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Admissions tests / ESAT / Maths 1 / Ratio, proportion and rates of change

Demanding. 15 questions, 15 marks, about 27 minutes.

ESAT Mathematics 1: Ratio, proportion and rates of change, set 3

Scale factors and diagrams, ratio notation and division in a ratio, percentages, direct and inverse proportion, growth and decay, and compound measures.

Download the questions (PDF) Download with worked solutions (PDF)

  • Answer all questions. No calculator.
  • Each question has exactly one correct answer.
  • This is a demanding set, pitched above real test standard: expect multi-step routes and less signposting than the real exam.
  1. 11 mark

    Two similar solid shapes have volumes in the ratio 8 : 27. What is the ratio of their surface areas, in its simplest form?

    1. A 2 : 3
    2. B 4 : 9
    3. C 8 : 27
    4. D 64 : 729
  2. 21 mark

    y is inversely proportional to x^2. If x is doubled, what is the percentage change in y?

    1. A A decrease of 50%
    2. B A decrease of 25%
    3. C An increase of 300%
    4. D A decrease of 75%
  3. 31 mark

    A shop increases the price of an item by 20%. In a later sale, the new price is reduced by 25%. The sale price is 270 pounds. What was the price before the increase?

    1. A 300 pounds
    2. B 225 pounds
    3. C 360 pounds
    4. D 180 pounds
  4. 41 mark

    z varies directly with x and inversely with y. When x = 4 and y = 3, z = 12. What is the value of z when x = 6 and y = 9?

    1. A 18
    2. B 4
    3. C 6
    4. D 24
  5. 51 mark

    A savings account pays compound interest at a rate of r% per year. After 2 years, 1000 pounds grows to 1210 pounds. What is the value of r?

    1. A 21
    2. B 10
    3. C 10.5
    4. D 110
  6. 61 mark

    A chemist mixes x litres of a 20% acid solution with y litres of a 50% acid solution to make 12 litres of a 30% acid solution. What is the value of x?

    1. A 8
    2. B 6
    3. C 4
    4. D 3.6
  7. 71 mark

    A rope is cut into three pieces in the ratio 2 : 3 : 4. The longest piece is 10 cm longer than the shortest piece. What is the length of the middle piece, in cm?

    1. A 7.5
    2. B 30
    3. C 45
    4. D 15
  8. 81 mark

    The time period T of a pendulum, in seconds, is directly proportional to the square root of its length L, in cm. A pendulum of length 40 cm has a period of 2 seconds. What length of pendulum has a period of 5 seconds?

    1. A 100 cm
    2. B 16 cm
    3. C 250 cm
    4. D 6.4 cm
  9. 91 mark

    Tap A alone would fill a tank in 6 hours. Tap B alone would fill the same tank in 3 hours. Working together, how long would the two taps take to fill the tank?

    1. A 4.5 hours
    2. B 2 hours
    3. C 9 hours
    4. D 0.5 hours
  10. 101 mark

    The number of tiles needed to tile a floor is inversely proportional to the square of the tile's side length. 720 tiles of side length 15 cm are needed to tile a floor. How many tiles of side length 20 cm would be needed to tile the same floor?

    1. A 540
    2. B 1280
    3. C 960
    4. D 405
  11. 111 mark

    A scale model of a statue is built at a scale of 1 : 20 (model : actual). The model has a volume of 15 cm^3. What is the volume of the actual statue, in cm^3?

    1. A 120000
    2. B 300
    3. C 6000
    4. D 900
  12. 121 mark

    A line has equation 4y = 3x + 12. A point (x, y) on this line satisfies x : y = 4 : 5. What is the value of x?

    1. A 60
    2. B 6
    3. C -6
    4. D 0
  13. 131 mark

    David starts with a sum of money. He spends 2/7 of it on a jacket, then spends 1/3 of what remains on a book. He has 20 pounds left. How much money did David start with?

    1. A 28 pounds
    2. B 52.5 pounds
    3. C 42 pounds
    4. D 84 pounds
  14. 141 mark

    A sequence is defined by the iterative rule a_(n+1) = 0.5 x a_n + 6, with a_1 = 4. What is the value of a_3?

