by computing 9% of the GST-inclusive price.
The 9% applies to the pre-GST price, so an inclusive $109 contains 9% of $100, not of $109.
Instead: Pre-GST price = total ÷ 1.09.
Mistake: Forgetting to match units before expressing one quantity as a percentage of another, e.g. $4.50 of $180 cents.
The division only makes sense when both quantities use the same unit.
Instead: Convert first ($4.50 and $18 are both dollars → 4.50/18 = 25%).
Mistake: Finding 5% of a quantity by dividing by 5, e.g. writing 5% of $240 as $48.
One fifth is 20%, not 5%. The per cent sign fixes the denominator at 100.
Instead: 5% = 5/100 = 0.05, so 5% of 240 = 240 × 0.05 = $12.
Mistake: Adding percentages that refer to different wholes, e.g. “20% of the boys plus 30% of the girls is 50% of the class”.
The two percentages use different bases, so they cannot be added directly.
Instead: Convert each percentage to an actual count first, then combine the counts.
Mistake: Computing simple interest without the years, e.g. $1 200 at 3% per year for 2 years “earns $36”.
Each year brings a fresh 3% of the principal, so the number of years must multiply in.
Instead: Interest = P × r% × n = 1 200 × 0.03 × 2 = $72.
Check yourself
Quick quiz
18 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Find a percentage of a quantity
What is 45% of 80?
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Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 6Foundation
- 5Standard
- 4Challenging
15 questions worth 51 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 6 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Chapter 6.
Mark scheme
PROChapter 6 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
- Term 2Coming soonWeighted Assessment 2 (WA2)
- Term 4Coming soonEnd-of-Year Examination (EOY)
Questions
Frequently asked questions
What is Sec 1 Math Chapter 6 about?
Chapter 6 covers percentages: converting between fractions, decimals and percentages, finding a percentage of a quantity, percentage increase and decrease, reverse percentages, and real-world contexts such as discount, GST and simple interest.
How do I find the original price after a percentage change?
Divide by the multiplier. A price that rose 25% is 125% of the original, so original = new ÷ 1.25. A price after a 20% discount is 80% of the original, so original = sale price ÷ 0.8. Adding the percentage back to the new value gives the wrong base.
What is the difference between simple interest and the amount?
Simple interest is the money earned: principal × rate × years. The amount is the principal plus the interest — the total in the account. Examiners mark the two as different answers.
Does a 20% decrease followed by a 20% increase return to the original value?
No. The multipliers compose: 0.8 × 1.2 = 0.96, so the final value is 96% of the original — a 4% net decrease, because the increase acts on a smaller base.
When is this chapter tested?
Percentage work usually appears in Weighted Assessment 2 and the end-of-year exam, often inside word problems. Every school sets its own scope, so check your school’s assessment outline.
Up next
Chapter 7: Ratio, Rate and Speed