Secondary 2 · Mathematics · Number
Percentages, rates and ratios
Solve problems involving percentage increase and decrease, profit and loss, best-buy comparisons and simplifying ratios, using everyday Australian contexts.
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These lessons were written around Australian Curriculum strands and matched to Singapore levels by age (Primary 1 is the first year of primary school). They are not aligned to MOE syllabuses or the PSLE, so check them against the syllabus your child is working towards.
This lesson uses some Australian examples (places, animals or contexts). Swap in local examples from Singapore as you teach.
Learning goals
Learners will
- Increase and decrease amounts by a percentage using multipliers.
- Calculate profit and loss, and express them as a percentage of the cost price.
- Compare best buys using unit prices, and simplify ratios including those with decimals or different units.
Success looks like
- I can increase $80 by 15% in one step by multiplying by 1.15.
- I can work out the profit or loss on a sale and write it as a percentage of the cost.
- I can use unit prices to choose the best buy, and simplify a ratio such as 2.5 : 4.
The big idea
Percent means “out of 100”, so 15% = 15100 = 0.15. To find a percentage of an amount, multiply by the decimal: 15% of $80 = 0.15 × 80 = $12.
To increase an amount by a percentage, you could find the percentage and add it on: $80 + $12 = $92. A faster way is to use a multiplier. The new amount is 100% + 15% = 115% of the original, so multiply by 1.15: 80 × 1.15 = $92.
To decrease an amount, subtract the percentage from 100%. A 25% discount leaves 75%, so the multiplier is 0.75: $60 × 0.75 = $45.
In business, profit = selling price − cost price, and loss = cost price − selling price. The percentage profit or loss is always calculated on the cost price:
- Percentage profit = profit ÷ cost price × 100%
- Percentage loss = loss ÷ cost price × 100%
Best buys: bigger packets are not always cheaper. To compare fairly, find the unit price of each option — the price per gram, per 100 g, per litre or per item — and choose the lowest.
Ratios are simplified by dividing both parts by their highest common factor. If a ratio has decimals, first multiply both parts by 10 (or 100) to make whole numbers: 2.5 : 4 = 25 : 40 = 5 : 8. If the quantities have different units, convert to the same unit first: 45 minutes : 2 hours = 45 : 120 = 3 : 8.
Worked examples
Percentage increase with a multiplier
A gym membership of $80 per month increases by 15%. What is the new price?
- The new price is 100% + 15% = 115% of the old price.
- The multiplier is 1.15.
- 80 × 1.15 = 92.
Answer: $92 per month
Percentage loss
A bike was bought for $250 and sold for $200. Find the loss and the percentage loss.
- The selling price is less than the cost, so this is a loss.
- Loss = $250 − $200 = $50.
- Percentage loss = 50 ÷ 250 × 100% = 20%.
Answer: A loss of $50, which is a 20% loss.
Best buy
Rice comes in a 750 g bag for $4.50 or a 1.2 kg bag for $6.60. Which is the better buy?
- Compare the price per 100 g. 750 g is 7.5 lots of 100 g, and 1.2 kg = 1200 g is 12 lots of 100 g.
- Small bag: $4.50 ÷ 7.5 = $0.60 per 100 g.
- Large bag: $6.60 ÷ 12 = $0.55 per 100 g.
- $0.55 is less than $0.60.
Answer: The 1.2 kg bag is the better buy at 55 cents per 100 g.
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
A $60 item is discounted by 25%. What is the sale price?
$45 — A 25% decrease leaves 75%: 60 × 0.75 = $45. ($15 is the discount, not the sale price.)
Which multiplier increases an amount by 12%?
1.12 — 100% + 12% = 112%, which is 1.12 as a decimal.
Simplify the ratio 2.5 : 4.
5 : 8 — Multiply both parts by 2 to remove the decimal: 5 : 8. These have no common factor, so 5 : 8 is simplest form.
Which bottle of juice is the best buy?
3 L for $4.80 — Price per litre: $1.70, $1.60, $1.75 and $1.80. The 3 L bottle is cheapest per litre.
Increase 250 by 8%.
270 — Multiplier 1.08: 250 × 1.08 = 270. (8% of 250 is 20, and 250 + 20 = 270.)
A shop buys a jacket for $40 and sells it for $58. What is the percentage profit?
45% — Profit = $58 − $40 = $18. Percentage profit = 18 ÷ 40 × 100% = 45%.
Simplify the ratio 45 minutes : 2 hours.
3 : 8 — Convert to the same unit: 2 hours = 120 minutes. 45 : 120, dividing by 15, gives 3 : 8.
A $1200 laptop is sold at a 15% loss. What is the selling price?
$1020 — A 15% loss leaves 85% of the cost: 1200 × 0.85 = $1020.
Watch out for
- Calculating percentage profit or loss on the selling price instead of the cost price. Always divide by what was originally paid.
- Thinking a 10% increase followed by a 10% decrease returns to the original. Try it: $100 → $110 → $99. The decrease is 10% of a bigger number.
- Assuming the largest packet is always the best buy. Work out unit prices — sometimes the bigger pack costs more per gram.
Tips for parents
- On your next shop, ask the learner to check two shelf unit prices with a calculator and decide which product is better value.
- Use sale signs (“30% off”) as quick mental-maths challenges: what is the new price, and what is the multiplier?
- Encourage the learner to estimate first (15% of $80 is a bit more than 10%, so a bit more than $8) to catch calculator slips.
Go further
- Plan a small market-stall or bake-sale budget: list costs, choose selling prices and calculate the percentage profit.
- Investigate successive discounts: is “20% off, then a further 10% off” the same as “30% off”? (No — it is 28% off.)
- Compare mobile phone or streaming plans using unit rates, such as cost per gigabyte or cost per month over a year.
About 50 minutes
You'll need
- Calculator
- Paper and pencil
- Shopping catalogues, receipts or online grocery prices (optional)
Key words
- Multiplier
- The decimal you multiply by to change an amount by a percentage. A 15% increase uses 1.15; a 15% decrease uses 0.85.
- Cost price
- The amount a seller paid for an item.
- Selling price
- The amount the item is sold for.
- Profit
- The amount gained when the selling price is more than the cost price: selling price − cost price.
- Loss
- The amount lost when the selling price is less than the cost price: cost price − selling price.
- Unit price
- The price for one unit of a quantity, such as the price per litre or per 100 g.
Still tricky?
For parents
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