    1. A 8
    2. B 11
    3. C 5.5
    4. D 10
  15. 151 mark

    Triangle ABC has a right angle at B, with AB = 9 cm and BC = 12 cm. Triangle ABC is similar to triangle XYZ, with the area of XYZ being 4 times the area of ABC. What is the length of XZ, which corresponds to AC?

    1. A 30 cm
    2. B 60 cm
    3. C 24 cm
    4. D 42 cm

Worked solutions

Every question below carries the reasoning, not just the answer. The official material for this test publishes a correct option letter and nothing else.

  1. Question 1Answer: B

    1. A volume ratio of 8 : 27 is a ratio of cubes, since 8 = 2^3 and 27 = 3^3, so the linear scale factor ratio is 2 : 3.
    2. Surface area is a two-dimensional measure, so it scales with the square of the linear scale factor: 2^2 : 3^2 = 4 : 9.
    3. The answer is B.
    • Why not A: Correctly converts the volume ratio to the linear scale factor ratio (the cube roots of 8 and 27), but forgets to square this to get an area ratio, leaving it as a length ratio instead.
    • Why not C: Uses the volume ratio directly as the surface area ratio, without converting it to a linear scale factor first.
    • Why not D: Squares the volume ratio itself (8^2 : 27^2), instead of first taking cube roots to find the linear scale factor and only then squaring that.

  2. Question 2Answer: D

    1. y is inversely proportional to x^2, so y = k/x^2 for some constant k.
    2. If x doubles to 2x, the new value of y is k/(2x)^2 = k/(4x^2) = y/4.
    3. Falling from y to y/4 is a fall of (3/4)y, which is a decrease of 75%.
    4. The answer is D.
    • Why not A: Treats y as inversely proportional to x itself rather than to x^2, so doubling x is thought to simply halve y.
    • Why not B: Correctly finds that the new value of y is one quarter of the original, but then reports this leftover fraction (a quarter) as the size of the decrease, instead of recognising that falling to one quarter is a decrease of three quarters.
    • Why not C: Treats y as directly, not inversely, proportional to x^2, so doubling x is thought to quadruple y, an increase of 300%.

  3. Question 3Answer: A

    1. Let the original price be P pounds.
    2. After the 20% increase, the price is 1.2P. After the 25% reduction in the sale, the price is 1.2P x 0.75 = 0.9P.
    3. This equals 270 pounds, so 0.9P = 270 and P = 270 / 0.9 = 300.
    4. The answer is A.
    • Why not B: Divides the sale price by 1.2 only, undoing the 20% increase but not the later 25% reduction, as if 270 pounds were the price straight after the increase.
    • Why not C: Divides the sale price by 0.75 only, undoing the 25% reduction but not the earlier 20% increase, as if 270 pounds were the original price before just the sale.
    • Why not D: Treats the 25% reduction as a further increase by mistake, dividing by 1.2 x 1.25 = 1.5 instead of by 1.2 x 0.75 = 0.9.

  4. Question 4Answer: C

    1. z = kx/y for some constant k, since z varies directly with x and inversely with y.
    2. Using x = 4, y = 3, z = 12: k = zy/x = (12 x 3)/4 = 9.
    3. When x = 6 and y = 9: z = kx/y = (9 x 6)/9 = 6.
    4. The answer is C.
    • Why not A: Treats z as varying only directly with x, ignoring the inverse relationship with y altogether, using k = z/x = 3.
    • Why not B: Treats z as varying only inversely with y, ignoring the direct relationship with x altogether, using k = zy = 36.
    • Why not D: Swaps the two relationships, treating z as directly proportional to y and inversely proportional to x.

  5. Question 5Answer: B

    1. Let the yearly multiplier be 1 + r/100.
    2. After 2 years: 1000 x (1 + r/100)^2 = 1210, so (1 + r/100)^2 = 1.21.
    3. Since 1.1^2 = 1.21, 1 + r/100 = 1.1, so r/100 = 0.1 and r = 10.
    4. The answer is B.
    • Why not A: States the total percentage increase seen over the whole 2 years (21%) as if it were the annual compound rate, rather than finding the single rate that compounds to give this total.
    • Why not C: Divides the total 21% increase evenly across the 2 years, treating the growth as simple (additive) rather than compound (multiplicative).
    • Why not D: Correctly finds that the annual growth factor is 1.1, but reports this factor itself as the percentage rate instead of subtracting 1 first, giving 110 instead of 10.

