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

Test standard. 15 questions, 15 marks, about 25 minutes.

ESAT Mathematics 1: Ratio, proportion and rates of change, set 2

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

    A scale drawing of a garden uses a scale of 1 : 50. A wall in the garden is 3.5 m long. How long is the wall on the drawing, in centimetres?

    1. A 17500 cm
    2. B 0.07 cm
    3. C 0.7 cm
    4. D 7 cm
  2. 21 mark

    A sum of money is shared between Ben and Chloe in the ratio 4 : 9. Chloe receives 85 pounds more than Ben. How much money was shared in total?

    1. A 221 pounds
    2. B 68 pounds
    3. C 153 pounds
    4. D 1105 pounds
  3. 31 mark

    A shop sells rice in two bag sizes: an 800 g bag for 240p, and a 1200 g bag for 300p. What is the price per 100 g of the cheaper (better value) bag, in pence?

    1. A 30p
    2. B 0.25p
    3. C 25p
    4. D 60p
  4. 41 mark

    The ratio of the length to the width of a flag is 5 : 3. Write this ratio in the form n : 1.

    1. A 3/5 : 1
    2. B 5/3 : 1
    3. C 8/3 : 1
    4. D 5/8 : 1
  5. 51 mark

    A quantity y is directly proportional to x. The graph of y against x is a straight line through the origin with gradient 3/4. When x = 24, what is the value of y?

    1. A 18
    2. B 32
    3. C 72
    4. D 6
  6. 61 mark

    In a fruit bowl, 3/8 of the fruits are apples, 1/4 are bananas and the rest are oranges. Write the ratio of apples : bananas : oranges in its simplest form.

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

    A cordial must be diluted with water in the ratio 1 : 6 (cordial : water) before drinking. A jug holds 2.1 litres of the diluted drink. How much cordial was used to make it?

    1. A 1.05 litres
    2. B 1.8 litres
    3. C 0.3 litres
    4. D 0.35 litres
  8. 81 mark

    A jacket's price is increased by 20%, and then in a later sale it is decreased by 20%. What is the overall percentage change in price, compared with the original price?

    1. A No change (0%)
    2. B A decrease of 4%
    3. C A decrease of 40%
    4. D An increase of 4%
  9. 91 mark

    y is inversely proportional to the square root of x. When x = 16, y = 5. What is the value of y when x = 100?

    1. A 31.25
    2. B 0.8
    3. C 12.5
    4. D 2
  10. 101 mark

    For x > 0, y is inversely proportional to x. Which of the following best describes the shape of the graph of y against x?

    1. A A curve that falls steadily, getting closer to both axes but never touching them
    2. B A straight line through the origin with a positive gradient
    3. C A straight line through the origin with a negative gradient
    4. D A parabola passing through the origin
  11. 111 mark

    Triangle DEF is similar to triangle GHI, with sides in the ratio DE : GH = 5 : 8. In triangle DEF, tan(angle D) = 3/4. What is tan(angle G), the corresponding angle in triangle GHI?

    1. A 15/32
    2. B 3/4
    3. C 6/5
    4. D 4/3
  12. 121 mark

    Two rectangular flags are mathematically similar. The larger flag's width is 1.5 times the smaller flag's width. The area of the smaller flag is 96 cm^2. What is the area of the larger flag?

    1. A 144 cm^2
    2. B 288 cm^2
    3. C 216 cm^2
    4. D 324 cm^2
  13. 131 mark

    A car is bought for 12000 pounds. Its value depreciates by 10% each year. What is the car's value after 3 years?

    1. A 8748 pounds
    2. B 8400 pounds
    3. C 10800 pounds
    4. D 15972 pounds
  14. 141 mark

    A population of bacteria increases by 8% every hour. If the population is P0 at time t = 0 (measured in hours), which expression correctly gives the population after t complete hours?

    1. A P0 x (1.8)^t
    2. B P0 x 1.08 x t
    3. C P0 + 0.08t
    4. D P0 x (1.08)^t
  15. 151 mark

    A small tank holds 45 litres and a large tank holds 60 litres. What percentage of the large tank's capacity is the small tank's capacity?

