Real-life graphs Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Real-life graphs questions. See exactly how to solve problems on distance-time graph, gradient as speed, interpreting flat sections, fractions of an hour.

distance-time graphgradient as speedinterpreting flat sectionsfractions of an hourconversion graphreading a graph
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
On a distance–time graph for Ravi’s car journey, a straight line joins (0,0)(0, 0) to (2,100)(2, 100), where time is measured in hours and distance from home in km\mathrm{km}. Work out the speed of the car.

Worked solution

  1. Sketch the graph from the description

    (0, 0)→(2, 100)(0,\ 0) \rightarrow (2,\ 100)

    The graph is one straight line from the origin to 100 km100\,\mathrm{km} after 22 hours, so the speed is constant.

  2. Use gradient = speed

    speed=change in distancechange in time=100−02−0\text{speed} = \frac{\text{change in distance}}{\text{change in time}} = \frac{100 - 0}{2 - 0}

    On a distance–time graph the gradient tells you how much distance is covered per unit of time — that is the speed.

  3. State the answer with units

    1002=50 km/h\frac{100}{2} = 50\,\mathrm{km/h}

    Distance is in km\mathrm{km} and time is in hours, so the speed comes out in km/h\mathrm{km/h}.

Answer
50 km/h50\,\mathrm{km/h}
Question 2
1 markeasy
A distance–time graph for Nia’s bike ride is a straight line from (0,0)(0, 0) to (3,36)(3, 36), where time is in hours and distance is in km\mathrm{km}. Work out her speed.

Worked solution

  1. Sketch the graph from the description

    (0, 0)→(3, 36)(0,\ 0) \rightarrow (3,\ 36)

    A single straight line means Nia rides at one steady speed for the whole 33 hours.

  2. Gradient of a distance–time graph is the speed

    speed=36−03−0\text{speed} = \frac{36 - 0}{3 - 0}

    Divide the distance travelled by the time taken.

  3. State the answer with units

    363=12 km/h\frac{36}{3} = 12\,\mathrm{km/h}

    Kilometres divided by hours gives km/h\mathrm{km/h}.

Answer
12 km/h12\,\mathrm{km/h}
Question 3
1 markeasy
A distance–time graph for a lorry is horizontal from (1,40)(1, 40) to (2,40)(2, 40), where time is in hours and distance from the depot is in km\mathrm{km}. Which statement describes the lorry between 11 hour and 22 hours?

Worked solution

  1. Sketch the flat section

    (1, 40)→(2, 40)(1,\ 40) \rightarrow (2,\ 40)

    The distance stays at 40 km40\,\mathrm{km} while the time goes from 11 hour to 22 hours.

  2. Find the gradient of that section

    gradient=40−402−1=01=0\text{gradient} = \frac{40 - 40}{2 - 1} = \frac{0}{1} = 0

    A zero gradient means zero speed.

  3. Interpret the zero gradient

    speed=0 km/h\text{speed} = 0\,\mathrm{km/h}

    The lorry is not moving — it stays parked 40 km40\,\mathrm{km} from the depot for that hour.

Answer
The lorry is stationary, 40 km from the depot\text{The lorry is stationary, 40 km from the depot}
Question 4
2 markseasy
Part of a distance–time graph is a straight line from (0,0)(0, 0) to (0.5,30)(0.5, 30), where time is in hours and distance is in km\mathrm{km}. Work out the speed in km/h\mathrm{km/h}.

Worked solution

  1. Sketch the section

    (0, 0)→(0.5, 30)(0,\ 0) \rightarrow (0.5,\ 30)

    30 km30\,\mathrm{km} are covered in half an hour.

  2. Divide distance by time

    speed=30−00.5−0=300.5\text{speed} = \frac{30 - 0}{0.5 - 0} = \frac{30}{0.5}

    Dividing by 0.50.5 is the same as multiplying by 22.

  3. State the answer with units

    300.5=60 km/h\frac{30}{0.5} = 60\,\mathrm{km/h}

    In a whole hour the car would cover twice 30 km30\,\mathrm{km}, so 60 km/h60\,\mathrm{km/h}.

Answer
60 km/h60\,\mathrm{km/h}
Question 5
1 markeasy
A conversion graph for miles and kilometres is a straight line through (0,0)(0, 0) and (10,16)(10, 16), where xx is the number of miles and yy is the number of kilometres. Use the graph to convert 55 miles into kilometres.

Worked solution

  1. Sketch the conversion line

    (0, 0)→(10, 16)(0,\ 0) \rightarrow (10,\ 16)

    The line passes through the origin, so miles and kilometres are in direct proportion.

  2. Go up from x=5x = 5 to the line

    5 miles is half of 10 miles\text{5 miles is half of 10 miles}

    Half of 1010 miles must be half of 16 km16\,\mathrm{km}.

  3. Read off the kilometres

    12×16=8 km\dfrac{1}{2} \times 16 = 8\,\mathrm{km}

    The point (5,8)(5, 8) lies on the line, so 55 miles is 8 km8\,\mathrm{km}.

Answer
8 km8\,\mathrm{km}

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