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Admissions tests / TMUA / Paper 1 / Number, ratio and proportion

Test standard. 12 questions, 12 marks, about 45 minutes.

TMUA Paper 1: Number, ratio and proportion, set 1

Higher-tier GCSE number: units and compound units, standard form, surds, percentages, ratio, direct and inverse proportion, growth and decay, bounds and rounding.

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

  • Answer all questions. No calculator.
  • Each question has exactly one correct answer.
  1. 11 mark

    Work out the value of 7 - 2 x (5 - 8)^2 + 6 / (-3).

    1. A 23
    2. B -13
    3. C 43
    4. D -9
  2. 21 mark

    Which of the following lists the four numbers -0.6, 3/5, -2/3 and 0.55 in order from smallest to largest?

    1. A -2/3, -0.6, 0.55, 3/5
    2. B -0.6, -2/3, 0.55, 3/5
    3. C -2/3, -0.6, 3/5, 0.55
    4. D -0.6, 0.55, 3/5, -2/3
  3. 31 mark

    Three lighthouses flash together at midnight. Lighthouse A then flashes every 18 seconds, lighthouse B every 24 seconds, and lighthouse C every 30 seconds. After how many seconds do all three lighthouses next flash together?

    1. A 72 seconds
    2. B 6 seconds
    3. C 360 seconds
    4. D 12960 seconds
  4. 41 mark

    A restaurant's set menu offers 4 starters, 5 main courses and 3 desserts. A customer must choose exactly one of each course. Two of the main courses contain nuts, and the customer is allergic to nuts, so those two mains are excluded. How many different three-course meals can the customer choose?

    1. A 60
    2. B 58
    3. C 10
    4. D 36
  5. 51 mark

    Simplify 27^(2/3) x 9^(-1/2), giving your answer as an integer.

    1. A -27
    2. B 3
    3. C 6
    4. D 1
  6. 61 mark

    A car travels 2.4 x 10^5 metres in 8 x 10^2 seconds at a constant speed. Given that 1 hour = 3600 seconds and 1 kilometre = 1000 metres, what is the car's average speed, in kilometres per hour?

    1. A 1080 km/h
    2. B 1080000 km/h
    3. C 18 km/h
    4. D 10800 km/h
  7. 71 mark

    Express the recurring decimal 0.4 recurring (0.4444...) as a percentage.

    1. A 44.4%
    2. B 40%
    3. C 4.44%
    4. D 44 4/9%
  8. 81 mark

    Simplify 6 / sqrt(3) + 2 sqrt(3), giving your answer in the form k sqrt(3), where k is an integer.

    1. A 8 sqrt(3)
    2. B 2 sqrt(3) + 2
    3. C 4 sqrt(3)
    4. D 12
  9. 91 mark

    A rectangle has length 6 cm and width 4 cm, each measured to the nearest whole centimetre. What is the upper bound for the area of the rectangle, in cm^2?

    1. A 29.25 cm^2
    2. B 19.25 cm^2
    3. C 24 cm^2
    4. D 26 cm^2
  10. 101 mark

    A fruit cordial is made by mixing concentrate and water in the ratio 2 : 7. A jug contains 3/5 litre of concentrate. How much water, in litres, must be added to make the cordial in the correct ratio?

    1. A 6/35 litres
    2. B 21/10 litres
    3. C 3 litres
    4. D 21/5 litres
  11. 111 mark

    A laptop's value depreciates by 20% each year. Its value when new was 500 pounds. What is its value, in pounds, after 2 years?

    1. A 300 pounds
    2. B 720 pounds
    3. C 320 pounds
    4. D 400 pounds
  12. 121 mark

    Two similar solid metal spheres are made from the same material. The larger sphere has a radius 3 times the radius of the smaller sphere. The smaller sphere has a mass of 5 kg. What is the mass of the larger sphere, in kg?

