Ratio and proportion – Year 8 maths test answers to download as PDF

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1. Answer to Question 1 (4 marks)

   1. Both 36 and 48 are multiples of 12, their HCF. Dividing each part by 12 leaves \(3:4\). (1 mark)
   2. To make the first part 1, divide both parts by 2.4: \(2.4\div2.4=1\) and \(6\div2.4=2.5\). The ratio is \(1:2.5\). (1 mark)
   3. The ratio has \(2+3+4=9\) parts, so one part is \(90\div9=£10\) (1 mark). The shares are \(2\times10=\)£20, \(3\times10=\)£30 and \(4\times10=\)£40 (1 mark).

   Check (c): \(20+30+40=90\). In (b) the form \(1:n\) often gives a decimal; that is allowed and makes ratios easy to compare. Check: \(1:2.5\) multiplied by 2.4 gives back \(2.4:6\).

2. Answer to Question 2 (4 marks)

   1. Find the cost of one notebook: \(£7.20\div6=£1.20\) (1 mark). Ten notebooks cost \(10\times£1.20=\)£12 (1 mark).
   2. Find how far the car goes on one litre: \(156\div12=13\) miles (1 mark). On 20 litres it goes \(20\times13=\)260 miles (1 mark).

   Both questions are direct proportion: when one quantity doubles, the other doubles. A sense check for (a): 10 notebooks are a bit less than twice 6 notebooks, so the cost should be a bit less than £14.40, and £12 fits. For (b), 20 litres is \(\dfrac{5}{3}\) of 12 litres, and \(156\times\dfrac{5}{3}=260\) gives the same distance.

3. Answer to Question 3 (4 marks)

   1. This is inverse proportion: fewer builders need more days. The total work is \(5\times12=60\) builder-days (1 mark). Shared between 4 builders: \(60\div4=\)15 days (1 mark).
   2. The scale 1 : 25 000 means 1 cm on the map is 25 000 cm on the ground. The 8 cm gap on the map is therefore \(8\times25\,000=200\,000\) cm in reality (1 mark). Divide by 100 to get metres (2 000 m), then by 1 000 to get kilometres: 2 km (1 mark).

   A typical error in (a) is to work out \(12\times\dfrac{4}{5}=9.6\) days, which treats the situation as direct proportion; with fewer builders the job must take longer, not shorter.

4. Answer to Question 4 (4 marks)

   1. The ratio 5 : 3 has \(5+3=8\) parts, and 72 cakes ÷ 8 = 9 cakes per part (1 mark). Sam bakes \(5\times9=\)45 cakes and Tia bakes \(3\times9=\)27 cakes (1 mark).
   2. All 72 cakes sell at £1.50 each: \(72\times1.50=\)£108. (1 mark)
   3. Tia gets 3 of the 8 parts of the money: \(108\div8=13.50\) per part, and \(3\times13.50=\)£40.50. (1 mark)

   Check: Sam receives \(5\times13.50=£67.50\), and \(67.50+40.50=108\). Tia's money can also be found directly from her cakes: \(27\times1.50=£40.50\), because the money is shared in the same ratio as the baking.

5. Answer to Question 5 (4 marks)

   1. Divide each price by the number of batteries: \(4.80\div6=£0.80\), \(7.60\div10=£0.76\) and \(17\div24=£0.708\ldots\), about £0.71 (1 mark for two correct, 2 marks for all three).
   2. The lowest price per battery is in the pack of 24. (1 mark)
   3. One pack of each size gives exactly \(24+10+6=40\) batteries for \(17+7.60+4.80=£29.40\). Other choices cost more: four packs of 10 cost £30.40 and two packs of 24 cost £34. The cheapest way is £29.40. (1 mark)

   Best value per battery does not always mean cheapest overall, because buying more than you need can waste money.

Total : 20 marks