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

Foundation. 15 questions, 15 marks, about 22 minutes.

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

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.
  1. 11 mark

    Write the ratio 56 : 42 : 70 in its simplest form.

    1. A 28 : 21 : 35
    2. B 4 : 3 : 5
    3. C 5 : 3 : 4
    4. D 8 : 6 : 10
  2. 21 mark

    195 pounds is shared between Amir, Priya and Dev in the ratio 2 : 3 : 8. How much does Dev receive?

    1. A 65 pounds
    2. B 45 pounds
    3. C 96 pounds
    4. D 120 pounds
  3. 31 mark

    A map has a scale of 1 : 25000. Two villages are 8 cm apart on the map. What is the actual distance between them, in kilometres?

    1. A 2 km
    2. B 20 km
    3. C 200 km
    4. D 0.2 km
  4. 41 mark

    A boy is 156 cm tall and his younger sister is 120 cm tall. Express the boy's height as a fraction of his sister's height, giving your answer in its simplest form.

    1. A 10/13
    2. B 39/30
    3. C 13/10
    4. D 3/10
  5. 51 mark

    A shade of green paint is made by mixing blue and yellow paint in the ratio 3 : 5. A customer wants to make 640 ml of this shade. How much yellow paint is needed?

    1. A 240 ml
    2. B 400 ml
    3. C 128 ml
    4. D 320 ml
  6. 61 mark

    In a class, 2/5 of the students are boys and the rest are girls. Express the ratio of boys to girls in its simplest form.

    1. A 2 : 3
    2. B 2 : 5
    3. C 3 : 2
    4. D 5 : 3
  7. 71 mark

    After a 20% pay rise, a worker's new hourly wage is 14.40 pounds. What was the wage before the rise?

    1. A 11.52 pounds
    2. B 17.28 pounds
    3. C 12.00 pounds
    4. D 14.20 pounds
  8. 81 mark

    y is directly proportional to x. When x = 8, y = 20. Which equation correctly gives y in terms of x?

    1. A y = 2.5x + 20
    2. B y = 2.5x
    3. C y = 0.4x
    4. D y = 4x
  9. 91 mark

    The cost, C pounds, of printing n leaflets is directly proportional to n. It costs 18 pounds to print 240 leaflets. How much does it cost to print 360 leaflets?

    1. A 27 pounds
    2. B 12 pounds
    3. C 9 pounds
    4. D 36 pounds
  10. 101 mark

    The time, T hours, taken to fill a tank is inversely proportional to the number of pumps used, p. Using 4 pumps takes 9 hours. How long would it take using 6 pumps?

    1. A 13.5 hours
    2. B 4.5 hours
    3. C 7 hours
    4. D 6 hours
  11. 111 mark

    Two similar cylinders have radii in the ratio 2 : 3. The volume of the smaller cylinder is 64 cm^3. What is the volume of the larger cylinder?

    1. A 96 cm^3
    2. B 144 cm^3
    3. C 216 cm^3
    4. D 72 cm^3
  12. 121 mark

    800 pounds is invested for 2 years at 5% compound interest per year. What is the value of the investment at the end of the 2 years?

    1. A 840 pounds
    2. B 882 pounds
    3. C 880 pounds
    4. D 842 pounds
  13. 131 mark

    Calculate the simple interest earned on 600 pounds invested for 4 years at a rate of 3% per year.

    1. A 72 pounds
    2. B 18 pounds
    3. C 672 pounds
    4. D 7200 pounds
  14. 141 mark

    The energy stored in a spring, E, is directly proportional to the square of its extension, x. When x = 3 cm, E = 18 J. What is E when x = 6 cm?

    1. A 36 J
    2. B 216 J
    3. C 12 J
    4. D 72 J
  15. 151 mark

    Triangle ABC is similar to triangle PQR, with AB corresponding to PQ and AC corresponding to PR. AB = 6 cm, PQ = 15 cm and AC = 8 cm. What is the length of PR?

    1. A 3.2 cm
    2. B 17 cm
    3. C 20 cm
    4. D 24 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. Write each term as a product of primes: 56 = 2^3 x 7, 42 = 2 x 3 x 7, 70 = 2 x 5 x 7.
    2. The factors common to all three terms are 2 and 7, so the highest common factor is 2 x 7 = 14.
    3. Divide every term by 14: 56/14 = 4, 42/14 = 3, 70/14 = 5.
    4. The ratio in its simplest form, keeping the original order, is 4 : 3 : 5, so the answer is B.
    • Why not A: Divides each term by 2, which is a common factor, but not the highest common factor of 14, so the ratio is not fully simplified.
    • Why not C: Reaches the correct three numbers but writes them in the wrong order, which changes what the ratio represents.
    • Why not D: Divides each term by 7, another common factor, but again stops short of dividing by the full highest common factor of 14.

