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Using Percentages: Paper 1 Challenge

This is the Challenge tier: ten harder Paper 1 questions that state a condition and leave the method to you. Expect unknown prices, discounts hidden inside offers, two banks to compare and tax bands read backwards, with several claims to prove or disprove. Work the standard Paper 1 set first. Detailed solutions can be requested by email.

10 questions · 55 marks · about 60 minutes · answers below each question

All questions are original, written for Math and AI Academy and modelled on past-paper style. They are not reproductions of HKEAA examination questions.

Question 1 · Modelled on 2013 P2 Q10 — the selling price is a letter, so the result has to be shown for every price; new relative to the standard set: working with an unknown amount throughout

A dealer sells two paintings for $$ P$ each. She gains $20%$ on one painting and loses $20%$ on the other.

(a) Express, in terms of $P$ , the total cost of the two paintings.

(b) Someone claims that after the two transactions, the dealer has no gain and no loss. Do you agree? Explain your answer.

(c) Find the overall percentage gain or percentage loss of the dealer.

(5 marks)

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(a) $$ \frac{25 P}{12}$

(b) Disagreed. The total cost $$ \frac{25 P}{12}$ is more than the total selling price $$2 P$ , so there is a loss.

(c) A loss of $4%$

The two $20%$ figures are taken from different costs, so they cannot cancel.

Question 2 · Modelled on 2018 P1 Q7 — the discount is the unknown and is bounded by a profit target; part (b) hides a discount inside an offer. New relative to the standard set: an inequality, and a discount that is not stated as a percentage

The marked price of a mug is $50%$ higher than its cost.

(a) The mug is sold at a discount of $d %$ on its marked price. If the percentage profit is at least $20%$ , find the greatest value of $d$ .

(b) The shop now sells the mugs at the marked price under the offer “buy three, get one free”. A customer takes four mugs under the offer. Is the percentage profit still at least $20%$ ? Explain your answer.

(5 marks)

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(a) $d = 20$

(b) No. The shop receives 3 marked prices for 4 mugs, which is a $25%$ discount, and the percentage profit is only $12.5%$ .

"Buy three, get one free" is a $25%$ discount on four mugs, more than the $20%$ found in (a).

Question 3 · Modelled on 2015 P2 Q9 — the two percentages are equal and unknown; part (a) is a proof for every value. New relative to the standard set: a general argument with a letter

The price of a share is increased by $p %$ and then decreased by $p %$ , where $0 < p < 100$ .

(a) Show that the final price is always lower than the original price.

(b) If the final price is $9%$ lower than the original price, find $p$ .

(5 marks)

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(a) Final price $=$ original price $\times (1 - \frac{p ^{2}}{10 000})$ , and $\frac{p ^{2}}{10 000} > 0$ .

(b) $p = 30$

The product $(1 + p %) (1 - p %)$ is a difference of two squares.

Question 4 · Modelled on 2026 P1 Q7 — neither the cost nor the marked price is given, only the profit at two different prices. New relative to the standard set: two unknowns from two conditions

A profit of $$100$ is made by selling a watch at a discount of $25%$ on its marked price. If the watch were sold at its marked price, the profit would be $$400$ .

(a) Find the percentage profit when the watch is sold at the discount of $25%$ .

(b) The shopkeeper wants the percentage profit to be exactly $20%$ . What discount per cent on the marked price should be offered?

(5 marks)

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(a) $12.5%$

(b) $20%$

The two profits differ by $$300$ , and that difference is exactly the $25%$ discount.

Question 5 · Modelled on 2022 P2 Q11 — two banks with different rates and different compounding are compared, and the comparison is then extended without calculation. New relative to the standard set: comparing growth factors

Bank $A$ offers an interest rate of $6%$ per annum, compounded monthly. Bank $B$ offers an interest rate of $6.1%$ per annum, compounded yearly. Mr Wong deposits $$200 000$ in each bank for 1 year.

(a) Find the interest from bank $A$ correct to the nearest dollar.

(b) Mr Wong claims that bank $B$ pays more interest because its interest rate is higher. Do you agree? Explain your answer.

(c) Would your answer to (b) be different if both deposits were kept for 5 years? Explain your answer.

(6 marks)

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(a) $$12 336$

(b) Disagreed. Bank $B$ pays $$12 200$ , which is less than $$12 336$ .

(c) No. Each year bank $A$ multiplies the amount by about $1.0617$ and bank $B$ by $1.061$ , so bank $A$ stays ahead for any number of years.

Compare what each bank multiplies the deposit by in one full year.

Question 6 · New sub-type: growth and depreciation — no direct past-paper model; one quantity grows while another shrinks and the question asks when they cross. New relative to the standard set: two rates at once, and a search year by year

At the start of 2026, the populations of town $X$ and town $Y$ are $50000$ and $40000$ respectively. The population of town $X$ decreases by $4%$ each year while the population of town $Y$ increases by $5%$ each year.

