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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 2

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

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

    Work out the value of 6 + 5 x (2 - 5)^2 - 18 / (-3).

    1. A -33
    2. B -27
    3. C 57
    4. D -18
  2. 21 mark

    A gardener has three lengths of rope: 84 m, 126 m and 210 m. She wants to cut all three into pieces that are the same length, with the pieces as long as possible and no rope left over. How long, in metres, should each piece be?

    1. A 42 metres
    2. B 6 metres
    3. C 1260 metres
    4. D 84 metres
  3. 31 mark

    A metal cube has a mass of 3.6 x 10^4 grams and a volume of 4.5 x 10^3 cm^3. Given that density = mass / volume, what is the density of the cube, in g/cm^3?

    1. A 0.125 g/cm^3
    2. B 0.08 g/cm^3
    3. C 0.8 g/cm^3
    4. D 8 g/cm^3
  4. 41 mark

    A padlock has a 3-digit code. Each digit is chosen freely from 0 to 9 and repeated digits are allowed, but the code is not allowed to start with the digit 0. How many different codes are possible?

    1. A 1000
    2. B 900
    3. C 729
    4. D 29
  5. 51 mark

    Simplify 16^(3/4) x 4^(-3/2), giving your answer as an integer.

    1. A 1
    2. B -64
    3. C 64
    4. D -72
  6. 61 mark

    Express the recurring decimal 0.363636... (0.36 recurring) as a fraction in its simplest form.

    1. A 11/4
    2. B 11/30
    3. C 4/11
    4. D 9/25
  7. 71 mark

    Simplify sqrt(45) + sqrt(125) - sqrt(20), giving your answer in the form k sqrt(5), where k is an integer.

    1. A 6 sqrt(5)
    2. B 10 sqrt(5)
    3. C 30 sqrt(5)
    4. D 26 sqrt(5)
  8. 81 mark

    A hiker walks a distance of 11 km, measured to the nearest kilometre, in a time of 3 hours, measured to the nearest hour. What is the lower bound for the hiker's average speed, in km/h?

    1. A 4.6 km/h
    2. B 11/3 km/h
    3. C 3 km/h
    4. D 4.2 km/h
  9. 91 mark

    A prize fund of 180 pounds is shared between three winners in the ratio 2 : 3 : 4. How much more does the winner with the largest share receive than the winner with the smallest share, in pounds?

    1. A 80 pounds
    2. B 120 pounds
    3. C 40 pounds
    4. D 20 pounds
  10. 101 mark

    After a 15% pay rise, an employee's salary is 27600 pounds a year. What was the employee's salary before the rise, in pounds?

    1. A 24000 pounds
    2. B 23460 pounds
    3. C 31740 pounds
    4. D 184000 pounds
  11. 111 mark

    A fixed distance is travelled at a constant speed. At a speed of 45 mph the journey takes 4 hours. Given that time is inversely proportional to speed for a journey of fixed distance, how long does the same journey take at a speed of 60 mph, in hours?

    1. A 16/3 hours
    2. B 3 hours
    3. C 1/3 hour
    4. D 8/3 hours
  12. 121 mark

    Amir invests 800 pounds in a savings account that pays compound interest at a rate of 5% per year. What is the value of his investment after 3 years, to the nearest pound?

    1. A 920 pounds
    2. B 926 pounds
    3. C 882 pounds
    4. D 2520 pounds

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: C

    1. Work inside the bracket first: 2 - 5 = -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: 5 x 9 = 45, and 18 / (-3) = -6.
    4. Work left to right through what remains: 6 + 45 - (-6) = 51 + 6 = 57.
    5. The answer is C.
    • Why not A: Loses the sign when squaring the negative bracket, treating (2 - 5)^2 as -9 instead of +9, which flips the sign of the whole multiplied term to -45.
    • Why not B: Ignores the priority of operations, working strictly left to right through the expression (addition, then multiplication, then subtraction, then division, in the order they are written) instead of carrying out the multiplication and division before the addition and subtraction.
    • Why not D: Confuses squaring with doubling, treating (2 - 5)^2 as (2 - 5) x 2 = -6 instead of (2 - 5) x (2 - 5) = 9.