    1. A 133 1/3%
    2. B 75%
    3. C 15%
    4. D 0.75%

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: D

    1. A scale of 1 : 50 means 1 cm on the drawing represents 50 cm of actual length, so actual length = drawing length x 50.
    2. To go the other way, drawing length = actual length / 50.
    3. Convert the actual length to centimetres first: 3.5 m = 350 cm.
    4. Drawing length = 350 / 50 = 7, so the wall is 7 cm long on the drawing, and the answer is D.
    • Why not A: Multiplies the actual length by the scale factor instead of dividing, inverting the direction needed to go from a real length to a drawing length.
    • Why not B: Divides the actual length by the scale factor without first converting the 3.5 m to centimetres, so the units no longer match the answer's label.
    • Why not C: Converts 3.5 m to centimetres using 1 m = 10 cm instead of the correct 1 m = 100 cm, an error of one power of ten.

  2. Question 2Answer: A

    1. The ratio 4 : 9 means Chloe's share (9 parts) exceeds Ben's share (4 parts) by 9 - 4 = 5 parts.
    2. This 5-part difference equals 85 pounds, so one part is worth 85 / 5 = 17 pounds.
    3. The total number of parts is 4 + 9 = 13, so the total shared is 13 x 17 = 221 pounds.
    4. The answer is A.
    • Why not B: Calculates Ben's individual share (4 parts) rather than the total amount that was shared out.
    • Why not C: Calculates Chloe's individual share (9 parts) rather than the total amount that was shared out.
    • Why not D: Multiplies the total number of parts (13) directly by the difference of 85, instead of first dividing 85 by the difference in parts (5) to find the value of one part.

  3. Question 3Answer: C

    1. To compare value for money, find each bag's price per 100 g.
    2. The 800 g bag costs 240p, so its price per 100 g is 240 / 8 = 30p (dividing by 800/100 = 8).
    3. The 1200 g bag costs 300p, so its price per 100 g is 300 / 12 = 25p (dividing by 1200/100 = 12).
    4. 25p is less than 30p, so the 1200 g bag is the better-value one, and its price per 100 g is 25p, so the answer is C.
    • Why not A: Gives the price per 100 g of the 800 g bag (240 / 8 = 30p), which is the MORE expensive bag per 100 g, not the better-value one the question asks for.
    • Why not B: Finds the price per gram for the 1200 g bag (300 / 1200 = 0.25p) but forgets to scale this up to a price per 100 g by multiplying by 100.
    • Why not D: Subtracts the two total prices (300 - 240 = 60p) and mistakes this price difference for a price per 100 g, rather than comparing each bag's own price per 100 g.

  4. Question 4Answer: B

    1. To write a ratio in the form n : 1, divide both terms by the second term.
    2. Here that means dividing both 5 and 3 by 3: 5/3 : 3/3, which simplifies to 5/3 : 1.
    3. The answer is B.
    • Why not A: Divides both terms by 5 instead of by 3, which gives the width-to-length ratio in n : 1 form rather than the length-to-width ratio asked for.
    • Why not C: Adds the two parts of the ratio (5 + 3 = 8) to form the numerator, instead of keeping the length as the numerator and dividing by the width.
    • Why not D: Divides the length by the sum of both parts (5 + 3 = 8) instead of by the width alone (3).

  5. Question 5Answer: A

    1. For direct proportion through the origin, y = (gradient) x x.
    2. Here y = (3/4) x 24.
    3. Dividing 24 by 4 gives 6, then multiplying by 3 gives 18.
    4. The answer is A.
    • Why not B: Uses the reciprocal gradient 4/3 instead of 3/4, giving 24 x 4/3 = 32.
    • Why not C: Multiplies x by the numerator of the gradient only (24 x 3 = 72), forgetting to divide by the denominator.
    • Why not D: Divides x by the denominator of the gradient only (24 / 4 = 6), forgetting to multiply by the numerator.

