Ratio, proportion and percentage change – Year 10 GCSE maths test answers to download as PDF

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1. Answer to Question 1 (4 marks)

   1. \(3:4\) (1) The HCF of 45 and 60 is 15, so divide both parts by 15. The simplest form is \(3:4\).
   2. \(2+3+7=12\) parts (1), \(360\div12=30\) (1), so £60, £90 and £210 (1) Check: \(60+90+210=360\). The shares are £60, £90 and £210.

   In (a), dividing by 5 only gives \(9:12\), which is not the simplest form and scores 0. In (b), a common error is to divide £360 by 3 because there are three people; the ratio has 12 equal parts, and the largest share, £210, belongs to the 7 parts.

2. Answer to Question 2 (4 marks)

   1. 1 kg \(=68\)p (1), so 8 kg \(=\) £5.44 (1) Divide £3.40 by 5 to find the cost of 1 kg, then multiply by 8. So 8 kg cost £5.44.
   2. Small jar: \(160\div4=40\)p per 100 g; large jar \(247\div6.5=38\)p per 100 g (1). The 650 g jar is better value (1) Compare the price per 100 g: the lower price gives more jam for the same money.

3. Answer to Question 3 (4 marks)

   1. \(7.2\times50\,000=360\,000\) cm (1) \(=\) 3.6 km (1) Multiply the map distance by 50 000, then divide by 100 000 to change centimetres into kilometres. The towns are 3.6 km apart.
   2. \(18\text{ km}=1\,800\,000\) cm (1), \(\div50\,000=\) 36 cm (1) Change kilometres into centimetres, then divide by 50 000. The road is 36 cm long on the map.

   A check for (b): 36 cm is five times the distance in (a), and 18 km is five times 3.6 km.

4. Answer to Question 4 (4 marks)

   1. \(480\times1.2=\) £576 (1) Adding 20% VAT gives the multiplier 1.2. The total price is £576.
   2. \(525\div0.7=\) £750 (1 for 70%, 1 for the answer) After a 30% reduction the sofa costs 70% of the original, so we divide by 0.7. The original price was £750.
   3. \(1.12\times0.88=0.9856\lt 1\) (1) The overall multiplier is less than 1, so the result is smaller than the original number.

5. Answer to Question 5 (4 marks)

   1. \(7k-4k=12\) (1), \(k=4\) (1) The difference between the ages is \(7k-4k=3k\), and this equals 12 years. So \(k=4\).
   2. 16 and 28 (1) Substitute \(k=4\): the cousins are \(4\times4=16\) and \(7\times4=28\) years old.
   3. \(\dfrac{16+n}{28+n}=\dfrac58\), \(128+8n=140+5n\), \(3n=12\), so 4 years (1) Both ages increase by \(n\) years. Cross-multiply, then solve. Check: \(20:32=5:8\).

   In (a), check that \(28-16=12\) years. In (c), \(16+4=20\) and \(28+4=32\), and dividing both by 4 gives \(5:8\).

Total : 20 marks