  2. Question 2Answer: A

    1. The largest piece length that divides evenly into all three ropes with none left over is the highest common factor of 84, 126 and 210.
    2. Write each length as a product of prime factors: 84 = 2^2 x 3 x 7, 126 = 2 x 3^2 x 7, and 210 = 2 x 3 x 5 x 7.
    3. The highest common factor takes the lowest power of each prime common to all three numbers: 2^1, 3^1 and 7^1 (5 is not common to all three, so it is excluded).
    4. Multiply these together: 2 x 3 x 7 = 42.
    5. As a check: 84 / 42 = 2, 126 / 42 = 3 and 210 / 42 = 5, all whole numbers, so 42 m pieces work exactly.
    6. The answer is A.
    • Why not B: Spots that 2 and 3 are common factors of all three lengths but misses the common factor of 7, so stops at 2 x 3 = 6 instead of the full highest common factor 2 x 3 x 7 = 42.
    • Why not C: Finds the lowest common multiple of the three lengths instead of their highest common factor, which answers a different question about the ropes.
    • Why not D: Uses the length of the shortest rope (84 m) as the answer, without checking that it actually divides evenly into the other two rope lengths (126 / 84 and 210 / 84 are not whole numbers).

  3. Question 3Answer: D

    1. Divide the mantissas and subtract the powers of ten: (3.6 x 10^4) / (4.5 x 10^3) = (3.6 / 4.5) x 10^(4 - 3).
    2. 3.6 / 4.5 = 0.8, since 4.5 x 0.8 = 3.6.
    3. 10^(4-3) = 10^1 = 10.
    4. Multiply the two parts together: 0.8 x 10 = 8.
    5. The density of the cube is 8 g/cm^3.
    6. The answer is D.
    • Why not A: Inverts the density formula, computing volume / mass instead of mass / volume: (4.5 x 10^3) / (3.6 x 10^4) = 0.125.
    • Why not B: Subtracts the powers of ten in the wrong order, using 10^(3-4) instead of 10^(4-3), so gets (3.6 / 4.5) x 10^(-1) = 0.08 instead of (3.6 / 4.5) x 10^1.
    • Why not C: Divides the two mantissas correctly (3.6 / 4.5 = 0.8) but drops the leftover power of ten entirely, forgetting that 10^4 / 10^3 = 10^1 still needs to be included.

  4. Question 4Answer: B

    1. The first digit cannot be 0, so it has 9 possible values: 1 to 9.
    2. The second and third digits can each be any of the 10 digits 0 to 9, since there is no restriction on them and repeats are allowed.
    3. By the counting principle, if there are m ways to choose the first digit, n ways to choose the second, and p ways to choose the third, the total number of codes is m x n x p.
    4. Here m = 9, n = 10 and p = 10, giving 9 x 10 x 10 = 900.
    5. The answer is B.
    • Why not A: Ignores the restriction on the first digit entirely, allowing 10 choices (0 to 9) for every position: 10 x 10 x 10 = 1000.
    • Why not C: Wrongly applies the 'cannot be 0' restriction to every digit rather than just the first, giving 9 choices at each of the three positions: 9 x 9 x 9 = 729.
    • Why not D: Adds the number of choices for each digit 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: 9 + 10 + 10 = 29.