  6. Question 6Answer: D

    1. Apples are 3/8 of the fruit and bananas are 1/4, which is 2/8.
    2. Oranges are the remainder: 8/8 - 3/8 - 2/8 = 3/8.
    3. The ratio apples : bananas : oranges, all over a common denominator of 8, is 3 : 2 : 3.
    4. No factor other than 1 divides all three numbers, so this is already in simplest form, and the answer is D.
    • Why not A: Correctly converts bananas' 1/4 to eighths (2/8), but makes an arithmetic slip in the final subtraction, computing 8/8 - 3/8 - 2/8 as 4/8 instead of the correct 3/8.
    • Why not B: Uses the numerators of 3/8 and 1/4 directly as 3 and 1, without converting 1/4 to eighths first, so bananas' share is understated and oranges is then found as 8 - 3 - 1 = 4.
    • Why not C: Correctly finds the three fractions as 3/8, 2/8 and 3/8, but lists them in the wrong order, giving bananas before apples.

  7. Question 7Answer: C

    1. The ratio cordial : water is 1 : 6, giving 1 + 6 = 7 total parts.
    2. One part is therefore 2.1 / 7 = 0.3 litres.
    3. Cordial is 1 part, so 0.3 litres of cordial was used.
    4. The answer is C.
    • Why not A: Splits the 2.1 litres equally between the two substances (2.1 / 2), ignoring the given ratio of 1 : 6 altogether.
    • Why not B: Finds the water's share instead of the cordial's, having reversed which ingredient corresponds to which part of the ratio.
    • Why not D: Divides by the water's 6 parts alone (2.1 / 6), forgetting that the cordial itself makes up a 7th part of the whole mixture.

  8. Question 8Answer: B

    1. Percentage changes are multiplicative, not additive. Take the original price as 100.
    2. A 20% increase gives 100 x 1.20 = 120.
    3. A 20% decrease on this new price gives 120 x 0.80 = 96.
    4. Compared with the original 100, this is a fall of 4, which is a 4% decrease, so the answer is B.
    • Why not A: Adds the two percentage changes together (+20% and -20%) and assumes they cancel out exactly, treating percentage change as additive rather than multiplicative.
    • Why not C: Subtracts both 20% amounts directly from the original price (100 - 20 - 20 = 60), applying both percentages to the original value instead of compounding the second change onto the increased price.
    • Why not D: Correctly finds the combined multiplier 1.20 x 0.80 = 0.96, but misreads a multiplier below 1 as representing an increase rather than a decrease.

  9. Question 9Answer: D

    1. y is inversely proportional to sqrt(x), so y = k / sqrt(x) for some constant k.
    2. When x = 16, sqrt(x) = 4, and y = 5, so k = 5 x 4 = 20.
    3. When x = 100, sqrt(x) = 10, so y = 20 / 10 = 2.
    4. The answer is D.
    • Why not A: Treats y as directly proportional to x itself, ignoring both the word 'inversely' and the square root, using k = y/x = 5/16 and then y = (5/16) x 100 = 31.25.
    • Why not B: Treats y as inversely proportional to x itself rather than to the square root of x, using k = xy = 5 x 16 = 80 and then y = 80/100 = 0.8.
    • Why not C: Treats y as directly proportional to the square root of x, forgetting that the relationship is inverse, using k = y/sqrt(x) = 5/4 and then y = 1.25 x sqrt(100) = 12.5.

  10. Question 10Answer: A

    1. Inverse proportion means y = k/x for some constant k > 0, with x > 0 so y > 0 too.
    2. As x increases, y decreases, but never reaches zero, and as x decreases towards 0, y grows without bound.
    3. This gives a smooth curve that falls steadily and approaches both axes without ever touching them, not a straight line or a parabola.
    4. The answer is A.
    • Why not B: Describes direct proportion, y = kx, not inverse proportion; confuses the two relationship types.
    • Why not C: Describes a linear relationship where y decreases at a constant rate, such as y = c - kx, which is not the same as y varying inversely with x, since inverse proportion is not linear at all.
    • Why not D: Describes y proportional to x^2, a different relationship entirely, not inverse proportion.

