Step-by-step worked examples and graded practice questions on real-life graphs — conversion graphs, cost and charge graphs, and container-filling graphs.
📚 Foundation & Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
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Real-life graphs model everyday situations — converting between units, working out the cost of a service, or tracking the depth of liquid in a container as it fills. The key skill is the same throughout: read values accurately from the graph, and connect the gradient and intercept to what they represent in context.
Three types come up regularly at GCSE:
Conversion graphs — straight lines through the origin linking two units
Cost/charge graphs — straight lines with a fixed charge (y-intercept) plus a rate (gradient)
Container-filling graphs — the shape of the container determines the shape of the graph
Conversion graphs
A conversion graph is a straight line through the origin, since the two quantities are directly proportional. To convert a value, find it on one axis, draw a line across to the graph, then read down (or across) to the other axis.
Worked Example 1 — Reading a conversion graph
A conversion graph shows that 5 miles is equivalent to 8 km, giving a line through the origin with gradient 1.6. Use the graph to convert 15 miles to km.
1
Find 15 on the miles axis and draw a line up to the graph, then across to the km axis
2
Check by calculation: km = 1.6 × miles = 1.6 × 15 = 24 km
Answer24 km
Worked Example 2 — Reading in the opposite direction
Using the same graph, convert 20 km back to miles.
1
Find 20 on the km axis and draw a line across to the graph, then down to the miles axis
2
Check by calculation: miles = km ÷ 1.6 = 20 ÷ 1.6 = 12.5 miles
Answer12.5 miles
Reading the conversion graph both ways: 15 miles → 24 km, and 20 km → 12.5 miles.
Cost and charge graphs
Many real-life cost graphs follow the pattern y = mx + c: a fixed charge (the y-intercept, c) plus a rate per unit (the gradient, m). Recognising this structure lets you read the fixed cost and the rate directly from the graph.
Worked Example 3 — A cost graph
A mobile phone plan costs a fixed £10 per month, plus £0.20 for each text sent beyond the free allowance. Find the total cost for a month with 50 extra texts.
1
The fixed charge is the y-intercept: £10 (this is the cost even with 0 extra texts)
2
Total cost = fixed charge + (rate × number of texts) = 10 + 0.2 × 50
3
= 10 + 10 = £20
Answer£20
The mobile phone cost graph: fixed charge £10 (y-intercept), rising at £0.20 per extra text.
Container-filling graphs
When a container is filled with liquid at a constant rate, the shape of the depth-time graph depends entirely on the shape of the container — specifically, how its cross-sectional width changes with height.
A cylinder has the same width all the way up, so the depth rises at a constant rate — a straight line.
A container that's narrow at the bottom and wide at the top fills quickly at first (small cross-section), then more slowly as it widens — the graph is steep, then levels off.
A container that's wide at the bottom and narrow at the top fills slowly at first (large cross-section), then more quickly as it narrows — the graph starts shallow and gets steeper.
How container shape determines graph shape, when filled at a constant rate.
Worked Example 4 — Matching a container to its graph
Water is poured into a cone-shaped flask at a constant rate. The flask has a narrow point at the bottom and widens towards the top. Describe the shape of the depth-time graph.
1
Near the bottom, the flask is narrow, so a small amount of water raises the depth a lot — the graph starts steep
2
As the flask widens near the top, the same rate of pouring raises the depth less — the graph levels off
AnswerA curve that is steep at first and gradually levels off (decreasing gradient)
Practice questions
Work through each question before checking the answers. Difficulty is shown for each question.
