(15) The time $t$ taken to buy fuel at a petrol station varies directly as the number of vehicles $V$ on queue and jointly varies inversely as the number of pumps $P$ available in the station. In a station, with 5 pumps, it took 10 minutes to fuel 20 vehicles. Find: (a) the relationship between $t, P$ and $V$; (b) the time it will take to fuel 50 vehicles in the station with 2 pumps; (c) the number of pumps required to fuel 40 vehicles in 20 minutes. (WAEC/SSCE)

Step 1 ：Given that the time $t$ taken to buy fuel at a petrol station varies directly as the number of vehicles $V$ on queue and jointly varies inversely as the number of pumps $P$ available in the station. This can be written as: \(t = k \frac{V}{P}\)

Step 2 ：We can find the value of $k$ by substituting the given values into the equation. We know that when $P = 5$ pumps, $V = 20$ vehicles, and $t = 10$ minutes. Substituting these values gives: \(10 = k \frac{20}{5}\)

Step 3 ：Solving for $k$ gives $k = 2.5$. So the relationship between $t, P$ and $V$ is: \(t = 2.5 \frac{V}{P}\)

Step 4 ：To find the time it will take to fuel 50 vehicles in the station with 2 pumps, we substitute $V = 50$ vehicles and $P = 2$ pumps into the equation: \(t = 2.5 \frac{50}{2}\)

Step 5 ：\(\boxed{t = 62.5}\) minutes

Step 6 ：To find the number of pumps required to fuel 40 vehicles in 20 minutes, we substitute $V = 40$ vehicles and $t = 20$ minutes into the equation: \(20 = 2.5 \frac{40}{P}\)

Step 7 ：Solving for $P$ gives \(\boxed{P = 5}\) pumps