  2. Question 2Answer: D

    1. The ratio 2 : 3 : 8 has 2 + 3 + 8 = 13 total parts.
    2. One part is therefore worth 195/13 = 15 pounds.
    3. Dev's share is 8 parts, so Dev receives 8 x 15 = 120 pounds.
    4. The answer is D.
    • Why not A: Divides 195 equally between the three people (195/3), ignoring the given ratio altogether.
    • Why not B: Correctly finds the value of one part as 15, but then multiplies by Priya's share of 3 instead of Dev's share of 8.
    • Why not C: Miscalculates the value of one part, treating 195/13 as 12 rather than 15, then scales up by Dev's share of 8.

  3. Question 3Answer: A

    1. The scale 1 : 25000 means 1 cm on the map represents 25000 cm in real life.
    2. The map distance is 8 cm, so the actual distance is 8 x 25000 = 200000 cm.
    3. Convert to kilometres using 1 km = 100000 cm: 200000 / 100000 = 2.
    4. The actual distance is 2 km, so the answer is A.
    • Why not B: Converts centimetres to kilometres using 1 km = 10000 cm instead of the correct 100000 cm, giving an answer ten times too large.
    • Why not C: Treats the actual distance in centimetres as if it were already in metres, then converts that figure to kilometres, skipping the centimetre-to-metre step.
    • Why not D: Divides by 1000000 to convert to kilometres, one power of ten too many, giving an answer ten times too small.

  4. Question 4Answer: C

    1. The boy's height as a fraction of his sister's height is 156/120.
    2. 156 = 2^2 x 3 x 13 and 120 = 2^3 x 3 x 5, so their highest common factor is 2^2 x 3 = 12.
    3. Dividing numerator and denominator by 12 gives 156/12 = 13 and 120/12 = 10.
    4. The boy's height is 13/10 of his sister's height, so the answer is C.
    • Why not A: Divides the sister's height by the boy's height instead of the boy's height by the sister's, inverting the fraction that was asked for.
    • Why not B: Reaches 39/30 by dividing both terms by 4, a common factor, but not the highest common factor of 12, so the fraction is not fully simplified.
    • Why not D: Uses the 36 cm difference in height as the numerator instead of the boy's own height of 156 cm.

  5. Question 5Answer: B

    1. The ratio of blue to yellow paint is 3 : 5, giving 3 + 5 = 8 total parts.
    2. One part is therefore 640/8 = 80 ml.
    3. Yellow paint corresponds to 5 parts, so the yellow paint needed is 5 x 80 = 400 ml.
    4. The answer is B.
    • Why not A: Correctly finds the amount of blue paint (3 parts) rather than the yellow paint (5 parts) that the question asks for.
    • Why not C: Finds the value of one part using an incorrect total of 5 (forgetting to add blue's 3 parts to it), and stops there without multiplying by yellow's share of 5.
    • Why not D: Splits the paint into two equal halves, 320 ml and 320 ml, ignoring the given ratio entirely.

  6. Question 6Answer: A

    1. Boys are 2/5 of the class, so girls make up 1 - 2/5 = 3/5 of the class.
    2. The ratio of boys to girls is therefore 2/5 : 3/5.
    3. Multiplying both sides by 5 to clear the denominators gives 2 : 3.
    4. 2 and 3 share no common factor, so this is already in simplest form, and the answer is A.
    • Why not B: Writes the ratio directly from the boys' fraction 2/5 as '2 : 5', without first working out that girls make up 3/5 of the class.
    • Why not C: Correctly works out the fractions 2/5 and 3/5 but writes the ratio girls to boys instead of the boys to girls ratio asked for.
    • Why not D: Treats the whole class, written as 5, and the girls' numerator, 3, as the two sides of the ratio, instead of comparing the boys' and girls' fractions directly.

  7. Question 7Answer: C

    1. A 20% pay rise means the new wage is 120% of the original, so new wage = 1.20 x original wage.
    2. Rearranging, the original wage = new wage / 1.20 = 14.40 / 1.20.
    3. 14.40 / 1.20 = 1440 / 120 = 12.
    4. The original wage was 12.00 pounds, so the answer is C.
    • Why not A: Finds 20% of the new wage (2.88 pounds) and subtracts it from 14.40, rather than dividing by the multiplier 1.20.
    • Why not B: Adds 20% of the new wage back on, treating the new wage as though it still needed to be increased.
    • Why not D: Subtracts 0.20 as if it were an amount in pounds rather than a percentage of the wage.

  8. Question 8Answer: B

    1. Direct proportion means y = kx for some constant k.
    2. Substituting x = 8 and y = 20 gives 20 = k x 8, so k = 20/8.
    3. 20/8 simplifies to 5/2, which is 2.5.
    4. The equation is y = 2.5x, so the answer is B.
    • Why not A: Assumes the relationship needs an added constant term, rather than recognising direct proportion as a straight line through the origin.
    • Why not C: Finds the constant of proportionality as x/y = 8/20 = 0.4 instead of y/x = 20/8, inverting the ratio.
    • Why not D: Miscalculates 20 divided by 8, arriving at 4 rather than the correct value of 2.5.

