Six hundred runners took part in a race.
The fastest runner took 8 minutes to complete the race while the slowest took 18 minutes.
The points (8, 1), (15, 300), (17, 450) and (18, 600) are points on the cumulative frequency curve for the runners as shown on the grids above. It is given that the interquartile time for the race is 4 minutes.

1. Find the cumulative frequency corresponding to the lower quartile.

There are $600$ runners.

Lower quartile cumulative frequency:
$\frac{1}{4}\times 600=150$

2. Use the given interquartile time to find the lower quartile time.

The upper quartile has cumulative frequency
$\frac{3}{4}\times 600=450.$

The point $(17,450)$ is given on the cumulative frequency curve, so
$Q_3=17.$

Given interquartile time is $4$ minutes,
$Q_3-Q_1=4$
$17-Q_1=4$
$Q_1=13.$

Hence the lower quartile point is
$(13,150).$

1. Identify all points that the cumulative frequency curve must pass through.

From part (i), the lower quartile point is $(13,150)$.

The given points are $(8,1)$, $(15,300)$, $(17,450)$ and $(18,600)$.

So plot the points
$(8,1),\ (13,150),\ (15,300),\ (17,450),\ (18,600).$

2. Draw the cumulative frequency curve through the plotted points.

Join the plotted points with a smooth, non-decreasing curve.

The curve should start near the fastest time at $8$ minutes and rise to cumulative frequency $600$ at $18$ minutes.