This set has twelve original questions, ordered from easier to harder, covering all five lessons in number sense and exact arithmetic. Questions 1 to 4 are warm-ups, 5 to 8 build exactness, and 9 to 12 mix skills.
Attempt each question on paper, without a calculator unless it says so, and write your working as you would in an exam. Only then open the answer. Mark the ones you got wrong and use the routing list at the end.
Questions
1. Write in ascending order: −0.6, −2/3, 0.2, −1/2, 0.
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Negatives: −2/3 ≈ −0.667, −0.6 and −1/2 = −0.5. By distance from zero, −2/3 is furthest (0.667), then 0.6, then 0.5, so the order from smallest is −2/3, −0.6, −1/2. Then 0, then 0.2.
−2/3, −0.6, −1/2, 0, 0.2
2. The temperature in a cold room is 5 °C. It falls by 9 °C, then rises by 2.5 °C. What is the final temperature?
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5 − 9 = −4 °C. Then −4 + 2.5 = −1.5 °C. The final temperature is −1.5 °C.
Check on a number line: from 5 move 9 left to −4, then 2.5 right to −1.5.
3. Write 36 and 60 as products of prime factors. Find their HCF and LCM.
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36 = 2² × 3² and 60 = 2² × 3 × 5. HCF uses the lower power of each shared prime: 2² × 3 = 12. LCM uses the highest power of every prime: 2² × 3² × 5 = 180.
Check: HCF × LCM = 12 × 180 = 2160 and 36 × 60 = 2160.
4. Simplify (45 × 28) / (21 × 30) using prime factors.
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45 = 3² × 5, 28 = 2² × 7, 21 = 3 × 7, 30 = 2 × 3 × 5. The numerator is 2² × 3² × 5 × 7. The denominator is 2 × 3² × 5 × 7. Cancelling leaves 2.
The answer is 2. Check: 45 × 28 = 1260 and 21 × 30 = 630, and 1260 ÷ 630 = 2.
5. Work out (30 + 18) / 6. Explain why you cannot cancel the 6 with the 18 only.
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The numerator is a sum, so 6 must divide all of it: 48 / 6 = 8. Factor first to see it: 6 × (5 + 3) / 6 = 5 + 3 = 8. Cancelling with only the 18 gives 30 + 3 = 33, which ignores that the 30 also needs dividing.
6. Work out 3/5 + 7/10 − 1/4 as an exact fraction.
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The lowest common multiple of 5, 10 and 4 is 20. 3/5 = 12/20, 7/10 = 14/20, 1/4 = 5/20. So 12/20 + 14/20 − 5/20 = 21/20.
The answer is 21/20, or 1 1/20. Check: 0.6 + 0.7 − 0.25 = 1.05.
7. Work out 3 1/3 ÷ 1 1/9.
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3 1/3 = 10/3 and 1 1/9 = 10/9. Flip the second and multiply: 10/3 × 9/10. Cancel the 10s and then 9/3 = 3.
The answer is 3. Check: 3.333 ÷ 1.111 = 3.
8. A bag holds 960 g of flour. A recipe uses 2/3 of the bag, and a cake uses 5/8 of that amount. How many grams does the cake use?
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2/3 of 960 = 640 g. Then 5/8 of 640 = 400 g.
Check with a single fraction: 5/8 × 2/3 = 10/24 = 5/12, and 960 × 5/12 = 400. The cake uses 400 g.
9. A circle has radius 4 cm. Give its circumference exactly, then to 1 decimal place.
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Circumference = 2 × π × 4 = 8π cm. On a calculator, 8π = 25.1327…, which is 25.1 cm to 1 decimal place.
Estimate check: 8 × 3 = 24, so a little over 24 fits.
10. Estimate 6.12 × 39.8 by rounding to 1 significant figure, then find the exact product.
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Estimate: 6 × 40 = 240. Exact: 6.12 × 39.8 = 6.12 × 40 − 6.12 × 0.2 = 244.8 − 1.224 = 243.576.
The exact value is close to the estimate, so there is no sign of a slip.
11. Work out 5 + 32 ÷ (7 − 3)² × 2.
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Bracket: 7 − 3 = 4. Power: 4² = 16. Then left to right: 32 ÷ 16 = 2, and 2 × 2 = 4. Finally 5 + 4 = 9.
12. Work out −3² + (−2)³.
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The power applies to 3 only in −3², so −3² = −9. Also (−2)³ = (−2) × (−2) × (−2) = −8. Then −9 + (−8) = −17.
If you got these wrong
Match the type of error to the lesson that repairs it.
- Questions 1 and 2, sign or order errors: go to order signed numbers on a number line.
- Questions 3, 4 and 5, cancelling and prime factors: go to use factors to simplify a numerical expression.
- Questions 6, 7 and 8, fraction working: go to work with fractions without decimal rounding.
- Questions 9 and 10, exact or approximate: go to compare exact and approximate answers.
- Questions 11 and 12, order of operations: go to check operation order in a multi-step calculation.
Record each slip in the mistake log and retest queue with the error type, so a fresh question can be retested later. The non-calculator working trainer gives extra fraction practice with step checking.
When the same error returns after a lesson, a teacher in online one-to-one Mathematics tuition can trace it through your written working and set questions that test exactly that step.