Topic 1 Number (9260): Practice Questions

These are original questions written for Marlbridge, in the style and at the standard of the examination. They are not reproduced past-paper questions — examination boards hold copyright in their own papers. Use these alongside the official past papers available free from your board.

Related: Topic 1 Number revision notes

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Section A

1. Write 72 and 108 as products of their prime factors, and hence find their HCF and LCM. [4]

2. After a 25% discount, a jacket costs $54. Find the original price. [2]

3. Simplify √50 + √18, giving your answer as a single surd in its simplest form. [3]

Section B

4. A recipe uses ingredients in direct proportion to the number of servings. 4 servings require 300 g of flour.

(a) Find the flour required for 7 servings. [2] (b) A different recipe takes 6 hours for 4 workers to complete, with time and workers in inverse proportion. How long would it take 3 workers? [3]

5. A rectangle has sides measured as 6.2 cm and 4.5 cm, both correct to 1 decimal place.

(a) State the upper and lower bounds of each side. [2] (b) Calculate the greatest possible perimeter of the rectangle. [3] (c) Calculate the greatest possible area of the rectangle. [2]

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Answers

1. 72 = 2³ × 3², 108 = 2² × 3³ [2]. HCF = 2² × 3² = 36 [1]. LCM = 2³ × 3³ = 216 [1].

2. Multiplier for a 25% discount = 0.75. Original price = 54 ÷ 0.75 = $72 [2].

3. √50 = 5√2, √18 = 3√2 [2]. Sum = 5√2 + 3√2 = 8√2 [1].

4. (a) Unit rate: 300 ÷ 4 = 75 g per serving [1]. 7 servings = 75 × 7 = 525 g [1]. (b) Inverse proportion: workers × time = constant = 4 × 6 = 24 [1]. For 3 workers: time = 24 ÷ 3 = 8 hours [2].

5. (a) 6.2 cm: 6.15 ≤ l < 6.25. 4.5 cm: 4.45 ≤ w < 4.55 [2]. (b) Greatest perimeter uses both upper bounds: 2 × (6.25 + 4.55) = 2 × 10.80 = 21.6 cm [3]. (c) Greatest area uses both upper bounds: 6.25 × 4.55 = 28.4375 cm² [2].

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Section C — additional questions

6. Write 0.000283 in standard form, and calculate (3 × 10⁴) × (5 × 10³), giving your answer in standard form. [3]

7. $3,500 is invested at 4% compound interest per year. Find its value after 3 years, to the nearest dollar. [3]

8. Calculate the least possible perimeter and least possible area of the rectangle in question 5, using the lower bounds instead of the upper bounds. [3]

Answers to Section C

8. Least perimeter uses both lower bounds: 2 × (6.15 + 4.45) = 2 × 10.60 = 21.2 cm [1]. Least area uses both lower bounds: 6.15 × 4.45 = 27.3675 cm² [2].

6. 0.000283 = 2.83 × 10⁻⁴ [1]. (3 × 10⁴) × (5 × 10³) = 15 × 10⁷, which must be rewritten in correct standard form as 1.5 × 10⁸ [2].

7. 3500 × 1.04³ [1] [1] = 3500 × 1.124864 = $3,937 (to the nearest dollar) [1].

Why bounds work in opposite directions for perimeter and area

Question 8 deliberately mirrors question 5 to make a point: for both perimeter (a sum) and area (a product) of two independently rounded measurements, the greatest possible result always uses the upper bound of every measurement involved, and the least possible result always uses the lower bound of every measurement involved. This is because both operations increase when either input increases, so pushing every input to its maximum gives the maximum result, and pushing every input to its minimum gives the minimum result. This is not true for every operation, though — a subtraction or division behaves differently, since the result of A − B is maximised by taking A’s upper bound together with B’s lower bound, not both upper bounds together, which is why bounds questions always require checking what type of calculation is actually being performed before deciding which bound to use for each value.

A note on reverse percentage problems

Question 2 tests a distinction that recurs across many percentage problems: when a value after a percentage change is given and the original value is required, the correct approach is to divide by the multiplier, not to multiply by it or to simply add back the stated percentage of the given value. A 25% discount leaves 75% of the original price, so the multiplier is 0.75, and since $54 represents that 75%, dividing 54 by 0.75 correctly recovers the full original price. A common error is instead calculating 25% of $54 and adding it back to $54, which produces a plausible-looking but incorrect answer, since 25% of the discounted price is not the same amount as 25% of the original price.

Where marks are usually lost

• Reverse percentage calculated by multiplying instead of dividing by the multiplier.
• Simplifying surds incorrectly by not extracting the largest square factor first.
• Confusing direct and inverse proportion — always check whether the two quantities increase together or move oppositely before setting up the relationship.
• Using a mix of upper and lower bounds inconsistently instead of choosing correctly for greatest/least sum, difference, product or quotient.
• Adding a percentage back onto a discounted price instead of dividing by the correct multiplier when working backwards to an original value.