ACER Level 3 Mathematics Practice

Try each question first, then open the worked solution to check your answer and see how to reach it.

1. 1.Find the missing number in the grid: 4(14)3, 5(22)4, 6(?)5

   • A28
   • B30
   • C32
   • D34
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   Answer: C (32)

   Find the rule linking the outside numbers to the middle one. 4 and 3: 4 × 3 = 12, plus 2 = 14. 5 and 4: 5 × 4 = 20, plus 2 = 22. The rule is 'multiply, then add 2'. For 6 and 5: 6 × 5 = 30, plus 2 = 32. Always confirm a rule on every example before applying it.

2. 2.A father is 40 and his son is 10. In how many years will the father be exactly twice as old as his son?

   • A10
   • B15
   • C20
   • D25
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   Answer: C (20)

   Let the number of years be t. In t years the father is 40 + t and the son is 10 + t. We need 40 + t = 2(10 + t) = 20 + 2t, which gives t = 20. Check: the father will be 60 and the son 30. The age gap stays 30 years, so the father is double when the son is 30.

3. 3.A mixture of 40L of milk and water contains 10% water. How much water must be added to make it 20% water?

   • A4L
   • B5L
   • C8L
   • D10L
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   Answer: B (5L)

   The mixture has 10% of 40 L = 4 L of water and 36 L of milk. Adding water keeps the milk at 36 L. For water to be 20%, milk must be 80% of the new total: 36 ÷ 0.8 = 45 L. So add 45 − 40 = 5 L of water. Check: 9 L water out of 45 L is 20%.

4. 4.In a race of 100m, A beats B by 10m and B beats C by 10m. By what distance does A beat C?

   • A18m
   • B19m
   • C20m
   • D21m
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   Answer: B (19m)

   When A runs 100 m, B runs 90 m. B beats C by 10 m over 100 m, so C runs 90% of whatever B runs. While B runs 90 m, C runs 0.9 × 90 = 81 m. So when A finishes, C has run 81 m, and A wins by 100 − 81 = 19 m. Simply adding 10 + 10 = 20 m ignores that the speeds are proportional.

5. 5.If x + y = 15 and xy = 50, what is the value of x² + y²?

   • A100
   • B125
   • C150
   • D225
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   Answer: B (125)

   Use the identity (x + y)² = x² + 2xy + y². So x² + y² = (x + y)² − 2xy = 15² − 2 × 50 = 225 − 100 = 125. You do not need to find x and y separately, which here would involve awkward surds. The answer 225 forgets to subtract the 2xy term.

6. 6.A solid cube is painted red on all sides and then cut into 27 smaller equal cubes. How many have exactly one red face?

   • A4
   • B6
   • C8
   • D12
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   Answer: B (6)

   Cutting the cube into 27 small cubes makes a 3 × 3 × 3 arrangement. Cubes with exactly one red face are in the centre of each large face, where they are not on any edge. Each of the 6 faces has exactly one centre cube, so there are 6. The 8 corner cubes have three red faces, and the 12 edge cubes have two.

7. 7.How many distinct ways can the letters of the word 'MATHS' be arranged?

   • A24
   • B60
   • C120
   • D240
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   Answer: C (120)

   MATHS has 5 different letters. The first position has 5 choices, the second 4, then 3, 2 and 1. Total arrangements: 5 × 4 × 3 × 2 × 1 = 5! = 120. If any letters repeated, you would divide by the repeats, but here all five letters are distinct.

8. 8.What is the unit digit of 7²⁰²³?

   • A1
   • B3
   • C7
   • D9
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   Answer: B (3)

   Powers of 7 have last digits that repeat in a cycle of four: 7, 9, 3, 1 (7¹ = 7, 7² = 49, 7³ = 343, 7⁴ = 2401). To find where 2023 falls, divide by 4: 2023 = 4 × 505 + 3, remainder 3. The third digit in the cycle is 3, so 7²⁰²³ ends in 3.

9. 9.If A is 50% more than B, then B is what percentage less than A?

   • A25%
   • B33.3%
   • C50%
   • D66.6%
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   Answer: B (33.3%)

   If A is 50% more than B, then A = 1.5B. The difference between them is 0.5B. As a fraction of A: 0.5B ÷ 1.5B = 1/3, which is about 33.3%. With numbers: if B = 100, then A = 150, and 50 is one third of 150. The trap answer 50% uses B as the base instead of A.

10. 10.What is the probability of getting exactly 2 heads when flipping 3 fair coins?

    • A1/8
    • B1/4
    • C3/8
    • D1/2
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    Answer: C (3/8)

    Three coins give 2³ = 8 equally likely outcomes. Exactly two heads happens in HHT, HTH and THH, which is 3 outcomes. So the probability is 3/8. The answer 1/8 counts only one arrangement, but the single tail can be in any of the three positions.