  5. Question 5Answer: A

    1. 16^(3/4) means take the 4th root of 16 and then cube the result: 16^(1/4) = 2, since 2^4 = 16, so 16^(3/4) = 2^3 = 8.
    2. 4^(-3/2) means take the square root of 4, cube it, then take the reciprocal because of the negative exponent: 4^(1/2) = 2, so 4^(3/2) = 2^3 = 8, giving 4^(-3/2) = 1/8.
    3. Multiply the two results together: 8 x 1/8 = 1.
    4. As a shorter route, note 16 = 4^2, so 16^(3/4) = (4^2)^(3/4) = 4^(3/2), and the product becomes 4^(3/2) x 4^(-3/2) = 4^0 = 1, confirming the same answer.
    5. The answer is A.
    • Why not B: Treats the negative exponent as making the result negative rather than as an instruction to take a reciprocal, computing 4^(-3/2) as -8 instead of 1/8, so gets 8 x (-8) = -64.
    • Why not C: Drops the negative sign in the exponent entirely, computing 4^(-3/2) as if it were 4^(3/2) = 8, so gets 8 x 8 = 64 instead of taking the reciprocal.
    • Why not D: Misreads the fractional and negative exponents as instructions to multiply the base by the exponent, computing 16^(3/4) as 16 x (3/4) = 12 and 4^(-3/2) as 4 x (-3/2) = -6, then multiplies these to get -72.

  6. Question 6Answer: C

    1. Let x = 0.363636... (the block '36' recurring). Since two digits recur, multiply by 100: 100x = 36.363636...
    2. Subtract the original equation from this, so the recurring parts cancel exactly: 100x - x = 36.363636... - 0.363636... = 36, so 99x = 36.
    3. Divide both sides by 99: x = 36/99.
    4. Simplify by dividing numerator and denominator by their highest common factor, 9: 36/99 = 4/11.
    5. The answer is C.
    • Why not A: Correctly sets up the equation 99x = 36 but then inverts the fraction when solving for x, computing 99/36 (which simplifies to 11/4) instead of 36/99.
    • Why not B: Misreads which digits recur, treating the decimal as 0.3 with only the 6 repeating (0.36666...) instead of both digits repeating (0.363636...), which changes which power of 10 should be used and gives 11/30 for the wrong decimal.
    • Why not D: Ignores the recurring nature of the decimal entirely, treating 0.363636... as if it terminated after two decimal places and converting 0.36 directly to 36/100 = 9/25.

  7. Question 7Answer: A

    1. Write each surd in terms of sqrt(5) by pulling out the largest square factor: sqrt(45) = sqrt(9 x 5) = 3 sqrt(5), sqrt(125) = sqrt(25 x 5) = 5 sqrt(5), and sqrt(20) = sqrt(4 x 5) = 2 sqrt(5).
    2. Substitute these back into the original expression: 3 sqrt(5) + 5 sqrt(5) - 2 sqrt(5).
    3. Since all three terms are like surds, combine the coefficients: 3 + 5 - 2 = 6.
    4. The simplified expression is 6 sqrt(5).
    5. The answer is A.
    • Why not D: Correctly spots that 125 = 25 x 5 but forgets to take the square root of the 25, writing sqrt(125) as 25 sqrt(5) instead of sqrt(25) x sqrt(5) = 5 sqrt(5), which inflates that term.
    • Why not B: Adds all three simplified surds instead of subtracting the last one, ignoring the minus sign in front of sqrt(20): 3 + 5 + 2 = 10 instead of 3 + 5 - 2 = 6.
    • Why not C: Simplifies each surd correctly (3 sqrt(5), 5 sqrt(5) and 2 sqrt(5)) but then combines them by multiplying the coefficients together (3 x 5 x 2 = 30) instead of adding and subtracting them as the expression requires.

  8. Question 8Answer: C

    1. A distance rounded to 11 km could truly be anywhere from 10.5 km up to 11.5 km, and a time rounded to 3 hours could truly be anywhere from 2.5 hours up to 3.5 hours.
    2. Speed is distance divided by time, and this quotient is smallest when the numerator is as small as possible and the denominator is as large as possible.
    3. So the lower bound of speed uses the LOWER bound of distance together with the UPPER bound of time: 10.5 / 3.5.
    4. 10.5 / 3.5 = 3.
    5. The answer is C.
    • Why not A: Pairs the upper bound of distance with the lower bound of time (11.5 / 2.5 = 4.6), which is the correct pairing for the UPPER bound of speed, not the lower bound.
    • Why not B: Uses the rounded measurements directly (11 km and 3 hours) without accounting for the possible rounding error in either quantity at all: 11 / 3 = 11/3.
    • Why not D: Pairs the lower bound of distance with the lower bound of time (10.5 / 2.5 = 4.2) instead of pairing the lower bound of distance with the UPPER bound of time, mistakenly moving both bounds in the same direction.

