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

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

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

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 metal cube has a volume of 8 cm^3 and a mass of 60 g. Find the density of the metal in kg/m^3.

    1. A 7500 kg/m^3
    2. B 7.5 kg/m^3
    3. C 0.0075 kg/m^3
    4. D 75 kg/m^3
  2. 21 mark

    Which of the following correctly places the numbers -2, -1.5, 5/8 and 0.7 in order, using the symbol < throughout?

    1. A -1.5 < -2 < 5/8 < 0.7
    2. B -2 < -1.5 < 0.7 < 5/8
    3. C 0.7 < 5/8 < -1.5 < -2
    4. D -2 < -1.5 < 5/8 < 0.7
  3. 31 mark

    Work out -2 1/3 + 3 3/4, giving your answer as a mixed number in its simplest form.

    1. A 2 1/12
    2. B 1 5/12
    3. C 6 1/12
    4. D 1 4/7
  4. 41 mark

    The prime factorisations of two numbers are 84 = 2^2 x 3 x 7 and 90 = 2 x 3^2 x 5. Find the highest common factor (HCF) of 84 and 90.

    1. A 1260
    2. B 36
    3. C 210
    4. D 6
  5. 51 mark

    Work out 20 - 3 x (4 - 1)^2 + sqrt(16).

    1. A 157
    2. B -3
    3. C -11
    4. D 6
  6. 61 mark

    A padlock code uses three different digits chosen from 1, 2, 3, 4 and 5 (no digit is repeated, and the order of the digits matters). How many different codes are possible?

    1. A 125
    2. B 60
    3. C 120
    4. D 10
  7. 71 mark

    Given that x^2 = 49 and y^3 = -8, where x is taken to be the positive square root of 49, find the value of x + y.

    1. A 5
    2. B 9
    3. C -9
    4. D -5
  8. 81 mark

    Simplify 27^(2/3) x 3^(-2), giving your answer as an integer or fraction in its simplest form.

    1. A 81
    2. B -1
    3. C 27
    4. D 1
  9. 91 mark

    Work out (4 x 10^5) x (3 x 10^-2), giving your answer in standard form.

    1. A 1.2 x 10^4
    2. B 1.2 x 10^3
    3. C 12 x 10^3
    4. D 1.2 x 10^8
  10. 101 mark

    Convert the recurring decimal 0.272727... (where the digits '27' recur forever) to a fraction in its simplest form.

    1. A 27/99
    2. B 3/11
    3. C 27/100
    4. D 11/3
  11. 111 mark

    Write 5/8 as a percentage.

    1. A 0.625%
    2. B 160%
    3. C 62.5%
    4. D 58%
  12. 121 mark

    Simplify 6 / sqrt(3), giving your answer in the form a sqrt(b), where b has no square factors.

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

    A rectangle has a length of 12 cm and a width of 5 cm, each measured to the nearest cm. What is the upper bound for the area of the rectangle, in cm^2?

    1. A 51.75
    2. B 68.75
    3. C 60
    4. D 78
  14. 141 mark

    A length is measured as 7.4 cm, correct to 1 decimal place. Write down the error interval for the true length, x, using inequality notation.

    1. A 7.3 <= x < 7.5
    2. B 7.35 <= x < 7.45
    3. C 7.35 <= x <= 7.45
    4. D 7.4 <= x < 7.5
  15. 151 mark

    By rounding each number to 1 significant figure, estimate the value of pi x 8.7 / 2.9.

    1. A 8
    2. B 27
    3. C 13.5
    4. D 9

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: A

    1. Density = mass / volume = 60 g / 8 cm^3 = 7.5 g/cm^3.
    2. To change g/cm^3 into kg/m^3, multiply by 1000: 1 g = 1/1000 kg and 1 cm^3 = 1/1,000,000 m^3, so the g/cm^3 value scales up by 1,000,000 / 1000 = 1000 when changed to kg/m^3.
    3. 7.5 x 1000 = 7500.
    4. So the density is 7500 kg/m^3, which is answer A.
    • Why not B: Correctly finds the density as 7.5 g/cm^3 (60 divided by 8) but does not convert the units at all, leaving the answer in g/cm^3 and simply relabelling it kg/m^3.
    • Why not C: Knows a conversion is needed but divides by 1000 instead of multiplying, treating a change to unit names that sound 'bigger' (kg, m) as always meaning a smaller number.
    • Why not D: Uses a conversion factor of 10, as if converting a length from cm to mm, instead of the correct factor of 1000 that applies when converting g/cm^3 to kg/m^3.

