Sec 3 E-Math Percentage & Interest Notes

Overview

What you’ll learn

Express one quantity as a percentage of another, including values greater than 100%.
Increase or decrease a quantity by a given percentage, and find a percentage change.
Find the original quantity before a known percentage change (reverse percentages).
Combine successive percentage changes into a single multiplier.
Calculate simple and compound interest, including non-annual compounding periods.
Solve personal-finance problems involving GST-inclusive bills, tax brackets, hire purchase and money exchange.

In 30 seconds

1Percentage of a quantity: write the percentage as a multiplier — 15% of 240 means 240 × 0.15.
2To increase by p% multiply by (1 + p/100); to decrease multiply by (1 − p/100).
3Reverse percentages undo a change: divide by the multiplier — a price after 20% increase is original × 1.2, so original = price ÷ 1.2.
4Successive changes multiply: +10% then −10% gives 1.1 × 0.9 = 0.99, a 1% decrease overall.
5Simple interest grows linearly: I = PRT/100. Compound interest multiplies: A = P(1 + R/100)ⁿ from the formula list.
6Match the compounding period: 6% p.a. for 18 months compounded half-yearly means 3 periods at 3%.
7Bills and taxes: add GST by multiplying by 1.09; work tax bracket by bracket, not on the whole income.
8Hire purchase total = deposit + instalments; every extra dollar over the cash price is interest.

Notes · 1 of 5

Percentages of quantities, and comparisons

A percentage is a fraction with a denominator of 100 — turn it into a multiplier and compute.

To express one quantity as a percentage of another, divide and scale by 100: 45 out of 150 is 45/150 × 100% = 30%. The comparison only makes sense with the same units and the same base — "45 is 30% of 150" and "150 is 333⅓% of 45" are both true, with different bases. Percentages above 100% are ordinary results: $250 is 250% of $100, because 2.5 × 100 = 250.

Fraction, decimal, percentage
Fraction	Decimal	Percentage
3/8	0.375	37.5%
9/40	0.225	22.5%
1.2 (as a multiplier)	1.2	120%

Percentage change compares a new value with the original: change ÷ original × 100%. A mark rising from 65 to 78 is a change of 13/65 = +20%. Keep the original in the denominator — dividing by the new value is the most common slip in this topic.

Notes · 2 of 5

Increases, decreases and reverse percentages

Multiply by (1 ± p/100) to apply a change; divide by the multiplier to undo it.

1Apply
Increase 240 by 15%: 240 × 1.15 = 276. Decrease 240 by 15%: 240 × 0.85 = 204.
2Combine
Successive changes multiply: +10% then −10% is 1.10 × 0.90 = 0.99 — a 1% decrease, not zero.
3Undo
A price of $324 is 20% above the original: original = 324 ÷ 1.2 = $270.

Two-stage changes also collapse into one multiplier: successive discounts of 30% and 20% give 0.70 × 0.80 = 0.56, a single 44% discount. Exam questions love asking for the "overall percentage change" — compute the product first, then convert.

Notes · 3 of 5

Simple and compound interest

Simple interest adds the same amount every period; compound interest multiplies every period.

The two interest models on $4 000 at 3% per year for 2 years
Model	Working	Interest	Total
Simple (I = PRT/100)	4000 × 3 × 2 ÷ 100	$240.00	$4 240.00
Compound (A = P(1 + R/100)ⁿ)	4000 × 1.03²	$243.60	$4 243.60

Compound interest earns interest on interest, so it always beats simple interest over multiple periods — here by $3.60. The compound formula sits on the printed formula list: A = P(1 + R/100)ⁿ, where n counts the compounding periods, not years.

Notes · 4 of 5

Bills, GST and tax brackets

Add GST with a multiplier; tax each slab of income at its own rate.

1Read the tariff tiers
A utilities bill charges different rates for different consumption blocks.
2Charge each block
412 kWh: first 200 kWh at $0.2722, next 200 at $0.3253, last 12 at $0.3682 → 54.44 + 65.06 + 4.4184 = $123.9184.
3Add GST
Multiply by 1.09: $123.9184 × 1.09 = $135.07 (nearest cent).

Income tax works in brackets: each rate applies only to the slice of income inside it. A chargeable income of $32 000 taxed at “0% on the first $20 000, 2% on the next $10 000, 3.5% on the next $10 000” owes 2% × 10 000 + 3.5% × 2 000 = $200 + $70 = $270 — never 3.5% of the whole $32 000.

Notes · 5 of 5

Money exchange

An exchange rate is a multiplier between currencies; the direction decides whether you multiply or divide.

A rate quoted as “$1 buys 3.72 units” converts dollars into the foreign currency by multiplication: $850 becomes 850 × 3.72 = 3 162 units. Going home, divide by the same rate: 930 units ÷ 3.72 = $250. Reading the quote in the wrong direction is the whole difficulty of the topic — write the units under every number (“$ × 3.72 units/$”) and the operation picks itself.

Which operation? Follow the units
You have	Quote	Operation	Result
dollars	3.72 units per $	multiply by 3.72	foreign units
foreign units	3.72 units per $	divide by 3.72	dollars
dollars	$1.45 per unit	multiply by 1.45	dollars-equivalent

Revise

Key terms

The words examiners expect you to use precisely.

