Number: Exam Practice and Problem Solving

This lesson brings together all the Number topics from the AQA GCSE Mathematics specification. It provides exam-style practice questions with detailed solutions, highlights common mistakes, and offers strategies for tackling Number questions in the exam. Use this lesson as a revision tool to consolidate your understanding.

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Exam Structure for Number Topics

Number topics appear across both Paper 1 (non-calculator) and Papers 2 and 3 (calculator). They can appear as standalone questions or as parts of larger problems.

Topic	Typical Marks	Papers
Place value and rounding	1-3	All
HCF and LCM	3-4	All
Fractions	2-5	Paper 1 especially
Percentages	3-5	All
Converting between forms	2-4	All
Indices	2-4	All
Surds [H]	3-5	All
Standard form	2-4	All
Bounds [H]	3-5	Papers 2, 3

Exam Tip: Number skills underpin many other topics. For example, you need fractions for probability, percentages for statistics, and indices for algebra. Mastering Number makes the whole exam easier.

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Practice Question 1: Estimation (3 marks)

Estimate the value of (19.7 x 4.13) / 0.489

Solution

Step 1: Round each number to 1 significant figure.

• 19.7 rounds to 20
• 4.13 rounds to 4
• 0.489 rounds to 0.5

Step 2: Calculate.

• (20 x 4) / 0.5
• = 80 / 0.5
• = 160

Common mistake: Forgetting to show the rounded values. Even if your final answer is correct, you may lose marks if the examiner cannot see your rounding.

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Practice Question 2: HCF and LCM (4 marks)

Using prime factorisation, find the HCF and LCM of 84 and 120.

Solution

Step 1: Prime factorise each number.

• 84 = 2 x 2 x 3 x 7 = 2 squared x 3 x 7
• 120 = 2 x 2 x 2 x 3 x 5 = 2 cubed x 3 x 5

Step 2: Venn diagram.

84 only	Overlap	120 only
7	2, 2, 3	2, 5

Step 3: Calculate.

• HCF = 2 x 2 x 3 = 12
• LCM = 7 x 2 x 2 x 3 x 2 x 5 = 840

Check: 12 x 840 = 10,080. Also 84 x 120 = 10,080. Correct.

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Practice Question 3: Fractions (4 marks)

Calculate 2 and 3/4 divided by 1 and 1/6. Give your answer as a mixed number in its simplest form.

Solution

Step 1: Convert to improper fractions.

• 2 and 3/4 = 11/4
• 1 and 1/6 = 7/6

Step 2: Flip and multiply.

• 11/4 x 6/7
• = 66/28

Step 3: Simplify.

• 66/28 = 33/14

Step 4: Convert to mixed number.

• 33 / 14 = 2 remainder 5

Answer: 2 and 5/14

Exam Tip: Always convert mixed numbers to improper fractions before dividing or multiplying. And always check whether your final fraction can be simplified.

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Practice Question 4: Percentage Change (3 marks)

A bicycle was bought for 340 pounds and sold for 289 pounds. Calculate the percentage loss.

Solution

• Loss = 340 - 289 = 51 pounds
• Percentage loss = (51 / 340) x 100
• = 0.15 x 100
• = 15%

Common mistake: Dividing by 289 (the selling price) instead of 340 (the original price). Always divide by the original.

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Practice Question 5: Reverse Percentages (3 marks)

After a 12% decrease in price, a television costs 440 pounds. What was the original price?

Solution

Step 1: The current price represents 88% of the original (100% - 12% = 88%).

Step 2: Original price = 440 / 0.88

Step 3: = 500 pounds

Check: 12% of 500 = 60. 500 - 60 = 440. Correct.

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Practice Question 6: Recurring Decimals [H] (3 marks)

Show that 0.363636... = 4/11.

Solution

Let x = 0.363636...

100x = 36.363636...

100x - x = 36.363636... - 0.363636...

99x = 36

x = 36/99

Simplify: 36/99 = 4/11 (divide numerator and denominator by 9)

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Practice Question 7: Indices [H] (3 marks)

Evaluate 64 to the power (-2/3).

Solution

Step 1: Deal with the negative — take the reciprocal.

• 1 / (64 to the power 2/3)

Step 2: Root first — cube root of 64 = 4.

Step 3: Then power — 4 squared = 16.

Step 4: Apply reciprocal — 1/16.

Answer: 1/16

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Practice Question 8: Surds [H] (4 marks)

Simplify (3 + root 5)(3 - root 5) + 2 root 20.

Solution

Step 1: Expand the brackets (difference of two squares).

• (3 + root 5)(3 - root 5) = 9 - 5 = 4

Step 2: Simplify root 20.

• Root 20 = root (4 x 5) = 2 root 5

Step 3: Calculate 2 root 20.

• 2 x 2 root 5 = 4 root 5

Step 4: Add results.

• 4 + 4 root 5

Answer: 4 + 4 root 5

Exam Tip: Look for the difference of two squares pattern in surd questions. It always eliminates the surd, leaving a rational number.

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Practice Question 9: Standard Form (3 marks)

Calculate (4.2 x 10 to the power 5) x (3 x 10 to the power (-2)). Give your answer in standard form.

Solution

Step 1: Multiply the "a" values: 4.2 x 3 = 12.6

Step 2: Add the powers: 5 + (-2) = 3

Step 3: 12.6 x 10 cubed

Step 4: Adjust to standard form: 12.6 = 1.26 x 10

So: 1.26 x 10 x 10 cubed = 1.26 x 10 to the power 4

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Practice Question 10: Bounds [H] (5 marks)

A runner completes a 100 m track (measured to the nearest metre) in 12.3 seconds (measured to the nearest 0.1 seconds). Calculate the upper and lower bounds of the runner's speed. State whether the runner could have been travelling faster than 8.2 m/s.

Solution

Step 1: Find bounds.

Measurement	Lower Bound	Upper Bound
Distance	99.5 m	100.5 m
Time	12.25 s	12.35 s

Step 2: Calculate speed bounds. (Speed = Distance / Time)

• Maximum speed = UB(distance) / LB(time) = 100.5 / 12.25 = 8.204... m/s
• Minimum speed = LB(distance) / UB(time) = 99.5 / 12.35 = 8.056... m/s

Step 3: Could the speed be more than 8.2 m/s?

The maximum possible speed is approximately 8.20 m/s. Since the upper bound exceeds 8.2, yes, the runner could have been travelling faster than 8.2 m/s.

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