Question

There are three cans and a bucket. The cans each have a capacity of 5 litres, but are partially filled with water. The bucket also has some water in it. The sum of the water in the bucket and the water in the first can is half of the total bucket capacity. When the first and third cans are emptied into the bucket, it contains 6 litres of water. Instead, when the second and the third cans are emptied into the bucket, it contains 7 litres of water. When the water in all the cans is poured into the bucket, it is filled to its capacity. The first and second cans contain a total of 7 litres. How many litres did the bucket already contain?

• A. 1 litre
• B. 2 litres
• C. 3 litres
• D. 4 litres

Hint:

Write one equation for each condition in terms of the bucket's initial water and the three cans, then solve the system.

Answer 1

The Correct Option is A

Step 1: Set up variables.
Let \(b\) be the water already in the bucket, \(c_1, c_2, c_3\) the water in the first, second and third cans, and \(C\) the bucket's full capacity.

Step 2: Write each condition as an equation.
\(b+c_1 = \dfrac{C}{2}\) from the first clue.
\(b+c_1+c_3 = 6\) from emptying cans 1 and 3.
\(b+c_2+c_3 = 7\) from emptying cans 2 and 3.
\(b+c_1+c_2+c_3 = C\) from emptying all three cans.
\(c_1+c_2 = 7\) from the last clue.

Step 3: Reduce using the first equation.
Since \(b+c_1 = \dfrac{C}{2}\), the second equation gives \(\dfrac{C}{2}+c_3=6\), so \(c_3 = 6-\dfrac{C}{2}\).
The fourth equation gives \(\dfrac{C}{2}+c_2+c_3=C\), so \(c_2+c_3=\dfrac{C}{2}\).
Substituting into the third equation, \(b+\dfrac{C}{2}=7\), so \(b=7-\dfrac{C}{2}\).

Step 4: Solve for the capacity \(C\).
From \(b+c_1=\dfrac{C}{2}\), \(c_1 = C-7\). Then from \(c_1+c_2=7\), \(c_2=14-C\).
Substituting \(c_2\) and \(c_3=6-\dfrac{C}{2}\) into \(c_2+c_3=\dfrac{C}{2}\) gives \(14-C+6-\dfrac{C}{2}=\dfrac{C}{2}\), so \(20=2C\) and \(C=10\).

Step 5: Back-substitute to find \(b\).
\(b = 7-\dfrac{C}{2} = 7-5 = 2\). The other values are \(c_1=3\), \(c_2=4\) and \(c_3=1\), and all satisfy the five original conditions.

Step 6: Note on the answer key.
This system of equations gives \(b=2\) litres uniquely, matching option (b), not option (a).
The official answer key marks option (a), 1 litre, so this row is flagged for review and the keyed option is kept unchanged as instructed.

Final Answer:
As per the official answer key the marked option is (a). \[ \boxed{1 \text{ litre (keyed option a)}} \]