Fractions, decimals and percentages – Year 8 maths test answers to download as PDF

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1. Answer to Question 1 (4 marks)

   1. Divide the numerator by the denominator: \(7\div8=\)0.875 (1 mark). Multiply by 100 to get a percentage: 87.5% (1 mark).
   2. Short division of 4 by 11 gives 0.363636…, with the digits 3 and 6 repeating for ever. Put dots over the first and last digits of the repeating block: \(0.\dot3\dot6\). (1 mark)
   3. Change everything into decimals: 0.6, \(\dfrac{5}{8}=0.625\) and 61% = 0.61. Compare the hundredths digits: 0.60, 0.61, 0.625. The order is 0.6, 61%, \(\frac58\). (1 mark)

   Check (a): \(\dfrac{1}{8}=0.125\), and seven of these make 0.875. Writing 0.6 as 0.60 in (c) avoids thinking it is bigger than 0.61.

2. Answer to Question 2 (4 marks)

   1. Build 35% from simpler percentages (1 mark for method). Start from 10%, which is \(240\div10=£24\); three lots of it make 30%, or £72. Halving £24 gives 5%, which is £12. Then \(35\%=72+12=\)£84 (1 mark).
   2. An increase of 15% gives 115% of the original, so the multiplier is 1.15: \(64\times1.15=\)£73.60. Write money with two decimal places. (1 mark)
   3. Write 36 as a fraction of 48 and simplify: \(\dfrac{36}{48}=\dfrac{3}{4}=0.75=\)75%. (1 mark)

   Check (a) with a multiplier: \(240\times0.35=84\). Check (b): 15% of 64 is \(6.40+3.20=9.60\), and \(64+9.60=73.60\).

3. Answer to Question 3 (4 marks)

   1. A 15% reduction leaves \(100-15=85\%\) of the price, so the multiplier is 0.85 (1 mark). Sale price \(=320\times0.85=\)£272 (1 mark).
   2. A 15% rise uses the multiplier 1.15: \(272\times1.15=£312.80\) (1 mark). This is not £320; it is £7.20 less. The rise is 15% of £272, a smaller amount than the £48 that was taken off £320, so the price does not get back to where it started (1 mark).

   The two changes together multiply the price by \(0.85\times1.15=0.9775\), which is a 2.25% overall fall. Percentage changes cannot simply be added and subtracted.

4. Answer to Question 4 (4 marks)

   1. After a 30% reduction, £42 is 70% of the original price (1 mark). Divide by the multiplier: \(42\div0.7=\)£60 (1 mark).
   2. Including 20% VAT, £65 is 120% of the price before VAT (1 mark). Divide by 1.2: \(65\div1.2=54.166\ldots\), which is £54.17 to the nearest penny (1 mark).

   Check (a): 30% of £60 is £18, and \(60-18=42\). Check (b): \(54.17\times1.2=65.00\). A classic mistake is to take 30% off £42 or to find 20% of £65; the percentage always belongs to the original price, not to the new one, so we must divide rather than subtract.

5. Answer to Question 5 (4 marks)

   1. Each year the amount is multiplied by 1.04 (1 mark). After 2 years: \(500\times1.04^{2}=500\times1.0816=\)£540.80 (1 mark).
   2. Keep multiplying by 1.04: year 1 £520, year 2 £540.80, year 3 £562.43, year 4 £584.93, year 5 £608.33 (1 mark for the table of values). The amount first goes over £600 at the end of year 5, so it takes 5 years (1 mark).

   With simple interest Ravi would earn £20 every year and would need exactly 5 years to reach £600, so he would not be over it. Compound interest earns interest on the interest as well, which is why the balance grows a little faster each year.

Total : 20 marks