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Admissions tests / ESAT / Maths 1 / Number, units and measures

Test standard. 15 questions, 15 marks, about 26 minutes.

ESAT Mathematics 1: Number, units and measures, set 2

Standard and compound units, ordering and operating on integers, decimals and fractions, primes and factors, powers and roots, standard form, surds, rounding, bounds and estimation.

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 car travels at a constant speed of 108 km/h. Work out this speed in m/s.

    1. A 1.8 m/s
    2. B 0.03 m/s
    3. C 30 m/s
    4. D 388.8 m/s
  2. 21 mark

    Exactly one of the following statements is true. Which one?

    1. A -0.6 <= -3/5
    2. B 7/8 != 0.875
    3. C -1/4 > -0.2
    4. D 3.5 < 7/2
  3. 31 mark

    Work out 2 5/6 - (-1 1/4), giving your answer as a mixed number in its simplest form.

    1. A 1 7/12
    2. B 3 7/12
    3. C 3 1/4
    4. D 4 1/12
  4. 41 mark

    The prime factorisations of two numbers are 36 = 2^2 x 3^2 and 48 = 2^4 x 3. Find the lowest common multiple (LCM) of 36 and 48.

    1. A 72
    2. B 144
    3. C 12
    4. D 1728
  5. 51 mark

    Work out (5 - 8)^2 - 4 x sqrt(9) + 6.

    1. A -15
    2. B 9
    3. C -12
    4. D 3
  6. 61 mark

    A bag contains 5 counters numbered 1 to 5. Two counters are drawn without replacement and their numbers are added together. By systematic listing of all the possible pairs, find how many pairs give a sum that is a multiple of 3.

    1. A 3
    2. B 10
    3. C 4
    4. D 5
  7. 71 mark

    Work out 2 x (the negative square root of 36) - (the cube root of -64).

    1. A -8
    2. B 16
    3. C -16
    4. D -2
  8. 81 mark

    Simplify 16^(3/4) / 2^5, giving your answer as a fraction in its simplest form.

    1. A 4
    2. B 1/4
    3. C 2
    4. D 32
  9. 91 mark

    Work out (5 x 10^-3) + (2.5 x 10^-2), giving your answer in standard form.

    1. A 1.25 x 10^-4
    2. B 3 x 10^-3
    3. C 3 x 10^-2
    4. D 7.5 x 10^-3
  10. 101 mark

    Convert the recurring decimal 0.41666... (where the digits 4 and 1 do not repeat, but the digit 6 recurs forever) to a fraction in its simplest form.

    1. A 41/99
    2. B 375/900
    3. C 3/8
    4. D 5/12
  11. 111 mark

    Which of the following is NOT equivalent to 0.35?

    1. A 27/80
    2. B 35%
    3. C 7/20
    4. D 350/1000
  12. 121 mark

    Rationalise the denominator of 4 / (sqrt(7) - 3), giving your answer in the form a + b sqrt(7), where a and b are integers.

    1. A 6 + 2 sqrt(7)
    2. B -2
    3. C 4 sqrt(7) - 6
    4. D -6 - 2 sqrt(7)
  13. 131 mark

    Two lengths are measured as 12.4 cm and 5.7 cm, each correct to 1 decimal place. Find the upper bound for the difference when the second length is subtracted from the first.

    1. A 6.6 cm
    2. B 6.8 cm
    3. C 6.7 cm
    4. D 7.7 cm
  14. 141 mark

    A number is truncated, not rounded, to 7.2 when truncated to 1 decimal place. Write down the error interval for the true value, x, using inequality notation.

    1. A 7.15 <= x < 7.25
    2. B 7.2 < x < 7.3
    3. C 7.2 <= x < 7.3
    4. D 7.1 < x <= 7.2
  15. 151 mark

    By using a suitable approximation for sqrt(48), and rounding 2.1 to the nearest integer, estimate the value of sqrt(48) x 2.1.

