A market and a stadium are 240 m apart. They are located between Paul's and Rani's houses as shown below. The market is exactly half-way in between the two houses. One morning, Paul and Rani started cycling from their houses at the same time and they arrived at the stadium together. Rani cycled at 100 m/min while Paul cycled at a speed 15 m/min faster than Rani.

1. Express the distances cycled by Paul and Rani in terms of the distance from their houses to the market.

Let the distance from the market to either house be $x$ m (since the market is exactly half-way between the two houses).
Paul's distance to the stadium = $x + 240$ m
Rani's distance to the stadium = $x - 240$ m

2. Calculate the difference in the distances cycled.

Difference = $(x + 240) - (x - 240)$
$= x + 240 - x + 240$
$= 480$ m

1. Find the difference in distance cycled by Paul and Rani (from part a)

Since the market is exactly halfway between their houses, Paul's distance to the market equals Rani's distance to the market. The stadium is $240$ m from the market towards Rani's house.
Difference in distance $= 240 + 240 = 480$ m

2. Calculate the time taken to arrive at the stadium

Difference in speed $= 15$ m/min
Time taken $= \frac{480}{15} = 32$ min

3. Calculate the distance from Rani's house to the stadium

Distance Rani cycled $= 100 \times 32 = 3200$ m