ACER Level 2 Mathematics Practice

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

1. 1.If 3 apples balance 2 pears, and 1 pear balances 2 plums, how many plums balance 3 apples?

   • A2
   • B3
   • C4
   • D6
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   Answer: C (4)

   Link the items through the pears. 3 apples balance 2 pears. Each pear balances 2 plums, so 2 pears balance 2 × 2 = 4 plums. Therefore 3 apples balance 4 plums. Replacing items with their equivalents one step at a time keeps the reasoning clear.

2. 2.Tom is 10 years old. In 5 years, he will be half his father's age. How old is his father now?

   • A20
   • B25
   • C30
   • D35
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   Answer: B (25)

   In 5 years Tom will be 10 + 5 = 15. At that time he will be half his father's age, so his father will be 15 × 2 = 30. That is 5 years from now, so the father is 30 − 5 = 25 now. The answer 20 doubles Tom's current age, which ignores the 5-year gap in the question.

3. 3.A piece of wood is cut into 3 pieces in a ratio of 2:3:4. If the longest piece is 20cm, what was the original total length?

   • A35cm
   • B40cm
   • C45cm
   • D50cm
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   Answer: C (45cm)

   The ratio 2:3:4 has 9 parts in total. The longest piece is 4 parts and measures 20 cm, so one part is 20 ÷ 4 = 5 cm. The whole piece of wood was 9 × 5 = 45 cm. Check: 10 + 15 + 20 = 45 cm. Using the known piece to find 'one part' is the key step.

4. 4.Sarah has twice as many coins as John. If Sarah gives John 5 coins, they have the same amount. How many did John start with?

   • A5
   • B10
   • C15
   • D20
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   Answer: B (10)

   Let John start with J coins, so Sarah has 2J. After Sarah gives 5 to John, they are equal: 2J − 5 = J + 5. Solving gives J = 10. Check: John has 10 and Sarah has 20. After the swap, both have 15. Remember that a gift changes both people's totals.

5. 5.If A = 2B and B = 3C, what is the ratio of A to C?

   • A1:6
   • B2:3
   • C5:1
   • D6:1
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   Answer: D (6:1)

   Express A in terms of C. B = 3C, and A = 2B = 2 × 3C = 6C. So A is six times C, giving the ratio A:C = 6:1. A quick check with numbers: if C = 1, then B = 3 and A = 6. The answer 1:6 reverses the ratio.

6. 6.A water tank is 1/4 full. Adding 10 litres makes it 1/2 full. What is the tank's total capacity?

   • A20L
   • B30L
   • C40L
   • D50L
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   Answer: C (40L)

   Going from 1/4 full to 1/2 full adds 1/2 − 1/4 = 1/4 of the tank. That quarter is 10 litres, so the whole tank holds 10 × 4 = 40 litres. Check: a quarter of 40 is 10 L, and adding 10 L gives 20 L, which is half. The answer 20 L confuses half the tank with the whole.

7. 7.A tap fills a bath in 10 mins. Another tap empties it in 15 mins. If both are open, how long to fill the bath?

   • A12.5 mins
   • B20 mins
   • C25 mins
   • D30 mins
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   Answer: D (30 mins)

   Compare what each tap does in one minute. The filling tap adds 1/10 of the bath and the emptying tap removes 1/15. Net: 1/10 − 1/15 = 3/30 − 2/30 = 1/30 of the bath per minute, so filling takes 30 minutes. It must take longer than 10 minutes, because the drain works against the tap.

8. 8.If x is 20% of y, and y is 50% of z, what percentage of z is x?

   • A10%
   • B20%
   • C30%
   • D40%
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   Answer: A (10%)

   Chain the percentages by multiplying. y is 50% of z, so y = 0.5z. x is 20% of y, so x = 0.2 × 0.5z = 0.1z, which is 10% of z. A check with numbers: if z = 100, then y = 50 and x = 10. Adding the percentages (70%) is not how chained percentages work.

9. 9.How many distinct 3-digit numbers can be formed from digits 1, 2, 3 without repeating?

   • A3
   • B6
   • C9
   • D27
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   Answer: B (6)

   Fill the three places one at a time. The hundreds digit has 3 choices, the tens digit has the 2 remaining choices, and the ones digit has 1 left. That gives 3 × 2 × 1 = 6 numbers: 123, 132, 213, 231, 312 and 321. Listing them in order confirms none are missed.

10. 10.The denominator of a fraction is 3 more than the numerator. If both are increased by 1, it equals 1/2. What was the original fraction?

    • A1/4
    • B2/5
    • C3/6
    • D4/7
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    Answer: B (2/5)

    Test the options against both conditions. The denominator must be 3 more than the numerator, and adding 1 to both must give 1/2. For 2/5: 5 is 3 more than 2, and (2 + 1)/(5 + 1) = 3/6 = 1/2. Option C, 3/6, already equals 1/2 and its denominator is not 3 more, so it fails.