  2. Question 2Answer: D

    1. Convert 5/8 to a decimal to compare it with the others: 5/8 = 0.625.
    2. Compare the negative numbers first: on a number line, -2 lies further to the left than -1.5, so -2 is the smaller (more negative) value, giving -2 < -1.5.
    3. Compare the positive values: 0.625 < 0.7, so 5/8 < 0.7.
    4. Putting these together gives -2 < -1.5 < 5/8 < 0.7, which is option D.
    • Why not A: Orders the negative numbers by their magnitude (distance from zero) rather than their true value, treating -2 as larger than -1.5 because 2 is greater than 1.5.
    • Why not B: Misreads the fraction 5/8 as its reciprocal 8/5 (= 1.6), overestimating its value and placing it above 0.7.
    • Why not C: Writes the list in descending order (largest to smallest) instead of ascending, effectively reading the '<' symbol as 'greater than' throughout.

  3. Question 3Answer: B

    1. Write each mixed number as an improper fraction: -2 1/3 = -(2 x 3 + 1)/3 = -7/3, and 3 3/4 = (3 x 4 + 3)/4 = 15/4.
    2. Find a common denominator of 3 and 4, which is 12: -7/3 = -28/12 and 15/4 = 45/12.
    3. Add the fractions: -28/12 + 45/12 = 17/12.
    4. Convert back to a mixed number: 17/12 = 1 5/12, so the answer is B.
    • Why not A: Treats -2 1/3 as (-2) + 1/3 rather than as the single negative value -(2 + 1/3), so only the whole-number part is made negative before adding.
    • Why not C: Drops the negative sign entirely, calculating 2 1/3 + 3 3/4 instead of -2 1/3 + 3 3/4.
    • Why not D: Combines the fraction parts by adding numerators and denominators directly (1/3 + 3/4 treated as (1+3)/(3+4) = 4/7) instead of finding a common denominator.

  4. Question 4Answer: D

    1. The HCF uses only the primes that appear in both factorisations, raised to the lower of the two powers.
    2. Both 84 and 90 contain the primes 2 and 3 (84 has no factor of 5, and 90 has no factor of 7, so these are not used).
    3. For 2: the lower power is 2^1 (84 has 2^2, 90 has 2^1). For 3: the lower power is 3^1 (84 has 3^1, 90 has 3^2).
    4. HCF = 2^1 x 3^1 = 2 x 3 = 6, so the answer is D.
    • Why not A: Finds the lowest common multiple instead of the highest common factor, using the highest power of every prime that appears in either factorisation: 2^2 x 3^2 x 5 x 7 = 1260.
    • Why not B: Uses the highest shared power of each common prime (2^2 x 3^2 = 36) instead of the lowest shared power (2^1 x 3^1).
    • Why not C: Multiplies every prime factor that appears in either factorisation once each (2 x 3 x 5 x 7 = 210), instead of using only the primes common to both numbers.

  5. Question 5Answer: B

    1. Work out the bracket first: 4 - 1 = 3.
    2. Apply the power: 3^2 = 9. Then the multiplication: 3 x 9 = 27. The square root: sqrt(16) = 4.
    3. The expression is now 20 - 27 + 4, worked left to right since addition and subtraction have equal priority: 20 - 27 = -7, then -7 + 4 = -3.
    4. So the answer is B.
    • Why not A: Performs the subtraction 20 - 3 before the multiplication, working strictly left to right instead of doing the multiplication (and the power inside it) first: (20 - 3) x (4 - 1)^2 + sqrt(16) = 17 x 9 + 4 = 157.
    • Why not C: Groups the final two terms together as if there were brackets around them, computing 20 - (27 + 4) = -11 instead of working left to right through 20 - 27 + 4.
    • Why not D: Reads the power (4 - 1)^2 as (4 - 1) x 2, doubling the bracket instead of squaring it, giving 3 x 6 = 18 and then 20 - 18 + 4 = 6.

