Drum X is 1/2 full of oil and drum Y, which has twice the capacity of drum X, is 2/3 full of oil. If all of the oil in drum X is poured into drum Y, then drum Y will be filled to what fraction of its capacity?
Let's denote the capacity of drum X as "CX" and the capacity of drum Y as "CY." We're given that drum Y has twice the capacity of drum X, so CY = 2 * CX.
We're also given that drum X is 1/2 full of oil, which means it contains (1/2) of its capacity. Therefore, the amount of oil in drum X is (1/2) * CX.
Drum Y is 2/3 full of oil, which means it contains (2/3) of its capacity. Therefore, the amount of oil in drum Y is (2/3) * CY.
Now, we're pouring all of the oil from drum X into drum Y. So, the amount of oil in drum Y after pouring will be the sum of its original oil content and the oil from drum X:
(2/3) * CY (original oil content in drum Y) + (1/2) * CX (oil from drum X)
Substitute CY = 2 * CX into the equation:
(2/3) * 2 * CX + (1/2) * CX
Now, simplify the equation:
(4/3) * CX + (1/2) * CX
To add these fractions, find a common denominator:
(8/6) * CX + (3/6) * CX
Combine the fractions:
(8/6 + 3/6) * CX
(11/6) * CX
Now, we know that drum Y is filled to (11/6) of its capacity. But we want to express it as a fraction of its capacity, which is CY. Recall that CY = 2 * CX:
So, drum Y is filled to:
(11/6) * CX / (2 * CX)
Simplify by canceling CX:
(11/6) / (2)
Now, simplify further:
(11/6) * (1/2)
Multiply the fractions:
(11/12)
So, drum Y will be filled to 11/12 of its capacity.
The correct answer is 11/12.
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