Percentage: A fraction of 100; x% means x/100.
Multiplier: The decimal form of a percentage change: ×1.15 increases by 15%, ×0.85 decreases by 15%.
Reverse percentage: Finding the original amount before a known percentage change, by dividing by the multiplier.
Percentage change: Change divided by the original value, expressed as a percentage.
Simple interest: Interest computed only on the original principal: I = PRT/100.
Compound interest: Interest computed on the running total, so the amount multiplies each period: A = P(1 + R/100)ⁿ.

Show 6 more key terms

Compounding period: The interval at which interest is added; n in the compound formula counts periods, not years.
GST: Goods and services tax, added to a bill by multiplying the pre-GST amount by (1 + rate).
Tax bracket: A range of chargeable income taxed at its own rate; brackets are computed slice by slice.
Hire purchase: Buying by deposit plus instalments; the excess over the cash price is the interest charged.
Exchange rate: The value of one currency in terms of another; multiply to convert, and keep the units.
Commission: A fee charged on an exchange or sale, often a percentage of the amount handled.

Revise

Common mistakes

Where students most often lose marks in this chapter — and what to do instead.

Mistake: Finding a percentage change with the new value in the denominator.
Percentage change is defined against the original: (78 − 65)/65, not /78.
Instead: Change ÷ original × 100%: the mark rise is +20%, not +16.7%.
Mistake: Undoing a 20% increase by subtracting 20% of the new amount.
The 20% was of the original; the new amount is 120% of it, so division is needed.
Instead: Reverse percentage: original = new ÷ 1.2. For $324 that is $270, not $259.20.

Show 10 more mistakes

Mistake: Assuming +10% then −10% cancels to no change.
The second percentage acts on a different base.
Instead: Multipliers multiply: 1.10 × 0.90 = 0.99 — a 1% decrease.
Mistake: Adding successive discounts of 30% and 20% to claim 50% off.
Percentages off different bases do not add.
Instead: 0.70 × 0.80 = 0.56: a total discount of 44%.
Mistake: Using simple interest when the question says compound (or vice versa).
The two models differ from the second period onwards.
Instead: "Compounded" or "grows at R% per year" signals A = P(1 + R/100)ⁿ; "simple" signals I = PRT/100.
Mistake: Raising 1.06 to the power 1.5 for 18 months of 6% p.a.
Compounding happens at discrete periods, not continuously.
Instead: Half-yearly compounding for 18 months means 3 periods at 3%: multiply by 1.03³.
Mistake: Forgetting to halve the annual rate when compounding half-yearly (or quarter the rate for quarterly).
The rate in the formula is the rate per period.
Instead: 6% p.a. half-yearly = 3% per period; quarterly = 1.5% per period.
Mistake: Rounding money to the nearest dollar mid-working.
Small roundings compound into wrong cents in later parts.
Instead: Carry full accuracy and round to the nearest cent once, at the final answer.
Mistake: Computing income tax on the whole salary at the top rate.
Each bracket rate applies only to its own slice of income.
Instead: Slice the income: $32 000 under the example bands owes $270, not 3.5% of $32 000.
Mistake: Adding GST by multiplying by 0.09 and reporting that as the total.
0.09 × amount is the GST alone, not the bill.
Instead: Total = amount × 1.09; GST alone is amount × 0.09.
Mistake: Comparing raw marks or raw savings instead of percentages.
Different totals make raw differences meaningless.
Instead: Convert to percentages of each person’s own base, then compare.
Mistake: Leaving a hire-purchase answer as the monthly payment when the question asks for the total paid.
Total paid = deposit + all instalments; the monthly figure is only one slice.
Instead: Add the deposit back: $372 + $1 636.80 = $2 008.80 in the worked example.

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Question 1 of 3

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Express one quantity as a percentage

45 expressed as a percentage of 150 is

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5 multiple-choice questions and structured questions with answer lines. Chapter 3 (N3 Percentage, with interest, GST, tax and hire-purchase contexts).

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Questions

Frequently asked questions

What does Sec 3 E-Math Chapter 3 cover?

It covers syllabus sub-topic N3 of O-Level 4052 / SEC G3 K310 — expressing quantities as percentages, percentage change, and reverse percentages — plus the personal-and-household-finance strand: simple and compound interest, GST bills, tax brackets, hire purchase and money exchange.

How do I reverse a percentage change without getting confused?

Write the multiplier first. A price after a 20% rise is the original × 1.2, so the original is the price ÷ 1.2. A price after a 15% discount is the original × 0.85, so divide by 0.85. Never subtract the percentage from the new amount.

Why do successive percentage changes multiply instead of add?

Because each change applies to a different base. A 30% discount then a 20% discount leaves 0.70 × 0.80 = 0.56 of the price — a 44% total discount, not 50%. Multiplying multipliers is the safe rule for any chain of changes.

When is this chapter usually tested in school?

Percentage work sits early in the Sec 3 year, so it anchors Weighted Assessment 1 and returns in every later assessment including the end-of-year examination. Finance contexts also feed the real-world question at the end of Paper 2 in the O-Level examination itself.

Sec 3 E-Math Percentage & Personal Finance notes