The capacity of a tank is 2400 cubic metres. A pump can be used both for filling and for emptying the tank. Its emptying capacity is 10 m^3 per minute higher than its filling capacity, and it takes 8 minutes less to empty the tank than to fill it. What is the filling capacity of the pump in cubic metres per minute?

Introduction / Context:
Here a single pump can either fill or empty a tank, with different capacities in each mode. The emptying capacity is known to be larger by a fixed amount, and the time difference for filling and emptying is given. We use the basic relationship time = volume / rate to set up an equation in one variable and then solve for the filling capacity.

Given Data / Assumptions:
- Tank capacity = 2400 m^3
- Let filling capacity of the pump = f m^3 per minute
- Then emptying capacity of the pump = f + 10 m^3 per minute
- Time to fill the tank = 2400 / f minutes
- Time to empty the tank = 2400 / (f + 10) minutes
- Emptying takes 8 minutes less than filling

Concept / Approach:
We use the equation based on the given time difference: time to empty = time to fill minus 8 minutes. Substituting time expressions in terms of f gives a rational equation. Solving this equation gives a positive value of f that matches one of the given options.

Step-by-Step Solution:
Step 1: Time to fill = 2400 / f minutes. Step 2: Time to empty = 2400 / (f + 10) minutes. Step 3: Given that emptying is 8 minutes faster: 2400 / (f + 10) = 2400 / f - 8. Step 4: Rearrange: 2400 / (f + 10) - 2400 / f = -8. Step 5: Combine fractions: 2400[f - (f + 10)] / [f(f + 10)] = -8 which simplifies to 2400 * (-10) / [f(f + 10)] = -8. Step 6: This gives 24000 / [f(f + 10)] = 8, so f(f + 10) = 24000 / 8 = 3000. Step 7: Solve f^2 + 10f - 3000 = 0. The positive root is f = 50 m^3 per minute.

Verification / Alternative check:
If f = 50, then emptying capacity is 60 m^3 per minute. Time to fill = 2400 / 50 = 48 minutes. Time to empty = 2400 / 60 = 40 minutes. The difference in times is indeed 48 - 40 = 8 minutes, so f = 50 m^3 per minute is consistent with all the conditions.

Why Other Options Are Wrong:
46, 44 or 48 cubic metres per minute: These values do not satisfy the quadratic equation f^2 + 10f - 3000 = 0 and fail to produce a time difference of exactly 8 minutes between filling and emptying when substituted into the time formulas.

Common Pitfalls:
Students may incorrectly write the time relation as filling time minus emptying time equals 8, reversing the sign. Another common error is algebraic manipulation when combining fractions or solving the quadratic equation. Careful stepwise substitution and simplification prevents such mistakes.

Final Answer:
The filling capacity of the pump is 50 cubic metres per minute.