OxfordAQA International GCSE Mathematics: Ordering, the four operations, primes, powers and calculators (9260) – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover section 3.1.1 Structure and calculation (references N1 to N5 and N10) of the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1 (for exams May/June 2018 onwards). All of it is Core content, which is also assessable on the Extension tier papers, so no question is “Extension only”. Where a question says “by hand”, work it out without a calculator to practise the skill. Read the study guide first, or use the revision notes. Also see the course hub, the checklist and the free diagnostics.

Questions

1. (by hand)

(a) Write these in order, smallest first: −0.7, −3/4, 0.07, 1/8. [2] (b) Write < , > or = between 5/12 and 0.41. [1] (c) Write down all the integers n such that −3 ≤ n < 2. [2]

2. (by hand) Work out, giving each answer as a fraction or mixed number in its simplest form.

(a) 3 2/5 + 1 3/4 [3] (b) 2 1/4 ÷ (−1 1/8) [3]

3. (by hand) You are given that 57 × 36 = 2052. Use this to write down the value of

(a) 5.7 × 3.6 [1] (b) 2052 ÷ 0.57 [1] (c) 0.057 × 360 [1]

4. Fabric costs £6.40 per metre.

(a) Maya buys 3 1/2 metres and pays with a £50 note. How much change does she get? [3] (b) She cuts the 3 1/2 metres into pieces of length 0.45 metres. How many complete pieces can she cut, and how much fabric remains? [3]

5. (by hand)

(a) Work out 40 − 3 × (7 − 2)². [2] (b) Work out (√49 − 3³) ÷ (−5). [2] (c) Write down the reciprocal of 1 1/4. [1]

6. (by hand)

(a) Insert one pair of brackets so that 4 + 6 × 3 − 2 = 28. [1] (b) I think of a number, subtract 5, multiply by 3, then take the square root. My answer is 6. Find my number. [3]

7. (by hand)

(a) List all the prime numbers between 40 and 60. [2] (b) Show that 161 is not a prime number. [2] (c) Write 450 as a product of prime factors, in index form. [3]

8. (by hand)

(a) Write 168 and 180 each as a product of prime factors in index form. [4] (b) Find the HCF and the LCM of 168 and 180. [2] (c) A baker has 168 chocolate biscuits and 180 plain biscuits. They are shared into identical boxes with none left over. What is the greatest number of boxes, and how many of each type go in a box? [2]

9. (by hand) A pump is serviced every 18 days and a filter every 24 days. Both are serviced on 1 March.

(a) After how many days are they next both serviced on the same day? [3] (b) On what date does this happen? [2]

10. (by hand)

(a) Write down the value of 5⁴ and of 3⁵. [2] (b) Work out (−2)⁵ and −2⁴. [2] (c) Work out ∛(−216) and ⁴√256. [2] (d) Explain why √(−16) has no real value. [1]

11. Give each length to 3 significant figures and each angle to 1 decimal place.

Triangle ABC has a right angle at B, AB = 8.4 cm and angle A = 36°. Triangle PQR has a right angle at Q, PQ = 6.5 cm and QR = 9.2 cm.

(a) Find BC. [2] (b) Find AC. [2] (c) Find angle QPR. [3]

12.

(a) Work out (7.35 + 2.8²) ÷ (√3.6 − 0.92), giving your answer to 3 significant figures. [2] (b) Work out 15 ÷ (1/0.8 + 1/0.5), giving your answer to 3 significant figures. [2]

Answers

1. (a) Convert: −3/4 = −0.75 and 1/8 = 0.125. [1] Order: −3/4, −0.7, 0.07, 1/8. [1] (b) 5/12 = 0.4166…, which is bigger than 0.41, so 5/12 > 0.41. [1] (c) The integers are −3, −2, −1, 0, 1. [2] (Include −3 but not 2; one mark for four correct and no extras.) Examiner insight: Show the decimal conversions on ordering questions, because a bare list with no working gives the reader nothing to follow if one value is wrong. Read the inequality ends carefully: ≤ includes the end value and < does not.

2. (a) 3 2/5 = 17/5 and 1 3/4 = 7/4. [1] Common denominator 20: 68/20 + 35/20 = 103/20. [1] 5 3/20. [1] (b) 9/4 ÷ (−9/8) = 9/4 × (−8/9). [1] Cancel: −8/4 [1] = −2. [1] Examiner insight: Convert mixed numbers before you do anything else, and show the line where you invert the second fraction. Check the sign of the answer separately from the size.

3. (a) Each number is 10 times smaller, so the product is 100 times smaller: 2052 ÷ 100 = 20.52. [1] (b) 0.57 is 57 ÷ 100, so the answer is 100 times bigger than 2052 ÷ 57 = 36: 3600. [1] (c) 0.057 is 1000 times smaller than 57, and 360 is 10 times bigger than 36, so 2052 ÷ 100 = 20.52. [1] Examiner insight: A line of reasoning about powers of 10 protects you when a question says “use the result”. Redoing the whole calculation by another route may not be what is asked for.

4. (a) Cost = 6.40 × 3.5 [1] = £22.40. [1] Change = 50 − 22.40 = £27.60. [1] (b) 3.5 ÷ 0.45 = 7.77…, so 7 whole pieces. [1] Used: 7 × 0.45 = 3.15 m. [1] Left: 3.5 − 3.15 = 0.35 m. [1] Examiner insight: Give money with two decimal places and include units in every context answer. In part (b), a rounded 8 pieces would not fit, so state that you rounded down and why.