  6. Question 6Answer: B

    1. The first digit of the code can be any of the 5 available digits: 5 choices.
    2. Since digits cannot repeat, the second digit is chosen from the 4 remaining digits: 4 choices.
    3. The third digit is then chosen from the 3 remaining digits: 3 choices.
    4. By the counting principle, the total number of codes is 5 x 4 x 3 = 60, so the answer is B.
    • Why not A: Allows digits to repeat, treating each of the 3 positions as independently able to take any of the 5 digits: 5^3 = 125.
    • Why not C: Forgets that the code only uses 3 of the 5 digits, and instead finds the number of ways to arrange all 5 digits: 5! = 120.
    • Why not D: Counts combinations instead of arrangements, ignoring that the order of the digits matters (so, for example, 123 and 321 are wrongly treated as the same code): 5C3 = 10.

  7. Question 7Answer: A

    1. 49 has two square roots, 7 and -7, but the question specifies the positive root, so x = 7.
    2. Unlike a square root, a negative number has exactly one real cube root, and it is negative: since (-2)^3 = -2 x -2 x -2 = -8, y = -2.
    3. Add the two values: x + y = 7 + (-2) = 5.
    4. So the answer is A.
    • Why not B: Treats the cube root of -8 as if it must be positive, like a square root, giving y = 2 instead of y = -2, so x + y = 7 + 2 = 9.
    • Why not C: Ignores the instruction that x is the positive square root, and instead takes x = -7 (correctly keeping y = -2): -7 + (-2) = -9.
    • Why not D: Takes x as the negative square root (x = -7) and also treats the cube root of -8 as positive (y = 2): -7 + 2 = -5.

  8. Question 8Answer: D

    1. 27^(1/3) = 3, because 3^3 = 27. A fractional index a^(2/3) means (a^(1/3))^2, so 27^(2/3) = 3^2 = 9.
    2. 3^(-2) means 1/(3^2) = 1/9, since a negative index gives the reciprocal of the positive-index value.
    3. Multiply the two results: 9 x 1/9 = 1.
    4. So the answer is D.
    • Why not A: Drops the negative sign on the second index, computing 3^(-2) as 3^2 = 9 instead of 1/9, giving 9 x 9 = 81.
    • Why not B: Treats the negative index as making the whole term negative rather than as a reciprocal, computing 3^(-2) as -(1/9), giving 9 x (-1/9) = -1.
    • Why not C: Misapplies the fractional index on 27, computing 27^(2/3) as 27^2 / 3 = 243 instead of (27^(1/3))^2 = 9, giving 243 x 1/9 = 27.

  9. Question 9Answer: A

    1. Multiply the coefficients: 4 x 3 = 12.
    2. Multiply the powers of ten by adding the indices: 10^5 x 10^-2 = 10^(5 + (-2)) = 10^3.
    3. This gives 12 x 10^3, but standard form needs a coefficient between 1 and 10, so rewrite 12 as 1.2 x 10, giving 1.2 x 10 x 10^3 = 1.2 x 10^4.
    4. So the answer is A.
    • Why not B: Correctly renormalises the coefficient (12 to 1.2) but forgets to increase the power of 10 to compensate, leaving the exponent unchanged at 3.
    • Why not C: Multiplies the coefficients and adds the indices correctly (giving 12 x 10^3) but does not rewrite the answer in standard form, where the coefficient must be between 1 and 10.
    • Why not D: Drops the negative sign on the second index, adding 5 + 2 = 7 instead of 5 + (-2) = 3, giving 12 x 10^7 = 1.2 x 10^8.

  10. Question 10Answer: B

    1. Let x = 0.272727..., where the two digits '27' repeat forever.
    2. Multiply both sides by 100 (since two digits recur): 100x = 27.272727...
    3. Subtract the original equation from this: 100x - x = 27.272727... - 0.272727..., so 99x = 27.
    4. Divide to find x: x = 27/99, which simplifies (dividing top and bottom by 9) to 3/11, so the answer is B.
    • Why not A: Correctly sets up 99x = 27 but stops there, giving the unsimplified fraction 27/99 instead of dividing top and bottom by their common factor of 9.
    • Why not C: Treats the decimal as if it terminated after two digits (0.27) rather than recurring forever, and simply writes it as 27/100.
    • Why not D: Solves the equation 99x = 27 by dividing 99 by 27 instead of 27 by 99, flipping the resulting fraction upside down.

