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Using Percentages: Paper 2 Challenge Multiple-Choice

This is the Challenge tier: ten harder multiple-choice questions where the tempting option is the right number for the wrong question. Each one turns on choosing the correct base for a percentage. Work the standard Paper 2 set first. Detailed solutions can be requested by email.

10 questions · about 20 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 2015 P2 Q9 — the second percentage is the unknown; the trap is the same percentage again

The price of a book is increased by $25%$ . To bring the price back to its original value, the new price should be decreased by

$20%$ .
$25%$ .
$75%$ .
$80%$ .

Show answer

A. From 100 to 125 and back is a fall of 25 out of 125, which is $20%$ . Option B is the right number for the wrong base.

Question 2 · New sub-type: profit on selling price — no direct past-paper model; the same profit is measured against two different bases

A trader finds that his profit on an item is $20%$ of its selling price. The percentage profit of the item is

$16 \frac{2}{3} %$ .
$20%$ .
$25%$ .
$80%$ .

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C. If the selling price is 100, the profit is 20 and the cost is 80. Percentage profit is measured on cost: $20 \div 80 = 25%$ .

Question 3 · Modelled on 2018 P2 Q9 — three statements about one deposit replace a single calculation; all three depend on seeing that monthly compounding beats the flat yearly rate

A sum of $$10 000$ is deposited at an interest rate of $12%$ per annum, compounded monthly. Which of the following are true?

I. The interest for the 1st year exceeds $$1200$ .
II. The amount after 2 years exceeds $$12 544$ .
III. The interest for the 2nd month is greater than the interest for the 1st month.

I and II only
I and III only
II and III only
I, II and III

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D. The rate is $1%$ per month. Interest for year 1 is about $$1268$ ; the amount after 2 years is about $$12 697$ , above $$10 000 (1.12)^{2} = $12 544$ ; and the 2nd month earns $$101$ against $$100$ .

Question 4 · New sub-type: repeated percentage decrease — no direct past-paper model; the number of periods is the unknown and the linear guess is a distractor

The price of a jacket is reduced by $10%$ at the end of each week. After how many weeks will the price first be less than $70%$ of the original price?

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B. $0. 9^{3} = 0.729$ is still above 0.7, and $0. 9^{4} = 0.6561$ is below it. Three cuts of $10%$ do not make $30%$ .

Question 5 · Modelled on 2018 P1 Q7 — the discount is the unknown and the sale ends in a small loss; the trap subtracts the percentages

The marked price of a toy is $40%$ above its cost. If the toy is sold at a discount of $x %$ on its marked price and the percentage loss is $2%$ , then $x =$

$28$ .
$30$ .
$38$ .
$42$ .

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B. $1.4 (1 - x %) = 0.98$ gives $1 - x % = 0.7$ , so $x = 30$ . Adding $40 + 2 = 42$ treats the discount as a percentage of the cost.

Question 6 · New sub-type: salaries tax — no direct past-paper model; a rise in income straddles two tax bands

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

on the first $$50 000$ : $2%$
on the next $$50 000$ : $6%$
on the next $$50 000$ : $10%$
on the remainder: $14%$

If the net chargeable income of a person increases from

$95 000

$105 000

, then the salaries tax payable increases by

$$300$ .
$$600$ .
$$800$ .
$$1000$ .

Show answer

C. The rise crosses $$100 000$ : $$5000$ at $6%$ and $$5000$ at $10%$ , giving $$300 + $500 = $800$ .

Question 7 · Modelled on 2026 P1 Q5 — the total changes as well as the part, so the percentage has a new base; set as multiple-choice

In a class, $40%$ of the students are girls. After 10 more girls join the class, $50%$ of the students are girls. Find the original number of students in the class.

$25$
$50$
$60$
$100$

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B. With $N$ students, $0.4 N + 10 = 0.5 (N + 10)$ , so $N = 50$ . Reading “10 girls are $10%$ ” gives 100, but the $50%$ is of the larger class.

Question 8 · Modelled on 2017 P2 Q10 — no sum of money is given; simple interest fixes the rate and compound interest is then compared with it

A sum of money deposited at simple interest doubles in 20 years. If the same sum is deposited at the same annual interest rate for 20 years, compounded yearly, then the amount is approximately

$1.63$ times the sum.
$2$ times the sum.
$2.05$ times the sum.
$2.65$ times the sum.

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D. Doubling by simple interest in 20 years means $5%$ a year. Compounded yearly, $(1.05)^{20} \approx 2.65$ .

Question 9 · New sub-type: loss of stock — no direct past-paper model; part of the stock is lost before the mark-up is applied

A merchant buys a batch of eggs. It is found that $10%$ of the eggs are broken and cannot be sold. He sells each of the remaining eggs at a price $20%$ above its cost. Find the percentage profit on the whole batch.

$8%$
$10%$
$18%$
$20%$

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A. Only $90%$ of the eggs bring in money, each at 1.2 times its cost: $0.9 \times 1.2 = 1.08$ , a profit of $8%$ .

Question 10 · New sub-type: depreciation — no direct past-paper model; the present value is given and the earlier value is asked

The value of a van depreciated by $20%$ each year over the last 2 years. If the value of the van is now $$32 000$ , find its value 2 years ago.

$$44 800$
$$46 080$
$$50 000$
$$53 333$

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C. $V (0.8)^{2} = 32000$ , so $V = $50 000$ . Dividing by 0.6 gives $$53 333$ , and multiplying by $1. 2^{2}$ gives $$46 080$ : neither undoes two falls of $20%$ .

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

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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