AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 13

📐 Step-by-Step Calculation

1. First interval (≤ 20 000): 44
2. Second interval (≤ 40 000): 44 + 22 = 66
3. Third interval (≤ 60 000): 66 + 14 = 80
4. Fourth interval (≤ 80 000): 80 + 5 = 85
5. Fifth interval (≤ 100 000): 85 + 2 = 87
6. Final interval (≤ 120 000): 87 + 3 = 90

✅ Completed Table Values

💡 Key Knowledge

• "Cumulative" means a running total.
• The final value in the cumulative frequency column must always equal the total frequency (here, 90 employees).

❌ Common Errors

• Copying the original frequencies rather than adding them up cumulatively.
• Arithmetic slips when adding single-digit frequencies near the end (e.g. 80 + 5 = 86).

💡 Where to Plot the Points

Always plot Cumulative Frequency (y-axis) against the Upper Class Boundary (x-axis):

• (20 000, 44)
• (40 000, 66)
• (60 000, 80)
• (80 000, 85)
• (100 000, 87)
• (120 000, 90)

Start the curve at (0, 0) or (20 000, 44).

🧠 Exam Technique & Scale Reading

• x-axis scale: 10 small squares = £20 000, so 1 small square = £2 000.
• y-axis scale: 10 small squares = 10 units, so 1 small square = 1 unit.
• Plot points with neat, small crosses (×) within ±0.5 small square.
• Join points with a single continuous smooth curve or straight line segments (polygon).

❌ Common Examiner Traps

• Plotting at midpoints: A very common error! Cumulative frequency must be plotted at the upper end of each interval, not the middle.
• Drawing a bar chart: If you draw bars, the examiner will ignore them, but you waste valuable time.
• Wobbly lines / double lines: Keep pencil lines neat and sharp. Joining the points like a "mountain range" with dips loses marks—the curve must be continuously non-decreasing.

📐 Step-by-Step Method

1. Locate £32 000 on the horizontal axis:
   Since each small square is £2 000, £32 000 is 6 small squares to the right of £20 000 (20 000 + 6 × 2 000 = 32 000).
2. Draw a vertical dashed line from £32 000 directly up to touch your curve.
3. Draw a horizontal dashed line from that intersection across to the vertical axis (Cumulative Frequency).
4. Read the y-axis value: The value falls in the range 56 to 60 (typically 58).

✅ Acceptable Answer Range

56 – 60

Any integer in the range [56, 60] (or the exact reading from your own drawn curve) is awarded full marks.

🧠 Why "Less than" Requires No Extra Step

Cumulative frequency graphs naturally measure values less than or equal to a given figure. Because the question asks for "less than £32 000", your reading is the final answer!

Note: If the question had asked for "more than £32 000", you would subtract your reading from the total: 90 − reading.

❌ Common Errors

• Misreading the scale: Assuming 1 small square = £1 000 instead of £2 000.
• Not showing construction lines: Always draw dashed lines on your graph; this guarantees the method mark (M1) even if you make a reading mistake.
• Unnecessary subtraction: Calculating 90 − 58 = 32 because of misreading "less than" as "more than".