Foundation (Grade 3–5)
Q1A conversion graph shows 5 miles = 8 km (a straight line through the origin). Use the graph to convert 10 miles to km.Foundation
Q2Using the same graph, convert 4 km to miles.Foundation
Q3A taxi charges a flat £3 plus £2 per mile. Find the cost of a 4-mile journey.Foundation
Q4A water tank fills at a constant rate, shown as a straight line on a depth-time graph. What does the straight line tell you about the rate of filling?Foundation
Q5A container is a cylinder. Describe the shape of its depth-time graph as it is filled at a constant rate.Foundation
Higher (Grade 5–7)
Q6A currency conversion graph shows £1 = $1.25 (a line through the origin with gradient 1.25). Find the dollar value of £60.Higher
Q7Using the same graph, find the pound value of $150.Higher
Q8A gym membership costs a joining fee of £25 plus £15 per month. Write an equation for the total cost y after x months, and find the cost after 6 months.Higher
Q9A cone-shaped container is filled at a constant rate, with the point of the cone at the bottom (narrow bottom, wide top). Describe how the depth changes over time.Higher
Q10A different container is a cone with the point at the top (wide bottom, narrow top). Describe how the depth changes over time.Higher
Higher — Hard (Grade 8–9)
Question 11 refers to the graph below, showing the taxi charge from Q3 (y = 2x + 3).
Graph for Question 11 — the taxi charge graph y = 2x + 3.
Q11Using the graph above, a taxi journey costs £17. Find the length of the journey in miles.Grade 8–9
Q12Using the gym membership equation from Q8 (y = 15x + 25), after how many whole months does the total cost first exceed £160?Grade 8–9
Q13A cylindrical container has radius 3 cm and is filled at a rate of 54π cm³ per minute. Find the rate at which the depth increases, in cm per minute.Grade 8–9
Q14A conical container (point at the bottom) reaches a depth of 4 cm after 2 minutes, and 7 cm after 6 minutes. Find the average rate of depth increase for each interval, and explain why the rate has decreased.Grade 8–9
Q15A container's depth-time graph is flat at 2 cm for the first 10 minutes, then rises steadily to 12 cm at 20 minutes. Suggest a reason for the flat section, and find the rate of depth increase during the second stage.Grade 8–9
Answers
Foundation (Q1–Q5)
Q116 km(10 × 1.6)
Q22.5 miles(4 ÷ 1.6)
Q3£11(2 × 4 + 3)
Q4The tank is filling at a constant rate — equal depth increases in equal time intervals
Q5A straight line, since the cylinder's width (and so its cross-sectional area) stays the same throughout
Q14First interval: 2 cm/min. Second interval: 0.75 cm/min. The rate decreases because the cone widens as it fills, so the same volume raises the depth less(4÷2 = 2; (7−4)÷(6−2) = 3÷4)
Q15The flat section suggests pouring was paused, or the container has a wide horizontal shelf at that depth. Rate during the second stage = 1 cm/min((12−2) ÷ (20−10) = 10÷10)
Common mistakes
These are the errors Alamin sees most frequently with real-life graphs at GCSE. Recognising them now will save you marks in the exam.
Common Mistake 1
Assuming every real-life graph passes through the origin
Conversion graphs pass through the origin (0 miles = 0 km), but cost graphs usually don't — a fixed charge means the line starts above zero on the y-axis.
Common Mistake 2
Reading the graph in the wrong direction
Always check which axis your starting value is on before drawing your reading lines. Converting "20 km to miles" starts on the km axis, not the miles axis — mixing these up gives the reciprocal of the right answer.
Common Mistake 3
Forgetting the fixed charge in a cost graph
A common error is calculating only rate × quantity and forgetting to add the fixed charge (the y-intercept). Always check whether the graph — or the question — mentions a starting cost.
Common Mistake 4
Mixing up which container shape gives which curve
A container that's narrow at the bottom fills quickly at first, so its graph starts steep. A container that's wide at the bottom fills slowly at first, so its graph starts shallow. These are opposite effects, and are very easy to swap under exam pressure.
Common Mistake 5
Drawing straight lines between points on a container-filling graph
Unless the container is a cylinder, the depth-time graph is a curve, not a series of straight segments — the rate of change is continuously changing as the container's width changes.
Common Mistake 6
Not linking the gradient back to the real-life context
Examiners often ask you to interpret the gradient or intercept, not just calculate it. Always state what the number means in context (e.g. "£0.20 per text", not just "gradient = 0.2").
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