    1. A 15 kg
    2. B 45 kg
    3. C 32 kg
    4. D 135 kg

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. Work inside the bracket first: 5 - 8 = -3.
    2. Apply the exponent: (-3)^2 = 9, since a negative number squared is positive.
    3. Carry out the multiplication and division, before any addition or subtraction: 2 x 9 = 18, and 6 / (-3) = -2.
    4. Work left to right through what remains: 7 - 18 + (-2) = -11 + (-2) = -13.
    5. The answer is B.
    • Why not A: Squares the negative bracket incorrectly, treating (-3)^2 as -9 rather than +9 (a lost sign when multiplying two negatives), which flips the sign of the term being subtracted.
    • Why not C: Ignores the convention that multiplication is carried out before subtraction, working strictly left to right: computes 7 - 2 = 5 first, then multiplies by (5 - 8)^2, instead of multiplying 2 by (5 - 8)^2 before subtracting from 7.
    • Why not D: Makes a sign error dividing a positive number by a negative one, computing 6 / (-3) as +2 instead of -2.

  2. Question 2Answer: A

    1. Convert every value to a decimal: -0.6 stays -0.6, 3/5 = 0.6, -2/3 = -0.666... (recurring), and 0.55 stays 0.55.
    2. Compare the two negative numbers first: since 0.666... is greater than 0.6, -2/3 is further from zero than -0.6, so -2/3 is the smaller (more negative) number: -2/3 < -0.6.
    3. Compare the two positive numbers: 0.55 is less than 0.6, so 0.55 < 3/5.
    4. Putting all four in order from smallest to largest gives -2/3, -0.6, 0.55, 3/5.
    5. The answer is A.
    • Why not B: Compares the two negative numbers by the size of their decimal part without reversing the inequality, so treats -0.6 as smaller than -2/3 simply because 0.6 is less than 0.666..., the opposite of the true order for negative numbers.
    • Why not C: Miscalculates the fraction 3/5 as a decimal smaller than 0.55, swapping the correct order of the two positive values.
    • Why not D: Drops the negative sign when converting -2/3 to a decimal, treating it as +0.666... and so placing it as the largest value instead of the smallest.

  3. Question 3Answer: C

    1. Write each interval as a product of prime factors: 18 = 2 x 3^2, 24 = 2^3 x 3, and 30 = 2 x 3 x 5.
    2. The lowest common multiple takes the highest power of each prime that appears in any of the numbers: 2^3 (from 24), 3^2 (from 18), and 5^1 (from 30).
    3. Multiply these highest powers together: 2^3 x 3^2 x 5 = 8 x 9 x 5 = 360.
    4. All three lighthouses next flash together after 360 seconds.
    5. The answer is C.
    • Why not A: Finds a common multiple of only two of the three flashing intervals (18 and 24), and stops there without checking that the result is also a multiple of the third interval, 30.
    • Why not B: Finds the highest common factor of the three intervals instead of their lowest common multiple, which answers a different question about the intervals.
    • Why not D: Multiplies all three intervals together (18 x 24 x 30) instead of finding their lowest common multiple, which double-counts the prime factors the three numbers share.

  4. Question 4Answer: D

    1. The customer is allergic to nuts, so only 5 - 2 = 3 main courses are safe to choose from.
    2. By the counting principle, if there are m ways to choose a starter, n ways to choose a main and p ways to choose a dessert, the total number of meals is m x n x p.
    3. Here m = 4 starters, n = 3 safe mains, and p = 3 desserts, giving 4 x 3 x 3 = 36.
    4. The answer is D.
    • Why not A: Multiplies the full number of starters, mains and desserts together without ever removing the two nut-containing mains, ignoring the allergy constraint completely.
    • Why not B: Finds the total number of meals without any restriction (4 x 5 x 3 = 60), then simply subtracts 2 for the excluded mains, rather than reducing the number of available mains before multiplying.
    • Why not C: Adds the number of choices in each course (4 + 3 + 3) instead of multiplying them, not applying the rule that if there are m ways to do one task and n ways to do another, there are m x n ways to do both in order.