    1. A 14
    2. B 21
    3. C 12
    4. D 14.7

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. Convert km/h to m/s by multiplying by 1000 (km to m) and dividing by 3600 (hours to seconds), which is the same as dividing by 3.6.
    2. 108 / 3.6 = 30.
    3. So the speed is 30 m/s, which is answer C.
    • Why not A: Divides by 60 instead of by 3.6, as if converting hours to minutes only and not also accounting for the change from kilometres to metres: 108 / 60 = 1.8.
    • Why not B: Divides by 3600 to convert hours to seconds, but forgets to also convert kilometres to metres first (should multiply by 1000 before dividing by 3600): 108 / 3600 = 0.03.
    • Why not D: Multiplies by 3.6 instead of dividing by it, treating the conversion from km/h to m/s as if it required scaling the speed up rather than down: 108 x 3.6 = 388.8.

  2. Question 2Answer: A

    1. Convert every value to a comparable form: -3/5 = -0.6, 7/8 = 0.875, -1/4 = -0.25, 7/2 = 3.5.
    2. Check each statement: -0.6 <= -0.6 is true, because <= is satisfied by equality as well as by 'less than'.
    3. 0.875 != 0.875 is false, since the two sides are equal. -0.25 > -0.2 is false, since -0.25 is the smaller number. 3.5 < 3.5 is false, since the two sides are equal.
    4. So the only true statement is A.
    • Why not B: 7/8 is equal to 0.875 exactly, so this statement is false; a candidate who treats a fraction and its decimal equivalent as inherently different values, rather than checking whether they are numerically equal, can wrongly judge the '!=' to hold here.
    • Why not C: -1/4 = -0.25, which is less than -0.2 (further from zero in the negative direction), so the statement is false; comparing only the size of the numbers while ignoring the negative sign (0.25 > 0.2) makes this look true.
    • Why not D: 7/2 = 3.5 exactly, so 3.5 < 3.5 is false, since the two sides are equal rather than one being strictly less than the other; misreading '<' as 'at most' (that is, as <=) makes this look true.

  3. Question 3Answer: D

    1. Subtracting a negative is the same as adding: 2 5/6 - (-1 1/4) = 2 5/6 + 1 1/4.
    2. Convert to improper fractions: 2 5/6 = 17/6 and 1 1/4 = 5/4.
    3. Use a common denominator of 12: 17/6 = 34/12 and 5/4 = 15/12. Add: 34/12 + 15/12 = 49/12.
    4. Convert back to a mixed number: 49/12 = 4 1/12, so the answer is D.
    • Why not A: Treats subtracting a negative as if it stayed a subtraction, dropping one of the two negative signs, and computes 2 5/6 - 1 1/4 = 17/6 - 5/4 = 34/12 - 15/12 = 19/12 = 1 7/12 instead of adding.
    • Why not B: Splits -1 1/4 into a negative whole number and a positive fraction (-1 and +1/4) rather than treating the whole mixed number as negative, effectively computing 2 5/6 + 1 - 1/4 instead of 2 5/6 + 1 1/4.
    • Why not C: Finds a common denominator by multiplying 6 and 4 to get 24, but forgets to scale the numerators to match the new denominator, adding 5/24 + 1/24 instead of the correctly scaled 20/24 + 6/24.

  4. Question 4Answer: B

    1. The LCM uses every prime that appears in either factorisation, each raised to the HIGHER of the two powers.
    2. For 2, the higher power is 2^4 (48 has 2^4, 36 has only 2^2). For 3, the higher power is 3^2 (36 has 3^2, 48 has only 3^1).
    3. LCM = 2^4 x 3^2 = 16 x 9 = 144.
    4. So the answer is B.
    • Why not A: Misreads the highest power of 2 needed, using 2^3 instead of 2^4, as if 48 = 2^3 x 6 rather than fully factorised as 2^4 x 3, giving 2^3 x 3^2 = 8 x 9 = 72.
    • Why not C: Finds the highest common factor instead of the lowest common multiple, using the LOWER power of each shared prime (2^2 and 3^1) instead of the higher power: 2^2 x 3 = 12.
    • Why not D: Multiplies the two numbers together directly (36 x 48 = 1728), a shortcut that only gives the LCM when the two numbers share no common factors, which 36 and 48 do.

