Mathematics using equations

Question

A family have some water in a tank. They filled a large barrel and a medium barrel with 195 liters of water from the tank. They wanted to fill another large barrel but they would have been short by 15 liters. They then filled another medium barrel with water and remained with 30 liters of water in the tank. Calculate how many liters of water the tank held initially.

Answer 1

Let's denote the T as the tank's capacity, L the large barrel's capacity and M the small barrel's capacity. In this case we have: L + M = 195. Now to fill a second large barrel, we'd need another 15 liters above the tank's capacity, or: 2L + M = T + 15. On the other hand, filling a second medium barrel would leave 30 liters extra in the tank, or: L + 2M = T -30. Putting all this together with a simple rearrangement gives us:

1) L + M = 195

2) 2L + M - 15 = T

3) L + 2M +30 = T

Now if we subtract the third equation from the second equation above, we obtain:

(2L + M -15) - (L + 2M + 30) = T - T , or equivalently:

4) L - M = 45

Subsequently, from equations 1) and 4) above we have:

L + M = 195

and

L - M = 45.

Now adding these two equations simply yields: 2L=240 or L=120. By replacing this value of L in either equations we find: M = 75. Finally, by using the found values in either equations 2) or 3) we obtain the capacity of the tank to be T = 300 Liters.

Answer 2