  5. Question 5Answer: B

    1. 27^(2/3) means take the cube root of 27 and then square the result: 27^(1/3) = 3, so 27^(2/3) = 3^2 = 9.
    2. 9^(-1/2) means take the square root of 9 and then take the reciprocal: 9^(1/2) = 3, so 9^(-1/2) = 1/3.
    3. Multiply the two results together: 9 x 1/3 = 3.
    4. The answer is B.
    • Why not A: Treats the negative exponent in 9^(-1/2) as making the whole value negative, computing -9^(1/2) = -3, instead of taking the reciprocal of the positive square root.
    • Why not C: Misreads the fractional exponent 2/3 as an instruction to multiply the base by 2/3 (27 x 2/3 = 18) instead of taking the cube root and squaring it.
    • Why not D: Takes the cube root of 27 (giving 3) but forgets to square it as the exponent 2/3 requires, stopping one step early.

  6. Question 6Answer: A

    1. Divide the standard-form quantities to find the speed in metres per second: (2.4 x 10^5) / (8 x 10^2) = (2.4 / 8) x 10^(5-2) = 0.3 x 10^3 = 300 m/s.
    2. Convert to metres per hour by multiplying by the number of seconds in an hour: 300 x 3600 = 1080000 m/h.
    3. Convert to kilometres per hour by dividing by 1000, since 1 km = 1000 m: 1080000 / 1000 = 1080.
    4. The car's average speed is 1080 km/h.
    5. The answer is A.
    • Why not B: Correctly finds the speed as 1080000 metres per hour but forgets the final step of dividing by 1000 to convert this into kilometres per hour.
    • Why not C: Uses 1 hour = 60 seconds, confusing it with 1 minute = 60 seconds, instead of the correct 3600 seconds, which understates the per-hour distance by a factor of 60.
    • Why not D: Divides by 100 rather than 1000 when converting metres to kilometres, a factor-of-10 slip in the metric conversion.

  7. Question 7Answer: D

    1. Let x = 0.4444... (the 4 recurring). Multiply by 10: 10x = 4.4444...
    2. Subtract the original equation from this: 10x - x = 4.4444... - 0.4444... = 4, so 9x = 4.
    3. Divide both sides by 9: x = 4/9.
    4. To express 4/9 as a percentage, multiply by 100: (4/9) x 100 = 400/9 = 44 and 4/9, so the recurring decimal is exactly 44 4/9%.
    5. The answer is D.
    • Why not A: Rounds the exact value of 44.444...% to one decimal place instead of expressing the recurring part exactly as a fraction, giving an approximation rather than the exact percentage.
    • Why not B: Treats 0.4 recurring as the terminating decimal 0.4, dropping the recurring part entirely before converting to a percentage.
    • Why not C: Misplaces the decimal point when converting the fraction to a percentage, multiplying by 10 instead of by 100.

  8. Question 8Answer: C

    1. Rationalise the denominator of 6 / sqrt(3) by multiplying the top and bottom by sqrt(3): 6 / sqrt(3) = (6 sqrt(3)) / (sqrt(3) x sqrt(3)) = (6 sqrt(3)) / 3.
    2. Divide through by 3: (6 sqrt(3)) / 3 = 2 sqrt(3).
    3. Add this to the second term, since both are like surds: 2 sqrt(3) + 2 sqrt(3) = (2 + 2) sqrt(3) = 4 sqrt(3).
    4. The answer is C.
    • Why not A: Multiplies the numerator and denominator of 6 / sqrt(3) by sqrt(3) to begin rationalising, reaching 6 sqrt(3) over 3, but forgets to complete the division by 3, so treats 6 sqrt(3) itself as the rationalised value.
    • Why not B: Cancels the sqrt(3) in the denominator directly against the 3 it becomes after squaring, incorrectly treating 6 / sqrt(3) as 6 / 3 = 2, a rational number with no surd, then simply writes this alongside the other term instead of combining like surds.
    • Why not D: Combines the two like surd terms by multiplying their coefficients and squaring the surd (2 x 2 x 3 = 12), instead of simply adding the coefficients to get 4 sqrt(3).