(a) Find the population of town $X$ at the start of 2028.

(b) At the start of which year will the population of town $Y$ first exceed that of town $X$ ? Explain your answer.

(5 marks)

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(a) $46080$

(b) 2029. At the start of 2028, $X$ has $46080$ and $Y$ has $44100$ ; at the start of 2029, $X$ has about $44237$ and $Y$ has $46305$ .

The gap closes from both sides, so it closes faster than either rate alone suggests.

Question 7 · Modelled on 2012 P2 Q8 — run backwards: the overall percentage is given and one of the group percentages is the unknown. New relative to the standard set: a percentage of a percentage, and a claim about a share within a subgroup

In a school, $45%$ of the students are boys. It is given that $20%$ of the boys and $x %$ of the girls wear glasses, and that $31%$ of the students in the school wear glasses.

(a) Find $x$ .

(b) The number of girls wearing glasses exceeds the number of boys wearing glasses by $104$ . Find the number of students in the school.

(c) Someone claims that among the students wearing glasses, fewer than $30%$ are boys. Do you agree? Explain your answer.

(6 marks)

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(a) $x = 40$

(b) $800$

(c) Agreed. Boys make up $\frac{9}{31} \approx 29.0%$ of the students wearing glasses.

Express every group as a percentage of the whole school before comparing.

Question 8 · New sub-type: salaries tax — no direct past-paper model; the table is used to test a common belief about tax bands, then solved for an income that meets a condition. New relative to the standard set: reasoning about the marginal rate

In a city, salaries tax is charged on the net chargeable income of a person at the following rates:

on the first $$40 000$ : $2%$
on the next $$40 000$ : $7%$
on the next $$40 000$ : $12%$
on the remainder: $16%$

(a) The net chargeable income of Mr Chan rises from $$90 000$ to $$120 000$ . Find the increase in his salaries tax payable.

(b) Ms Ho says, “If a pay rise pushes my net chargeable income into a higher band, I may end up with less money after tax than before.” Explain why she is wrong.

(c) The salaries tax payable by Mr Yip is exactly $10%$ of his net chargeable income. Find his net chargeable income.

(7 marks)

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(a) $$3600$

(b) A higher rate applies only to the extra dollars inside the higher band. Each extra dollar is taxed at $16%$ at most, so at least 84 cents of it is kept and the income after tax still rises.

(c) $$180 000$

The rate in a band is charged on the slice in that band, never on the whole income.

Question 9 · Modelled on 2017 P2 Q10 — run backwards: both interests are given and the principal and the rate are unknown. New relative to the standard set: two unknowns, and an explanation of where the extra interest comes from

A sum of $$ P$ is deposited at an interest rate of $r %$ per annum for 2 years. If simple interest is paid, the interest is $$4800$ . If the interest is compounded yearly, the interest is $$5088$ .

(a) Find $r$ and $P$ .

(b) Explain, without working out any amount, why the two interests differ by exactly $$288$ .

(5 marks)

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(a) $r = 12$ , $P = 20000$

(b) Both methods pay $$2400$ in the first year. In the second year, compound interest also pays $12%$ on that $$2400$ , which is $$288$ .

The gap between the two interests is the interest earned on the first year's interest.

Question 10 · Modelled on 2014 P1 Q6 — two discounts are stacked, the order is questioned, and a fixed-amount coupon is compared with a percentage. New relative to the standard set: an inequality in the marked price

A shop sells every item at a discount of $20%$ on its marked price. A member of the shop gets a further discount of $15%$ on the discounted price.

(a) A member buys an item. Find the single discount per cent on the marked price that gives the same price.

(b) A member claims that she would pay less if the $15%$ were taken off first and the $20%$ afterwards. Do you agree? Explain your answer.

(c) Instead of the further $15%$ , a member may use a coupon which takes $$60$ off the discounted price. The marked price of a coat is $$ M$ , where $M > 100$ . Find the range of values of $M$ for which the coupon gives the lower price.

(6 marks)

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(a) $32%$

(b) Disagreed. $0.8 \times 0.85 = 0.85 \times 0.8 = 0.68$ , so the price is the same in either order.

(c) $100 < M < 500$

A fixed $$60$ is worth more than $15%$ only while $15%$ of the discounted price is below $$60$ .

Continue practising this topic: Go back to the Paper 1 long questions for the standard set, or move to Paper 2 multiple-choice — also available as Paper 2 Challenge MC.

Where this sits in the syllabus

Using Percentages is junior secondary content that the HKDSE keeps examining. In most years from 2012 to 2026, Paper 1 Section A(1) carries a short profit-and-discount or percentage-change question, and Paper 2 often carries a compound interest item. These marks are among the most reachable on the paper, and they are usually lost on one thing: taking a percentage of the wrong amount.

Full coverage of this topic, and the rest of the course, is on the Mathematics page. More sets are listed on the practice index.

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