  6. Question 6Answer: A

    1. Let x be the volume of the 20% solution and y the volume of the 50% solution, so x + y = 12.
    2. The total acid content must be 30% of 12 litres: 0.20x + 0.50y = 3.6.
    3. Substituting y = 12 - x: 0.20x + 0.50(12 - x) = 3.6, so 0.20x + 6 - 0.50x = 3.6, giving -0.30x = -2.4.
    4. So x = 2.4 / 0.30 = 8.
    5. The answer is A.
    • Why not B: Assumes the two solutions are mixed in equal volumes, ignoring the different concentrations and the specific 30% target, giving x = y = 6.
    • Why not C: Correctly finds y, the volume of the 50% solution, but reports this value (4 litres) as x, the quantity the question actually asks for.
    • Why not D: Correctly calculates the total amount of pure acid needed (0.30 x 12 = 3.6 litres) but stops there and gives this intermediate quantity as x, instead of continuing to solve for it.

  7. Question 7Answer: D

    1. Let the three pieces be 2k, 3k and 4k cm.
    2. The longest exceeds the shortest by 4k - 2k = 2k, and this difference is 10 cm, so 2k = 10 and k = 5.
    3. The middle piece is 3k = 3 x 5 = 15 cm.
    4. The answer is D.
    • Why not A: Divides the 10 cm difference by the longest piece's ratio number (4) instead of the difference between the longest and shortest ratio numbers (4 - 2 = 2), then finds the middle piece from this wrong value of k.
    • Why not B: Uses the difference between the longest and middle ratio numbers (4 - 3 = 1) instead of the longest and shortest (4 - 2 = 2), making k twice as large as it should be.
    • Why not C: Correctly finds k = 5 but then gives the rope's total length (2k + 3k + 4k = 45 cm) instead of the middle piece's length that the question asks for.

  8. Question 8Answer: C

    1. T is directly proportional to sqrt(L), so T = k x sqrt(L), which rearranges to L = (T/k)^2, meaning L is proportional to T^2.
    2. So L'/L = (T'/T)^2 = (5/2)^2 = 25/4.
    3. L' = 40 x 25/4 = 250.
    4. The answer is C.
    • Why not A: Treats L as directly proportional to T itself, not to T^2, so scales L by the same factor (5/2) as T instead of by its square.
    • Why not B: Uses the reciprocal ratio (T/T') without squaring, scaling L by 2/5 instead of by (5/2)^2.
    • Why not D: Treats the relationship as inverse rather than direct, scaling L by (T/T')^2 = (2/5)^2 instead of (T'/T)^2.

  9. Question 9Answer: B

    1. Tap A fills 1/6 of the tank each hour; Tap B fills 1/3 = 2/6 of the tank each hour.
    2. Working together, the combined rate is 1/6 + 2/6 = 3/6 = 1/2 of the tank per hour.
    3. The time to fill the whole tank is 1 / (1/2) = 2 hours.
    4. The answer is B.
    • Why not A: Averages the two individual times ((6 + 3)/2 = 4.5), treating the combined time as a simple mean rather than combining the rates.
    • Why not C: Adds the two individual times together (6 + 3 = 9), as if working together took longer than either tap alone.
    • Why not D: Correctly finds the combined rate as 1/2 of the tank per hour, but reports this rate itself (0.5) as the time taken, instead of taking its reciprocal.

  10. Question 10Answer: D

    1. The number of tiles N is inversely proportional to the square of the side length s, so N x s^2 is constant.
    2. N1 x s1^2 = N2 x s2^2, so N2 = N1 x (s1/s2)^2 = 720 x (15/20)^2 = 720 x 9/16.
    3. 720 / 16 = 45, and 45 x 9 = 405.
    4. The answer is D.
    • Why not A: Scales the number of tiles by the ratio of side lengths (15/20) directly, forgetting that the number of tiles varies with the square of the side length, not the side length itself.
    • Why not B: Forgets that the relationship is inverse, treating the number of tiles as directly proportional to the square of the side length, so scales by (20/15)^2 instead of (15/20)^2.
    • Why not C: Forgets both the square and the inverse relationship, scaling the number of tiles directly by the ratio of side lengths (20/15).

