Water is leaking out of a barrel. In the first hour, it lost half the number of litres it held, plus one litre. In the second hour, it lost half the remainder plus another litre. In the third hour, it lost half the remainder again, plus a further litre; and during the fourth hour, half the remainder leaked out, plus one litre more. At the end of the fourth hour, the barrel was completely empty. How many litres did it contain to start with?
work backwards:
2(2(2(2(0+1)+1)+1)+1) = 30 liters
To solve this problem, we will work backward from the end of the fourth hour.
Let's denote the number of liters in the barrel at the end of the fourth hour as X.
During the fourth hour, half the remainder leaked out, plus one liter more. This means that at the end of the third hour, there must have been 2X + 2 liters in the barrel.
During the third hour, the barrel lost half the remainder again, plus a further liter. This means that at the end of the second hour, there must have been 2(2X + 2) + 2 liters in the barrel.
During the second hour, the barrel lost half the remainder plus another liter. This means that at the end of the first hour, there must have been 2(2(2X + 2) + 2) + 2 liters in the barrel.
Finally, during the first hour, the barrel lost half the number of liters it held, plus one liter. This gives us the equation:
2(2(2X + 2) + 2) + 2 = X
Now, let's solve for X.
Expanding the equation, we get:
8X + 20 + 2 = X
Combining like terms, we have:
8X + 22 = X
Subtracting X from both sides, we get:
7X + 22 = 0
Subtracting 22 from both sides, we have:
7X = -22
Dividing both sides by 7, we find:
X = -22/7
Based on this calculation, we find that X is a negative fraction, which doesn't make sense in the context of the problem. Therefore, it appears that there is no solution to this problem.
It's possible that there might be a mistake in the problem statement or that there is some missing information.