5. (a) 7 − 2 = 5, then 5² = 25. [1] 40 − 3 × 25 = 40 − 75 = −35. [1] (b) √49 = 7 and 3³ = 27, so 7 − 27 = −20. [1] −20 ÷ (−5) = 4. [1] (c) 1 1/4 = 5/4, so the reciprocal is 4/5. [1] Examiner insight: Write the bracket value on its own line so that a slip later still shows correct earlier working. Do not multiply 40 − 3 together before the power.

6. (a) (4 + 6) × 3 − 2 = 28. [1] (b) Work backwards. Square: 6² = 36. [1] Divide by 3: 12. [1] Add 5: 17. [1] Examiner insight: Check by running forward: 17 − 5 = 12, 12 × 3 = 36, √36 = 6. Reversing the order of the operations but keeping the original ones is the usual slip.

7. (a) The primes are 41, 43, 47, 53, 59. [2] (One mark for at least three correct with no composite numbers.) (b) √161 is about 12.7, so test primes up to 11. [1] 161 = 7 × 23, so it has a factor other than 1 and itself. [1] (c) 450 = 2 × 225 = 2 × 3 × 75 = 2 × 3 × 3 × 25 [1] = 2 × 3² × 5 × 5 [1] = 2 × 3² × 5². [1] Examiner insight: On “show that”, state the factor pair you found, since the factor is the evidence. In part (c), index form means each prime appears once, with its power.

8. (a) 168 = 2 × 84 = 2 × 2 × 42 = 2 × 2 × 2 × 21 = 2 × 2 × 2 × 3 × 7. [1] So 168 = 2³ × 3 × 7. [1] 180 = 2 × 2 × 3 × 3 × 5 [1] = 2² × 3² × 5. [1] (b) HCF: lower powers of 2 and 3, so 2² × 3 = 12. [1] LCM: higher powers, so 2³ × 3² × 5 × 7 = 2520. [1] (c) Greatest number of boxes = HCF = 12 boxes. [1] Each has 168 ÷ 12 = 14 chocolate and 180 ÷ 12 = 15 plain. [1] Examiner insight: Name the HCF or LCM you are using in a context part, because the number alone does not show the method. Check HCF × LCM = 12 × 2520 = 30 240 = 168 × 180.

9. (a) 18 = 2 × 3² and 24 = 2³ × 3. [1] LCM = 2³ × 3² [1] = 72 days. [1] (b) 72 days after 1 March: March has 31 days, so 1 April is 31 days on, and 1 May is 61 days on. [1] 72 − 61 = 11 more days, so 12 May. [1] Examiner insight: Counting dates is a place where answers are often one day out. Write the checkpoint (1 May is 61 days later) so a small slip is easy to spot.

10. (a) 625 and 243. [2] (b) (−2)⁵ = −32 and −2⁴ = −16. [2] (c) (−6)³ = −216 so ∛(−216) = −6. [1] 4⁴ = 256, so ⁴√256 = 4. [1] (d) Any real number squared is zero or positive, so no real number squares to give −16. [1] Examiner insight: For “explain”, a bare statement such as “it is negative” is thin. Give the reason about squares. For part (b), write the bracket form in your answer so the sign rule is visible.

11. (a) tan 36° = BC ÷ 8.4. [1] BC = 8.4 × tan 36° = 6.1029… = 6.10 cm. [1] (b) cos 36° = 8.4 ÷ AC. [1] AC = 8.4 ÷ cos 36° = 10.38… = 10.4 cm. [1] (c) tan(QPR) = 9.2 ÷ 6.5 (opposite over adjacent). [1] QPR = tan⁻¹(9.2 ÷ 6.5) [1] = 54.757…° = 54.8°. [1] Examiner insight: Label the sides opposite, adjacent and hypotenuse first. Use full unrounded values from your calculator and only round the final answer. In part (b), 8.4 ÷ cos 36° is not the same as 8.4 × cos 36°.

12. (a) Top: 7.35 + 7.84 = 15.19. [1] Bottom: √3.6 − 0.92 = 0.9774…; 15.19 ÷ 0.9774… = 15.54… = 15.5. [1] (b) 1 ÷ 0.8 = 1.25 and 1 ÷ 0.5 = 2, so the bracket is 3.25. [1] 15 ÷ 3.25 = 4.615… = 4.62. [1] Examiner insight: When you enter these on a calculator, bracket the top and bottom of the fraction. Write the intermediate values you used, because if a bracket is misplaced in the keystrokes, your written values show where it went wrong.

Where marks are usually lost

• Ordering negative numbers by their digits alone, ignoring that −0.75 is smaller than −0.7.
• Multiplying or dividing mixed numbers without first changing them to improper fractions.
• Forgetting the sign in division and multiplication of fractions.
• Rounding up when a question asks for whole items that fit.
• Treating −2⁴ as 16, or (−2)⁵ as positive.
• Writing 1 as a prime, or stopping a factor tree before every branch ends in a prime.
• Taking the higher powers for the HCF, or the lower powers for the LCM.
• Counting dates from the wrong starting day.
• Pressing a long division on a calculator without brackets round the denominator.
• Rounding in the middle of a calculation, or giving 4.6 where 3 significant figures asks for 4.62.

Next steps

• Review the topic with the revision notes.
• Re-read the study guide for any question you missed.
• Visit the OxfordAQA IGCSE Mathematics hub and tick off topics on the checklist.
• Try all the free 10-minute diagnostics.
• Book a free trial class.

Official syllabus

OxfordAQA International GCSE Mathematics (9260), Version 5.1, for exams May/June 2018 onwards, published by OxfordAQA: section 3.1.1 Structure and calculation, references N1 to N5 and N10.