  5. Question 5Answer: D

    1. Work out the bracket first: 5 - 8 = -3.
    2. Apply the power: (-3)^2 = 9, since a negative number squared is positive. The root: sqrt(9) = 3.
    3. Multiply: 4 x 3 = 12.
    4. Work left to right: 9 - 12 + 6 = -3 + 6 = 3, so the answer is D.
    • Why not A: Computes (5 - 8)^2 as -(3^2) = -9, mistakenly keeping the result negative as if squaring a negative number gives a negative answer, giving -9 - 12 + 6 = -15.
    • Why not B: Treats 4 x sqrt(9) as sqrt(4 x 9) = sqrt(36) = 6, wrongly moving the multiplication inside the square root, giving 9 - 6 + 6 = 9.
    • Why not C: Reads (5 - 8)^2 as (5 - 8) x 2 rather than (5 - 8) squared, computing -3 x 2 = -6, giving -6 - 12 + 6 = -12.

  6. Question 6Answer: C

    1. List all 10 possible pairs and their sums systematically: (1,2)=3, (1,3)=4, (1,4)=5, (1,5)=6, (2,3)=5, (2,4)=6, (2,5)=7, (3,4)=7, (3,5)=8, (4,5)=9.
    2. The sums that are multiples of 3 are 3, 6, 6 and 9, coming from the pairs (1,2), (1,5), (2,4) and (4,5).
    3. That is 4 pairs in total.
    4. So the answer is C.
    • Why not A: Lists pairs starting only from the smaller numbers, such as those including 1 or 2, and stops before checking the pair (4,5), missing one of the four valid pairs and so finding only 3.
    • Why not B: Counts the total number of possible pairs from the 5 counters, which is 10, but forgets to check which of them actually sum to a multiple of 3.
    • Why not D: Makes an arithmetic slip, adding 3 and 5 as 9 instead of 8, and so wrongly includes the pair (3,5) as an extra multiple of 3 alongside the 4 genuine pairs.

  7. Question 7Answer: A

    1. 36 has two square roots, 6 and -6, but the question asks for the negative one, so the negative square root of 36 is -6.
    2. A negative real number has exactly one real cube root, and it is negative: since (-4)^3 = -4 x -4 x -4 = -64, the cube root of -64 is -4.
    3. Substitute these values: 2 x (-6) - (-4) = -12 - (-4) = -12 + 4 = -8.
    4. So the answer is A.
    • Why not B: Ignores the instruction to take the negative square root of 36 and uses the positive root instead, taking the root as 6: 2 x 6 - (-4) = 12 + 4 = 16.
    • Why not C: Correctly uses -6 for the square root, but wrongly assumes a cube root can take two values like a square root, taking the cube root of -64 as 4 instead of the one real cube root, -4: 2 x (-6) - 4 = -12 - 4 = -16.
    • Why not D: Forgets to double the square root term before subtracting, computing (-6) - (-4) instead of 2 x (-6) - (-4): -6 + 4 = -2.

  8. Question 8Answer: B

    1. 16 = 2^4, so 16^(3/4) = (2^4)^(3/4) = 2^3 = 8, since 16^(1/4) = 2 and 2 cubed is 8.
    2. Divide by 2^5: 2^3 / 2^5 = 2^(3-5) = 2^-2.
    3. 2^-2 = 1/(2^2) = 1/4.
    4. So the answer is B.
    • Why not A: Rewrites 16^(3/4) correctly as 2^3, but then subtracts the exponents the wrong way round when dividing powers, computing 2^(5-3) = 2^2 = 4 instead of 2^(3-5) = 2^-2.
    • Why not C: Mistakes the fourth root in 16^(1/4) for a square root, taking 16^(1/4) as 4 instead of 2, then cubes this to get 4^3 = 64, before dividing by 32 to get 2.
    • Why not D: Misapplies the fractional index 3/4 by cubing 16 first and then dividing by 4, rather than taking the fourth root first and cubing that, giving 4096 / 4 = 1024, then dividing by 32 to get 32.