  9. Question 9Answer: A

    1. Both measurements are given to the nearest whole centimetre, so the length of 6 cm could actually be anywhere from 5.5 cm up to 6.5 cm, and the width of 4 cm from 3.5 cm up to 4.5 cm.
    2. The area of a rectangle is length x width, and this product is largest when both dimensions take their largest possible value, so the upper bound of the area uses the upper bound of each measurement.
    3. Upper bound area = 6.5 x 4.5 = (6.5 x 4) + (6.5 x 0.5) = 26 + 3.25 = 29.25 cm^2.
    4. As a check, using the rounded measurements directly gives an estimate of 6 x 4 = 24 cm^2; the true upper bound of 29.25 cm^2 is sensibly a little larger than this estimate, confirming the bound has been applied in the right direction.
    5. The answer is A.
    • Why not B: Uses the lower bound of each measurement (5.5 cm and 3.5 cm) instead of the upper bound, which gives the smallest possible area rather than the largest.
    • Why not C: Uses the rounded measurements of 6 cm and 4 cm directly, ignoring the fact that each has a possible error of up to 0.5 cm and so does not have one exact value.
    • Why not D: Applies the upper bound to the length (6.5 cm) but forgets to also apply it to the width, leaving the width at its rounded value of 4 cm instead of its upper bound of 4.5 cm.

  10. Question 10Answer: B

    1. The ratio of concentrate to water is 2 : 7, so the given 3/5 litre of concentrate represents 2 parts of the mixture.
    2. Divide to find the value of one part: (3/5) / 2 = 3/10 litre per part.
    3. Water is 7 parts, so the amount of water needed is 7 x 3/10 = 21/10 litres.
    4. As a check, since the ratio 2 : 7 has 2 + 7 = 9 total parts, concentrate makes up 2/9 of the whole mixture and water makes up 7/9, so it is sensible that the water needed is noticeably more than the concentrate given.
    5. The answer is B.
    • Why not A: Confuses which number in the ratio belongs to concentrate, treating water's share (7) as if it were the concentrate's share, so divides 3/5 by 7 instead of by 2 to find the value of one part, then scales up using 2 as though it were water's share.
    • Why not C: Scales the concentrate amount by the difference between the ratio parts (7 - 2 = 5) rather than by the ratio of water's parts to concentrate's parts, which is not how a part : part ratio converts an amount.
    • Why not D: Treats the given 3/5 litre as if it already represented a single ratio part, skipping the division by 2 needed to find the value of one part before scaling up to water's 7 parts.

  11. Question 11Answer: C

    1. A 20% decrease each year means the value is multiplied by (1 - 20/100) = 0.8 for each year that passes.
    2. After 1 year: 500 x 0.8 = 400 pounds.
    3. After 2 years, apply the same multiplier again to the new value: 400 x 0.8 = 320 pounds.
    4. This is compound depreciation, since each year's decrease is calculated on the previous year's value, not on the original 500 pounds.
    5. The answer is C.
    • Why not A: Uses simple percentage decrease, taking 20% of the original 500 pounds each year and subtracting it twice (500 - 100 - 100), instead of compounding the decrease on the previous year's reduced value.
    • Why not B: Treats the 20% change as an increase rather than a decrease, multiplying by 1.2 each year instead of 0.8.
    • Why not D: Applies the 20% decrease for only one year instead of two, stopping after the first year's calculation.

  12. Question 12Answer: D

    1. The two spheres are similar and made of the same material, so their masses are directly proportional to their volumes (mass = density x volume, and density is constant here).
    2. The linear scale factor between the spheres is 3, since the larger radius is 3 times the smaller radius.
    3. Volume scales with the cube of the linear scale factor, so the volume scale factor is 3^3 = 27.
    4. Since mass is proportional to volume, the larger sphere's mass is 27 times the smaller sphere's mass: 5 x 27 = 135 kg.
    5. The answer is D.
    • Why not A: Scales the mass by the linear scale factor of 3 directly, treating mass as proportional to radius rather than to volume.
    • Why not B: Uses the scale factor squared (3^2 = 9, giving 5 x 9 = 45), which is how area scales with a linear factor, not how volume scales.
    • Why not C: Adds the volume scale factor to the original mass (5 + 27 = 32) instead of multiplying by it, misapplying the scale factor as an increase rather than a multiplier.

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