  11. Question 11Answer: A

    1. The model is built at a scale of 1 : 20, so lengths on the actual statue are 20 times the corresponding lengths on the model.
    2. Volume scales with the cube of the linear scale factor, so the volume scale factor is 20^3 = 8000.
    3. The actual volume is 15 x 8000 = 120000 cm^3.
    4. The answer is A.
    • Why not B: Scales the volume by the linear scale factor only (15 x 20), forgetting that volume scales with the cube of the linear scale factor.
    • Why not C: Scales the volume by the square of the linear scale factor (15 x 20^2), as if scaling an area rather than a volume.
    • Why not D: Treats 'cubed' as multiplying the scale factor by 3 rather than raising it to the power 3, using an effective factor of 20 x 3 = 60.

  12. Question 12Answer: B

    1. x : y = 4 : 5 means y = (5/4)x.
    2. Substituting into 4y = 3x + 12 gives 4 x (5/4)x = 3x + 12, which simplifies to 5x = 3x + 12.
    3. So 2x = 12, and x = 6.
    4. The answer is B.
    • Why not A: Uses the ratio the wrong way round, taking y = (4/5)x instead of y = (5/4)x from x : y = 4 : 5.
    • Why not C: Makes a sign error when rearranging, solving 5x = 3x - 12 instead of 5x = 3x + 12.
    • Why not D: Drops the constant term when rearranging, solving 5x = 3x instead of 5x = 3x + 12.

  13. Question 13Answer: C

    1. After the jacket, David has 1 - 2/7 = 5/7 of his money left.
    2. Spending 1/3 of this on the book leaves 2/3 of it: (2/3) x (5/7) = 10/21 of the original amount.
    3. This 10/21 share equals 20 pounds, so the original amount was 20 x 21/10 = 42 pounds.
    4. The answer is C.
    • Why not A: Finds the amount remaining after the first purchase correctly, but stops there, ignoring that a further 1/3 of this amount was then spent on the book.
    • Why not B: Treats the two fractions as if they simply add together (2/7 + 1/3), subtracting their sum from the whole, instead of applying the second fraction to what already remained after the first spend.
    • Why not D: Assumes that spending 1/3 of what remained leaves 1/3 of what remained, rather than the 2/3 that is actually left.

  14. Question 14Answer: D

    1. Apply the rule once: a_2 = 0.5 x 4 + 6 = 2 + 6 = 8.
    2. Apply the rule again: a_3 = 0.5 x 8 + 6 = 4 + 6 = 10.
    3. The answer is D.
    • Why not A: Stops after only one application of the rule, giving a_2 instead of continuing to apply it a second time to reach a_3.
    • Why not B: Applies the rule one time too many, computing a_4 instead of a_3.
    • Why not C: Misapplies the order of operations in the rule, adding 6 to a_n before halving instead of halving a_n first and then adding 6.

  15. Question 15Answer: A

    1. Since angle B = 90 degrees, AC can be found using Pythagoras' theorem: AC = sqrt(9^2 + 12^2) = sqrt(81 + 144) = sqrt(225) = 15 cm.
    2. Areas of similar shapes scale with the square of the linear scale factor, and the area ratio here is 4 : 1, so the linear scale factor is sqrt(4) = 2.
    3. XZ corresponds to AC, so XZ = 15 x 2 = 30 cm.
    4. The answer is A.
    • Why not B: Uses the area ratio itself (4) as the linear scale factor, instead of taking its square root (2) to convert an area ratio into a length ratio.
    • Why not C: Correctly finds the linear scale factor of 2, but applies it to the given side BC instead of the side XZ actually asked for, which corresponds to the hypotenuse AC.
    • Why not D: Finds AC by adding the two legs (9 + 12 = 21) instead of applying Pythagoras' theorem, then scales this incorrect length by the correct scale factor of 2.

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