  9. Question 9Answer: C

    1. To add numbers in standard form, first write them with the same power of ten: 2.5 x 10^-2 = 25 x 10^-3.
    2. Add the coefficients: 5 x 10^-3 + 25 x 10^-3 = 30 x 10^-3.
    3. Renormalise to standard form: 30 x 10^-3 = 3 x 10 x 10^-3 = 3 x 10^-2.
    4. So the answer is C.
    • Why not A: Confuses adding two standard-form numbers with multiplying them, multiplying the coefficients (5 x 2.5 = 12.5) and adding the indices (-3 + -2 = -5), then renormalising 12.5 x 10^-5 to 1.25 x 10^-4.
    • Why not B: Correctly converts 2.5 x 10^-2 to 25 x 10^-3 and adds to get 30 x 10^-3, but then renormalises the coefficient to 3 without also increasing the power of ten to compensate, writing 3 x 10^-3 instead of 3 x 10^-2.
    • Why not D: Adds the two coefficients directly (5 + 2.5 = 7.5) without first converting either number to a matching power of ten, and simply keeps the smaller of the two powers, -3, unchanged.

  10. Question 10Answer: D

    1. Let x = 0.41666..., where 4 and 1 do not recur but 6 does, starting from the third decimal place.
    2. Multiply by 10 to move past the non-recurring digits: 10x = 4.1666.... Multiply by 100 to move one recurring digit further: 100x = 41.666....
    3. Subtract: 100x - 10x = 41.666... - 4.1666... = 37.5, so 90x = 37.5.
    4. x = 37.5/90 = 375/900, which simplifies to 5/12, so the answer is D.
    • Why not A: Applies the standard method for a decimal where a two-digit block recurs from the start, like 0.414141..., treating this decimal as if '41' were the recurring block, giving 41/99, when in fact only the single digit 6 recurs, and only after the non-recurring digits 4 and 1.
    • Why not B: Correctly sets up 100x - 10x = 37.5, so 90x = 37.5, and correctly finds x = 37.5/90 = 375/900, but stops without simplifying this fraction to its lowest terms.
    • Why not C: Correctly reaches 90x = 37.5, but then divides 37.5 by 100 instead of by 90, forgetting which coefficient of x the subtraction actually produced, giving x = 37.5/100 = 3/8.

  11. Question 11Answer: A

    1. 0.35 as a percentage: 0.35 x 100 = 35%, so 35% is equivalent.
    2. 0.35 as a fraction: 35/100, which simplifies (dividing by 5) to 7/20, so 7/20 is equivalent. Also, 350/1000 = 0.35 directly, so that is equivalent too.
    3. 27/80: dividing 27 by 80 gives 0.3375, which does not equal 0.35.
    4. So 27/80 is the one that is NOT equivalent, and the answer is A.
    • Why not B: 35% does correctly equal 0.35, since dividing 35 by 100 gives 0.35; a candidate who instead divides by 1000 when converting a percentage to a decimal reaches 0.035, and so wrongly flags 35% as not equivalent.
    • Why not C: 7/20 does correctly equal 0.35, since 35/100 simplifies to 7/20 by dividing top and bottom by 5; a candidate who simplifies incorrectly, for example dividing only the numerator by 5, fails to reach 7/20 and so doubts its equivalence.
    • Why not D: 350/1000 does correctly equal 0.35; a candidate who does not recognise that both the numerator and denominator have simply been multiplied by 10 compared with 35/100 may wrongly suspect this fraction represents a different value.