  9. Question 9Answer: A

    1. Cost is directly proportional to the number of leaflets, so C = kn for some constant k.
    2. Using C = 18 when n = 240 gives k = 18/240 = 3/40.
    3. For n = 360, C = (3/40) x 360 = 3 x 9 = 27.
    4. The cost is 27 pounds, so the answer is A.
    • Why not B: Sets up the proportion backwards, calculating 18 x 240/360, as if cost decreased with more leaflets rather than increasing with them.
    • Why not C: Correctly finds the extra cost for the additional 120 leaflets (9 pounds) but forgets to add this to the original 18 pounds.
    • Why not D: Rounds the scale factor 360/240 up to 2 instead of using its exact value of 1.5, doubling the cost instead of scaling it correctly.

  10. Question 10Answer: D

    1. Inverse proportion means T = k/p for some constant k.
    2. Using T = 9 when p = 4 gives k = 9 x 4 = 36.
    3. For p = 6, T = 36/6 = 6.
    4. It would take 6 hours, so the answer is D.
    • Why not A: Treats T and p as directly proportional, scaling 9 hours up by 6/4 to get 13.5, instead of using the inverse relationship.
    • Why not B: Assumes a constant saving of 9/4 = 2.25 hours for each extra pump, giving a linear rather than inverse relationship between time and pumps.
    • Why not C: Assumes each of the 2 extra pumps saves exactly 1 hour, subtracting a flat 2 hours from the original 9 rather than applying inverse proportion.

  11. Question 11Answer: C

    1. For similar solids, volumes scale with the cube of the linear scale factor.
    2. The radii are in ratio 2 : 3, so the linear scale factor from the smaller to the larger cylinder is 3/2.
    3. The volume scale factor is (3/2)^3 = 27/8.
    4. The larger volume is 64 x 27/8 = 8 x 27 = 216 cm^3, so the answer is C.
    • Why not A: Scales the volume by the linear radius ratio 3/2 directly, without cubing it as volumes of similar solids require.
    • Why not B: Scales the volume by the squared ratio (3/2)^2, as if finding a scaled area rather than a scaled volume.
    • Why not D: Computes 3^3 as 3 x 3 = 9 rather than 3 x 3 x 3 = 27, understating the volume scale factor as 9/8 instead of 27/8.

  12. Question 12Answer: B

    1. Compound interest means each year's interest is calculated on the new balance, so the multiplier for one year is 1.05.
    2. After year 1: 800 x 1.05 = 840.
    3. After year 2: 840 x 1.05 = 882.
    4. The investment is worth 882 pounds, so the answer is B.
    • Why not A: Applies the 5% interest for one year only, forgetting to compound for the second year.
    • Why not C: Uses simple interest, adding 5% of the original 800 pounds (40 pounds) in each of the two years, rather than compounding on the new balance.
    • Why not D: In the second year, applies 5% to only the first year's interest of 40 pounds rather than to the whole new balance of 840 pounds.

  13. Question 13Answer: A

    1. Simple interest is calculated only on the original principal each year, using I = Prt/100.
    2. Here P = 600, r = 3 and t = 4.
    3. I = (600 x 3 x 4)/100 = 7200/100 = 72.
    4. The simple interest earned is 72 pounds, so the answer is A.
    • Why not B: Calculates the interest for a single year only (600 x 3/100 = 18) and forgets to multiply by the 4 years invested.
    • Why not C: Finds the total amount in the account, principal plus interest, rather than the interest alone that the question asks for.
    • Why not D: Forgets to divide by 100 when applying the percentage rate, treating 3% as a factor of 3 rather than 0.03.

  14. Question 14Answer: D

    1. E is proportional to x^2, so E = kx^2 for some constant k.
    2. Using E = 18 when x = 3 gives 18 = k x 3^2 = 9k, so k = 2.
    3. When x = 6, E = 2 x 6^2 = 2 x 36 = 72.
    4. The answer is D.
    • Why not A: Assumes E is directly (linearly) proportional to x, so doubling x from 3 to 6 simply doubles E from 18 to 36.
    • Why not B: Finds the constant using k = 18/3 = 6, as if E = kx, rather than the correct k = 18/9 = 2 from E = kx^2, then still squares x correctly and calculates E = 6 x 6^2 = 216.
    • Why not C: Correctly finds k = 2 from E = kx^2, but then calculates E = 2 x 6 = 12, forgetting to square x again in the final step.

  15. Question 15Answer: C

    1. Since the triangles are similar with AB corresponding to PQ, the scale factor from triangle ABC to triangle PQR is PQ/AB = 15/6 = 5/2.
    2. AC corresponds to PR, so PR = AC x scale factor.
    3. PR = 8 x 5/2 = 20.
    4. The length of PR is 20 cm, so the answer is C.
    • Why not A: Forms the scale factor as AB/PQ = 6/15 instead of PQ/AB = 15/6, inverting which triangle is being scaled up to which.
    • Why not B: Treats the similarity as additive, adding the fixed difference of 9 cm between AB and PQ onto AC, rather than multiplying by a common scale factor.
    • Why not D: Rounds the scale factor 15/6 = 2.5 up to 3 before multiplying, an arithmetic slip that overstates the scaled length.

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