Ratio, Proportion and Rate of Change

1. 1

   Share £120 in the ratio $2 : 3 : 5$. What is the largest share?

   • A

     $£24$

   • B

     $£36$

   • C

     $£60$

   • D

     $£50$

   Check answer

   Answer & explanation

   Correct: option C

   Total parts $= 2 + 3 + 5 = 10$. One part $= £120 \div 10 = £12$. Largest share $= 5 \times £12 = £60$.

2. 2

   $y$ is directly proportional to $x^2$. When $x = 4$, $y = 48$. What is $y$ when $x = 5$?

   • A

     $60$

   • B

     $75$

   • C

     $80$

   • D

     $100$

   Check answer

   Answer & explanation

   Correct: option B

   $y = kx^2$. Substituting: $48 = k(16) \implies k = 3$. When $x = 5$: $y = 3(25) = 75$.

3. 3

   A jacket is sold for £85 after a 15% reduction. What was the original price?

   • A

     $£97.75$

   • B

     $£100$

   • C

     $£72.25$

   • D

     $£102$

   Check answer

   Answer & explanation

   Correct: option B

   Multiplier for 15% reduction $= 0.85$. Original price $= £85 \div 0.85 = £100$.

4. 4

   £3 000 is invested at 5% compound interest per annum. How much is the investment worth after 3 years?

   • A

     $£3 450.00$

   • B

     $£3 457.88$

   • C

     $£3 472.88$

   • D

     $£3 465.00$

   Check answer

   Answer & explanation

   Correct: option B

   $A = 3000 \times 1.05^3 = 3000 \times 1.157625 = £3 472.88$. Wait — $1.05^3 = 1.157625$; $3000 \times 1.157625 = £3 472.88$. The correct answer is C.

5. 5

   A car travels 150 km in 2.5 hours. What is its average speed in km/h?

   • A

     $50 \text{ km/h}$

   • B

     $55 \text{ km/h}$

   • C

     $60 \text{ km/h}$

   • D

     $65 \text{ km/h}$

   Check answer

   Answer & explanation

   Correct: option C

   $S = D \div T = 150 \div 2.5 = 60 \text{ km/h}$.

6. 6

   A metal block has a mass of 270 g and a volume of 100 cm³. What is its density?

   • A

     $2.5 \text{ g/cm}^3$

   • B

     $2.7 \text{ g/cm}^3$

   • C

     $27 \text{ g/cm}^3$

   • D

     $0.37 \text{ g/cm}^3$

   Check answer

   Answer & explanation

   Correct: option B

   $\text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{270}{100} = 2.7 \text{ g/cm}^3$.

7. 7

   $y$ is inversely proportional to $x$. When $x = 6$, $y = 4$. What is $x$ when $y = 12$?

   • A

     $x = 2$

   • B

     $x = 3$

   • C

     $x = 8$

   • D

     $x = 18$

   Check answer

   Answer & explanation

   Correct: option A

   $y = k/x$. Substituting: $4 = k/6 \implies k = 24$. When $y = 12$: $12 = 24/x \implies x = 2$.

8. 8

   A population of bacteria doubles every hour. Starting with 500, how many are there after 4 hours?

   • A

     $2 000$

   • B

     $4 000$

   • C

     $8 000$

   • D

     $16 000$

   Check answer

   Answer & explanation

   Correct: option C

   $A = 500 \times 2^4 = 500 \times 16 = 8 000$.

Exam-style multi-part questions. Try each part on paper, then reveal the model answer to compare your working.

1. Q16 marks total

   Show all answers

   Scenario

   Orange paint is made by mixing red paint and yellow paint in the ratio $3 : 2$.

   1. a

      How much yellow paint is needed to mix with 12 litres of red paint?

      [2]
      Show answer

      Model answer · 2 marks

      $\frac{\text{yellow}}{\text{red}} = \frac{2}{3}$. Yellow $= \frac{2}{3} \times 12 = 8$ litres.

   2. b

      A different shade uses red, yellow and white in the ratio $4 : 3 : 1$. A total of 40 litres of this paint is made. Find the amount of each colour.

      [3]
      Show answer

      Model answer · 3 marks

      Total parts $= 4 + 3 + 1 = 8$. One part $= 40 \div 8 = 5$ litres. Red $= 20$ litres, yellow $= 15$ litres, white $= 5$ litres.

   3. c

      Is it possible to make exactly 7 litres of the original orange paint ($3:2$ ratio) using a whole number of litres of each colour? Justify your answer.

      [1]
      Show answer

      Model answer · 1 mark

      Total parts $= 5$; one part $= 7 \div 5 = 1.4$ litres. Since $1.4$ is not a whole number, it is not possible to use whole number amounts of each colour.

   Examiner note

   Part (c) requires a clear justification — candidates must show that the total parts do not divide exactly into the given total.

2. Q27 marks total

   Show all answers

   Scenario

   The value $V$ (£) of a car $t$ years after purchase is modelled by $V = 18\,000 \times 0.75^t$.

   1. a

      Write down the purchase price of the car.

      [1]
      Show answer

      Model answer · 1 mark

      When $t = 0$: $V = 18\,000 \times 0.75^0 = 18\,000 \times 1 = £18\,000$.

   2. b

      Find the value of the car after 3 years. Give your answer to the nearest pound.

      [2]
      Show answer

      Model answer · 2 marks

      $V = 18\,000 \times 0.75^3 = 18\,000 \times 0.421875 = £7\,594$ (nearest £).

   3. c

      What percentage does the car lose in value each year? Explain how you can tell from the formula.

      [2]
      Show answer

      Model answer · 2 marks

      The multiplier is $0.75 = 1 - 0.25$, so the car loses $25\%$ of its value each year. This is read directly from the base of the exponential: a multiplier of $0.75$ represents a $25\%$ decrease per year.

   4. d

      After how many complete years is the car worth less than £5 000?

      [2]
      Show answer

      Model answer · 2 marks

      Solve $18\,000 \times 0.75^t < 5\,000 \implies 0.75^t < 0.2778$. Trial: $0.75^4 = 0.3164$; $0.75^5 = 0.2373$. Since $0.2373 < 0.2778$, the car is worth less than £5 000 after 5 complete years.

   Examiner note

   Part (c) tests understanding of the structure of the exponential model — not just a numerical calculation.

3. Q37 marks total

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   Scenario

   The graph shows the distance (km) travelled by a cyclist plotted against time (minutes).

   1. a

      The cyclist travels at a constant speed for the first 20 minutes, covering 8 km. Calculate the speed in km/h.

      [2]
      Show answer

      Model answer · 2 marks

      Speed $= \frac{8 \text{ km}}{20 \text{ min}} = \frac{8}{20/60} \text{ km/h} = \frac{8 \times 60}{20} = 24 \text{ km/h}$.

   2. b

      Between 20 and 30 minutes the distance remains constant at 8 km. Describe what the cyclist is doing and state the gradient of the graph in this section.

      [2]
      Show answer

      Model answer · 2 marks

      The cyclist is stationary (resting). The gradient is $0$ km/min — no distance is being covered.

   3. c

      Between 30 and 50 minutes the cyclist travels a further 16 km. Calculate the average speed for the entire 50-minute journey.

      [3]
      Show answer

      Model answer · 3 marks

      Total distance $= 8 + 0 + 16 = 24$ km. Total time $= 50$ min $= \frac{50}{60}$ h. Average speed $= \frac{24}{50/60} = \frac{24 \times 60}{50} = 28.8$ km/h.

   Examiner note

   Part (c) requires total distance divided by total time — candidates who average the two non-zero speeds will lose marks.