  12. Question 12Answer: D

    1. Multiply the top and bottom by the conjugate of the denominator, sqrt(7) + 3: [4(sqrt(7) + 3)] / [(sqrt(7) - 3)(sqrt(7) + 3)].
    2. The denominator becomes (sqrt(7))^2 - 3^2 = 7 - 9 = -2. The numerator becomes 4 sqrt(7) + 12.
    3. Divide every term of the numerator by -2: (4 sqrt(7))/(-2) + 12/(-2) = -2 sqrt(7) - 6.
    4. So the answer, in the form a + b sqrt(7), is -6 - 2 sqrt(7), which is D.
    • Why not A: Multiplies out (sqrt(7) - 3)(sqrt(7) + 3) as 9 - 7 = 2 instead of 7 - 9 = -2, getting the difference of two squares the wrong way round, and so finds 4(sqrt(7) + 3) / 2 = 2 sqrt(7) + 6.
    • Why not B: Multiplies only the denominator by its conjugate to rationalise it, forgetting that the numerator must be multiplied by the same conjugate too, and so simply divides 4 by the rationalised denominator -2 to get -2.
    • Why not C: Correctly expands the numerator to 4 sqrt(7) + 12 and identifies the denominator as -2, but divides only the constant term by -2, getting -6, while leaving the sqrt(7) term undivided, giving 4 sqrt(7) - 6.

  13. Question 13Answer: B

    1. To maximise a difference a - b, use the largest possible value of a and the smallest possible value of b.
    2. The upper bound of 12.4 (rounded to 1 dp) is 12.4 + 0.05 = 12.45. The lower bound of 5.7 is 5.7 - 0.05 = 5.65.
    3. Upper bound of the difference = 12.45 - 5.65 = 6.8.
    4. So the answer is B.
    • Why not A: Pairs the bounds the wrong way round, taking the lower bound of the first length and the upper bound of the second (12.35 - 5.75 = 6.6), the opposite of the pairing that maximises the difference.
    • Why not C: Uses the two measurements exactly as given (12.4 - 5.7 = 6.7), without allowing for the fact that each has only been rounded to 1 decimal place and so is not exact.
    • Why not D: Treats each measurement as accurate only to the nearest whole number rather than the nearest 0.1, using a tolerance of 0.5 either side instead of 0.05: 12.9 - 5.2 = 7.7.

  14. Question 14Answer: C

    1. Truncating to 1 decimal place means discarding all digits after the first decimal place without rounding, so every value from 7.2 up to, but not reaching, 7.3 truncates down to 7.2.
    2. The true value can be exactly 7.2 itself, since that truncates to 7.2, so the lower bound is included.
    3. The true value cannot reach 7.3, since at that point it would truncate to 7.3, not 7.2, so the upper bound is excluded.
    4. The error interval is 7.2 <= x < 7.3, so the answer is C.
    • Why not A: Applies the error interval method used for ROUNDING, where the true value can lie up to half a unit either side of the given value, giving 7.15 <= x < 7.25, instead of recognising that truncation always cuts value away rather than rounding to the nearest.
    • Why not B: Assumes truncation, like rounding, can never produce a value exactly equal to the given figure, and so uses a strict inequality at the lower end too, excluding the possibility that x itself is exactly 7.2.
    • Why not D: Gets the direction of truncation backwards, treating 7.2 as an upper limit that the true value approaches from below, rather than a lower limit that truncation cuts down to from above.

  15. Question 15Answer: A

    1. 48 lies between the perfect squares 36 and 49, and is much closer to 49, so sqrt(48) can be approximated as sqrt(49) = 7.
    2. Rounding 2.1 to the nearest integer gives 2, since 2.1 is closer to 2 than to 3.
    3. Estimate: 7 x 2 = 14.
    4. So the answer is A.
    • Why not B: Rounds 2.1 to the nearest integer incorrectly as 3 instead of 2, even though 2.1 is closer to 2, then computes 7 x 3 = 21.
    • Why not C: Approximates sqrt(48) using the perfect square below 48, sqrt(36) = 6, instead of the perfect square above it, sqrt(49) = 7, even though 48 is much closer to 49, then computes 6 x 2 = 12.
    • Why not D: Correctly approximates sqrt(48) as 7 using the nearest perfect square, but does not also round 2.1 as instructed, using the exact value 2.1 to compute 7 x 2.1 = 14.7.

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