Mathematics • Year 8 • Unit 1 • Lesson 11
Successive % Changes in the Real World
Use combined multipliers where they really show up: stacked store discounts, pay rises followed by cuts, stock-price swings, sale-tax sequences. Then explain WHY two equal-and-opposite percentages don't cancel.
1. Word problems
Each problem uses the “multiply, don't add” idea. Show your working — a single final answer with no working only earns half marks.
1.1 — Stacked sale. A $\$120$ pair of headphones is marked down 25%. At the till, a student card gives a further 10% off the discounted price.
(a) Use multipliers to find the final price the student pays.
(b) What single percentage discount on $\$120$ would give the same final price?
(c) Is this the same as “35% off”? Explain in one sentence. 4 marks
1.2 — Pay rise then pay cut. Asha earns $\$50\,000$ per year. She receives a 40% pay rise, then six months later her company cuts everyone's salary by 40%.
(a) Find her salary after each change.
(b) Calculate the overall percentage change from her original salary.
(c) In one sentence, explain WHY +40% then −40% doesn't put her back at $\$50\,000$. 4 marks
1.3 — Stock swing. A share price of $\$25$ rises 15% in March, then falls 10% in April, then rises 8% in May.
(a) Find the price at the end of each month.
(b) Find the combined multiplier for the three months.
(c) Find the equivalent single percentage change from $\$25$ to the May price. 3 marks
1.4 — Two shops, same product. A backpack costs $\$80$ at both stores. Store A advertises “30% off, then 20% off at the till”. Store B advertises a flat “50% off”.
(a) Find the final price at each store.
(b) Which is cheaper, and by how much?
(c) Which has the bigger equivalent single percentage discount? 3 marks
1.5 — House price boom and bust. A house bought for $\$600\,000$ rises 20% in year 1, falls 15% in year 2, then rises 5% in year 3.
(a) Find the price at the end of year 3.
(b) Has the house gained or lost value overall? By what percentage?
(c) If the same chain of changes (+20%, −15%, +5%) repeats over the next 3 years, what would the price be after 6 years? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A friend insists that “a 30% rise followed by a 30% fall just cancels out — you end up where you started”. In your own words, explain (i) why this is wrong, (ii) what actually happens to a starting price of $\$100$ under +30% then −30%, (iii) why mathematicians say the two percentages are “on different bases”, and (iv) what the equivalent single percentage change is. Use the word “multiplier” somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Stacked sale
(a) $\$120 \times 0.75 \times 0.90 = \$120 \times 0.675 = \textbf{\$81}$.
(b) Equivalent single discount = 1 − 0.675 = 32.5% off.
(c) No — 35% off would give $\$78$, not $\$81$. Adding percentages overstates the saving because the second 10% is on a SMALLER price.
1.2 — Pay rise then cut
(a) After +40%: $\$50\,000 \times 1.40 = \$70\,000$. After −40%: $\$70\,000 \times 0.60 = \textbf{\$42\,000}$.
(b) Combined multiplier 1.40 × 0.60 = 0.84, so overall 16% decrease.
(c) The 40% rise gave $\$20\,000$ (40% of $\$50\,000$). The 40% cut took $\$28\,000$ (40% of the LARGER $\$70\,000$). The cut acts on a bigger base, so it removes more dollars than the rise added.
1.3 — Stock swing
(a) March: $25 \times 1.15 = \$28.75$. April: $28.75 \times 0.90 = \$25.875$. May: $25.875 \times 1.08 = \textbf{\$27.945}$ (about $\$27.95$).
(b) Combined multiplier = $1.15 \times 0.90 \times 1.08 = \textbf{1.1178}$.
(c) Equivalent single change = +11.78% (not +13%).
1.4 — Two shops
(a) Store A: $\$80 \times 0.70 \times 0.80 = \$80 \times 0.56 = \textbf{\$44.80}$. Store B: $\$80 \times 0.50 = \textbf{\$40}$.
(b) Store B is cheaper, by $\$4.80$.
(c) Store A's equivalent single discount is 1 − 0.56 = 44% off. Store B's is 50% off. Store B has the bigger discount. (So “30% + 20%” sounds bigger than “50%” but isn't — that's the trap.)
1.5 — House price
(a) $\$600\,000 \times 1.20 \times 0.85 \times 1.05 = \$600\,000 \times 1.0710 = \textbf{\$642\,600}$.
(b) Gained value: overall change = +7.10%.
(c) After 6 years: $\$600\,000 \times 1.0710 \times 1.0710 = \$600\,000 \times 1.0710^2 = \$600\,000 \times 1.14704 \approx \textbf{\$688\,225}$ (overall +14.70%).
2.1 — Explain your thinking (sample response)
The friend is wrong because percentages don't add — multipliers do. A 30% rise turns each dollar into 1.30 dollars; the multiplier is 1.30. A 30% fall multiplies by 0.70. The combined multiplier is 1.30 × 0.70 = 0.91, which is less than 1, so the value drops. Starting at $\$100$, the price rises to $\$130$, then a 30% cut on the bigger $\$130$ removes $\$39$ — much more than the $\$30$ it gained — leaving $\$91$. We say the two percentages are “on different bases” because the +30% is applied to $\$100$ but the −30% is applied to the new $\$130$. The equivalent single change is 0.91 − 1 = −0.09, or a 9% overall decrease, not 0%.
Marking: 1 mark for explaining the “multiply not add” rule with the word “multiplier”; 1 mark for the working $\$100 \to \$130 \to \$91$; 1 mark for explaining the “different bases” idea (the second % is on the bigger value); 1 mark for stating the equivalent single change as 9% decrease.