  9. Question 9Answer: C

    1. The ratio 2 : 3 : 4 has 2 + 3 + 4 = 9 parts in total.
    2. Divide the prize fund by the total number of parts to find the value of one part: 180 / 9 = 20 pounds.
    3. The smallest share (2 parts) is 2 x 20 = 40 pounds, and the largest share (4 parts) is 4 x 20 = 80 pounds.
    4. The difference between the largest and smallest shares is 80 - 40 = 40 pounds.
    5. The answer is C.
    • Why not A: Gives the largest share (80 pounds) as the final answer, forgetting that the question asks for the DIFFERENCE between the largest and smallest shares, not the largest share on its own.
    • Why not B: Divides the total by the number of winners, 3, instead of the total number of ratio parts, 9, giving an incorrect value of 60 pounds per part instead of 20, then finds a difference of (4 - 2) x 60 = 120 using this wrong per-part value.
    • Why not D: Finds the difference between the smallest share and the MIDDLE share (60 - 40 = 20) instead of the difference between the smallest and largest shares.

  10. Question 10Answer: A

    1. A 15% pay rise means the new salary is 115% of the original salary, so new salary = original salary x 1.15.
    2. Rearranging, original salary = new salary / 1.15.
    3. 27600 / 1.15 = 24000, since 24000 x 1.15 = 24000 + (0.15 x 24000) = 24000 + 3600 = 27600.
    4. The salary before the rise was 24000 pounds.
    5. The answer is A.
    • Why not B: Subtracts 15% of the NEW salary from the new salary, treating the final value as if it were 100% rather than 115% of the original: 27600 - (0.15 x 27600) = 27600 - 4140 = 23460.
    • Why not C: Misreads the direction of the problem and adds a further 15% to the given salary instead of finding what it grew from: 27600 x 1.15 = 31740.
    • Why not D: Divides the salary by 0.15 (the size of the increase alone) instead of by 1.15 (100% plus the 15% increase): 27600 / 0.15 = 184000.

  11. Question 11Answer: B

    1. Since time is inversely proportional to speed for a fixed distance, speed x time is constant and equal to the distance.
    2. At 45 mph taking 4 hours, the distance is 45 x 4 = 180 miles.
    3. At 60 mph, the time taken is distance / speed = 180 / 60.
    4. 180 / 60 = 3.
    5. The answer is B.
    • Why not A: Treats time as DIRECTLY proportional to speed instead of inversely proportional, scaling the time up in the same direction as the speed: 4 x (60/45) = 16/3, when a faster speed should give a shorter time.
    • Why not D: Notices that the speed increases by a factor of 1/3 (a 33.3% increase) but then wrongly applies this same fraction as a percentage DECREASE in time: 4 - (1/3) x 4 = 8/3, instead of using the inverse proportion relationship correctly.
    • Why not C: Correctly finds the fixed distance as 180 miles but then divides speed by distance instead of distance by speed when finding the new time: 60 / 180 = 1/3.

  12. Question 12Answer: B

    1. Compound interest at 5% per year means the balance is multiplied by 1.05 at the end of each year.
    2. After year 1: 800 x 1.05 = 840.
    3. After year 2: 840 x 1.05 = 882.
    4. After year 3: 882 x 1.05 = 926.1, which rounds to 926 pounds to the nearest pound.
    5. The answer is B.
    • Why not A: Uses SIMPLE interest instead of compound interest, adding 5% of the original 800 pounds in each of the 3 years rather than compounding on the growing balance: 800 + 3 x (0.05 x 800) = 800 + 120 = 920.
    • Why not C: Compounds the interest for only 2 years instead of 3, stopping one year early: 800 x 1.05 x 1.05 = 882.
    • Why not D: Misapplies the rate, multiplying the original amount by 3 x 1.05 = 3.15 instead of multiplying by 1.05 three times over: 800 x 3.15 = 2520.

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