  11. Question 11Answer: C

    1. To convert a fraction to a percentage, first write it as a decimal: 5/8 = 0.625 (8 goes into 50 six times remainder 2, into 20 twice remainder 4, into 40 five times exactly).
    2. Multiply the decimal by 100 to express it as a percentage: 0.625 x 100 = 62.5.
    3. So 5/8 = 62.5%.
    4. The answer is C.
    • Why not A: Correctly converts 5/8 to the decimal 0.625 but forgets the final step of multiplying by 100 to express it as a percentage.
    • Why not B: Inverts the fraction before converting, computing 8/5 = 1.6 and then multiplying by 100, instead of using 5/8 as given.
    • Why not D: Divides 5 by 8 incorrectly (getting 0.58 instead of 0.625), then converts this wrong decimal to a percentage.

  12. Question 12Answer: C

    1. To rationalise 6 / sqrt(3), multiply the top and bottom by sqrt(3): (6 x sqrt(3)) / (sqrt(3) x sqrt(3)) = 6 sqrt(3) / 3.
    2. sqrt(3) x sqrt(3) = 3, since squaring a square root removes the root.
    3. Divide the coefficient by the denominator: 6 / 3 = 2, so 6 / sqrt(3) = 2 sqrt(3).
    4. Since 3 has no square factors, this is fully simplified, so the answer is C.
    • Why not A: Multiplies the top and bottom by sqrt(3) to start rationalising (giving 6 sqrt(3) over 3) but then forgets to actually divide by the resulting denominator of 3.
    • Why not B: Correctly rationalises to 6 sqrt(3) / 3 but then miscalculates 6 divided by 3 as 3 instead of 2.
    • Why not D: Wrongly treats the division as taking place under a single square root, writing 6 / sqrt(3) as sqrt(6) / sqrt(3) = sqrt(6/3) = sqrt(2), which is not a valid way to rewrite the plain number 6.

  13. Question 13Answer: B

    1. A length rounded to the nearest cm could have been up to half a cm larger before rounding, so the upper bound of 12 cm is 12.5 cm, and the upper bound of 5 cm is 5.5 cm.
    2. The upper bound of the area is found by multiplying the upper bounds of both dimensions, since a larger length and a larger width both increase the area.
    3. 12.5 x 5.5 = 12.5 x 5 + 12.5 x 0.5 = 62.5 + 6.25 = 68.75.
    4. So the answer is B.
    • Why not A: Finds the lower bound of the area instead of the upper bound, using the lower bounds of both measurements: 11.5 x 4.5 = 51.75.
    • Why not C: Uses the given measurements directly (12 x 5), ignoring that they have only been rounded to the nearest cm and so are not exact.
    • Why not D: Adjusts each measurement by a whole unit (+1) instead of half a unit (+0.5), computing 13 x 6 = 78.

  14. Question 14Answer: B

    1. A value given correct to 1 decimal place (the nearest 0.1) could have been rounded from any true value within 0.05 of the stated value.
    2. Lower bound: 7.4 - 0.05 = 7.35. Upper bound: 7.4 + 0.05 = 7.45.
    3. The lower bound is included, since 7.35 rounds to 7.4, but the upper bound is not included, since 7.45 would round up to 7.5, not down to 7.4.
    4. So the error interval is 7.35 <= x < 7.45, which is answer B.
    • Why not A: Uses a tolerance of 0.1 either side of the given value (a whole rounding unit) instead of half a unit (0.05), giving too wide an interval.
    • Why not C: Uses <= at both ends of the interval, not recognising that the upper bound value of 7.45 would itself round up to 7.5, not down to 7.4, and so must be excluded.
    • Why not D: Treats the given value, 7.4, as the lower bound of the interval and adds a full tolerance to find the upper bound, instead of finding a lower bound below 7.4.

  15. Question 15Answer: D

    1. Round each number to 1 significant figure: pi = 3.14159... rounds to 3; 8.7 rounds to 9 (since 8.7 is closer to 9 than to 8); 2.9 rounds to 3 (since 2.9 is closer to 3 than to 2).
    2. Substitute the rounded values into the expression: 3 x 9 / 3.
    3. 3 x 9 = 27, and 27 / 3 = 9.
    4. So the estimate is 9, and the answer is D.
    • Why not A: Rounds 8.7 down to 8 to 1 significant figure instead of up to 9, since 8.7 is closer to 9.
    • Why not B: Correctly rounds each number (pi to 3, 8.7 to 9, 2.9 to 3) but forgets to divide, computing only the numerator: 3 x 9 = 27.
    • Why not C: Rounds 2.9 down to 2 to 1 significant figure instead of up to 3, since 2.9 is closer to 3, giving 3 x 9 / 2 = 13.5.

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