  11. Question 11Answer: B

    1. Similar triangles have equal corresponding angles; only their side lengths scale, by the linear scale factor.
    2. A trigonometric ratio such as tan of an angle depends only on the size of the angle, not on the size of the triangle it sits in.
    3. Angle G corresponds to, and equals, angle D, so tan(angle G) = tan(angle D) = 3/4.
    4. The side ratio 5 : 8 is irrelevant here; it would only matter if the question asked for an actual side length. The answer is B.
    • Why not A: Multiplies tan(angle D) by the linear scale factor 5/8, treating the trigonometric ratio as if it scales like a side length.
    • Why not C: Multiplies tan(angle D) by the inverse scale factor 8/5, making the same error as A but in the opposite direction.
    • Why not D: Inverts 3/4 to 4/3, confusing tan(angle D) with the tangent of its complementary angle.

  12. Question 12Answer: C

    1. For similar shapes, areas scale with the square of the linear scale factor.
    2. The linear scale factor from the smaller to the larger flag is 1.5, so the area scale factor is 1.5^2 = 2.25.
    3. The larger flag's area is 96 x 2.25 = 216 cm^2.
    4. The answer is C.
    • Why not A: Applies the linear scale factor of 1.5 directly to the area (96 x 1.5), without accounting for the fact that area scales with the square of the linear factor.
    • Why not B: Confuses 'squared' with 'doubled', using an area scale factor of 2 x 1.5 = 3 instead of 1.5 x 1.5 = 2.25.
    • Why not D: Cubes the linear scale factor (1.5^3 = 3.375), as if finding a volume scale factor, rather than squaring it for an area scale factor.

  13. Question 13Answer: A

    1. Compound depreciation means each year's 10% reduction is calculated on the previous year's value, so the multiplier for one year is 0.90.
    2. After 3 years, the value is 12000 x 0.90^3 = 12000 x 0.729.
    3. 12000 x 0.729 = 8748.
    4. The answer is A.
    • Why not B: Depreciates by 10% of the ORIGINAL value in each of the 3 years (simple, straight-line depreciation: 12000 - 3 x 1200), rather than compounding the reduction onto each year's new balance.
    • Why not C: Applies the 10% depreciation for one year only, forgetting to repeat it for all 3 years.
    • Why not D: Applies a growth multiplier of 1.10 instead of a decay multiplier of 0.90, increasing the value each year rather than decreasing it.

  14. Question 14Answer: D

    1. An 8% increase each hour means the population is multiplied by 1 + 0.08 = 1.08 every hour.
    2. Repeating this multiplication for t hours means multiplying by 1.08 a total of t times, which is 1.08 raised to the power t.
    3. The population after t hours is therefore P0 x (1.08)^t.
    4. The answer is D.
    • Why not A: Adds the 8% directly into the multiplier as if it were 80%, mistaking 0.08 for 0.8 when converting the percentage to a decimal.
    • Why not B: Treats the growth as being added repeatedly rather than compounded, multiplying by a constant factor once per hour instead of raising that factor to the power of the number of hours.
    • Why not C: Treats percentage growth as a fixed constant amount added to the original population each hour, ignoring that the growth compounds on an ever-larger population.

  15. Question 15Answer: B

    1. 'What percentage of Y is X' is calculated as (X/Y) x 100.
    2. Here X = 45 (small tank) and Y = 60 (large tank), so the percentage is (45/60) x 100.
    3. 45/60 simplifies to 3/4, and 3/4 x 100 = 75.
    4. The answer is B.
    • Why not A: Inverts which quantity is the whole, calculating the large tank's capacity as a percentage of the small tank's instead of the reverse.
    • Why not C: Confuses the 15-litre difference between the two tanks with the percentage itself.
    • Why not D: Correctly finds the decimal 45/60 = 0.75 but forgets to multiply by 100 to express it as a percentage.

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