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 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 water in all the cans are poured into the bucket, it is filled to its capacity. The first and second can contain a total of 7 litres. What is the capacity of the bucket?

• A. 6 litres
• B. 7 litres
• C. 8 litres
• D. 10 litres
• E. 12 litres

Answer 1

The Correct Option is D

The correct answer is option (D):
10 litres

Let's break the problem down step by step and produce a clean algebraic solution.

Unknowns

• B = capacity of the bucket (litres)
• c1, c2, c3 = amounts of water in can 1, can 2 and can 3 (litres)

Given relations (translated to equations)

1. When the first and third cans are emptied into the bucket it contains 6 litres:
   B + c1 + c3 = 6
2. When the second and third cans are emptied into the bucket it contains 7 litres:
   B + c2 + c3 = 7
3. When all cans are emptied into the bucket it is exactly full:
   B + c1 + c2 + c3 = B ⇒  c1 + c2 + c3 = 0
4. The first and second cans together contain 7 litres:
   c1 + c2 = 7

Solve the system

From c1 + c2 + c3 = 0 and c1 + c2 = 7 we get
7 + c3 = 0 ⇒ c3 = −7.

Substitute c3 = −7 into the two bucket equations:
B + c1 + c3 = 6 ⇒ B + c1 − 7 = 6 ⇒ B + c1 = 13.
B + c2 + c3 = 7 ⇒ B + c2 − 7 = 7 ⇒ B + c2 = 14.

Use c1 + c2 = 7. Express c2 = 7 − c1 and substitute into B + c2 = 14:
B + (7 − c1) = 14 ⇒ B − c1 = 7.

Now we have two linear equations in B and c1:
B + c1 = 13
B − c1 = 7
Add them: 2B = 20 ⇒ B = 10 litres.

Answer: The bucket's capacity is 10 litres.