AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 14

7 marks · Medium difficulty · Multi-step Problem

Use upper bounds of rounded masses to determine if a plane can take off safely, and draw a speed-time graph for a journey of 1250 miles in 2 hours 30 minutes at constant speed.

Practise this question

Question

Question 14 consists of two parts. Part (a) gives three rounded masses for a small plane: empty plane is 800 kg to the nearest 100 kg, fuel is 190 kg to the nearest 5 kg, and passengers is 163 kg to the nearest kg. Students are asked whether the plane can definitely take off safely if the maximum safe mass is 1200 kg, ticking 'Yes' or 'No' and showing working (4 marks). Part (b) states a different plane travels 1250 miles in 2 hours 30 minutes at a constant speed, requiring students to draw a speed/time graph on an unlabelled grid with axes (3 marks).

Mark scheme

Show the mark scheme Mark scheme for Question 14. Part (a) awards marks for identifying bounds (B1 for 850, 192.5, or 163.5; B1 for all three upper bounds), adding the upper bounds (M1), and concluding 1206 and ticking 'No' (A1), or alternatively using any values exceeding 1200 (B4). Part (b) awards B1 for finding speed = 500 mph, B1ft for correctly labelling and scaling the axes ('Speed' and 'Time'), and B1ft for a horizontal line from time 0 to 2.5 hours at speed 500.

How to answer it

Upper Bounds & Drawing Speed-Time Graphs

📋 What this question tests

This 7-mark question assesses two distinct Higher Tier mathematical skills:

  • Part (a) [4 marks]: Calculating upper bounds for values rounded to varying degrees of accuracy (nearest 100, nearest 5, nearest integer) and combining them to evaluate a worst-case scenario.
  • Part (b) [3 marks]: Converting units of time into decimal hours, calculating constant speed via Speed = Distance ÷ Time , and creating a properly scaled and labelled speed-time graph.

Question 14 (a) — Limits of Accuracy / Upper Bounds

4 Marks

💡 Key Knowledge: Upper Bounds

To find bounds, divide the rounding interval by 2, then add or subtract:

  • Empty plane (nearest 100 kg): Interval is 100 kg. Half is 50 kg.
    Upper Bound = 800 + 50 = 850 kg
  • Fuel (nearest 5 kg): Interval is 5 kg. Half is 2.5 kg.
    Upper Bound = 190 + 2.5 = 192.5 kg
  • Passengers (nearest 1 kg): Interval is 1 kg. Half is 0.5 kg.
    Upper Bound = 163 + 0.5 = 163.5 kg

🧠 Exam Technique: "Can it definitely...?"

  • The question asks if the plane can definitely take off safely (maximum safe limit = 1200 kg).
  • To prove it can definitely take off, the worst-case maximum possible mass (the sum of all upper bounds) must be ≤ 1200 kg.
  • If the upper bound sum exceeds 1200 kg, there is a chance the plane is overloaded, meaning it cannot definitely take off safely.
  • Always complete your working with a clear conclusion and tick the box!

📐 Step-by-Step Calculation

Step 1: Calculate the maximum possible total mass

Total Upper Bound = 850 + 192.5 + 163.5 = 1206 kg

Step 2: Compare against the maximum safety threshold

1206 kg > 1200 kg (it can exceed safety limit by 6 kg)

Step 3: State conclusion and tick box

Because 1206 kg > 1200 kg, the plane may be overweight.

✅ Correct Answer & Marks

Box ticked: No

Supporting working: Shows total upper bound of 1206 kg

Mark Breakdown:
• B1: Any one correct bound seen (750 / 850, 187.5 / 192.5, 162.5 / 163.5)
• B1: All three upper bounds correct: 850, 192.5, and 163.5
• M1: Adding three values within the accepted intervals: (800, 900] + (190, 195] + (163, 164]
• A1: 1206 and tick 'No' (or conclusion that it is 6 kg too heavy)

❌ Common Errors to Avoid

  • Miscalculating the fuel bound: Adding 5 instead of half of 5, writing 195 kg instead of 192.5 kg. This capped students to at most 1 mark.
  • Using lower bounds or original values: Adding 800 + 190 + 163 = 1153 kg and concluding "Yes" ignores the word definitely.
  • Forgetting to tick the box: Full method marks can be earned, but the final mark requires a clear decision (ticking 'No' or writing No).

Question 14 (b) — Speed-Time Graph

3 Marks

💡 Key Knowledge: Speed & Graphs

  • Time conversion: 2 hours 30 minutes = 2.5 hours (NOT 2.3 hours!).
  • Speed formula: Speed = Distance ÷ Time = 1250 ÷ 2.5 = 500 mph .
  • Nature of a speed-time graph: A constant speed is represented by a flat horizontal line, NOT a sloping diagonal line.

🧠 Exam Technique: Drawing the Axes

  • Axis Labels: The vertical axis must be labelled Speed and the horizontal axis must be labelled Time.
  • Linear Scale: Choose regular, even intervals that allow you to reach at least 500 on the vertical axis and at least 2.5 hours (or 150 minutes) on the horizontal axis.
  • Line Placement: Draw a straight horizontal line starting at (0, 500) and stopping precisely at (2.5, 500) .

📐 Step-by-Step Construction

Step 1: Calculate the speed

Speed = 1250 ÷ 2.5 = 500 mph

(Alternative in minutes: 1250 ÷ 150 = 8.33 miles/min)

Step 2: Scale the axes

  • Horizontal axis: Label "Time" (e.g. 1 major division = 0.5 hours or 30 minutes, extending to at least 2.5 hours / 150 mins).
  • Vertical axis: Label "Speed" (e.g. 1 major division = 100 mph, extending to at least 500 mph).

Step 3: Plot the line

Draw a horizontal line using a ruler from Time = 0 to Time = 2.5 hours at a constant height of 500.

✅ Correct Answer & Marks

Mark Breakdown:
• B1: Calculates (Speed =) 500 [or 8.33... miles per minute with correct units stated]
• B1ft: Vertical axis labelled 'Speed' with a linear scale to at least 500 AND Horizontal axis labelled 'Time' with a linear scale to at least 2.5 hours (or 150 mins)
• B1ft: Horizontal line drawn from (0, 500) stopping at (2.5 hours, 500) or (150 mins, 500)

❌ Common Errors to Avoid

  • Drawing a distance-time graph: Drawing a diagonal sloping line from (0,0) to (2.5, 1250). A distance-time graph scores a maximum of 1 mark out of 3 (only if 500 is shown in working).
  • Decimal time trap: Calculating 1250 ÷ 2.3 = 543.5... by incorrectly treating 30 minutes as 0.3 hours. This loses the first mark.
  • Missing axis labels: Omitting the words 'Speed' and 'Time' on the axes loses the second mark even if scales are correct.
  • Extending the line too far: Continuing the horizontal line past 2.5 hours (150 minutes). The plane only flew for 2 hours 30 minutes!

Topics

Number · Algebra · Ratio, proportion and rates of change · 3.1.3 Measures and accuracy · 3.2.2 Graphs · 3.3 Ratio, proportion and rates of change

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 3 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.