Question

One vessel contains milk and water in the ratio a: While another vessel contains milk and water in the ratio b: In what ratio must the contents of the first vessel be mixed with the contents of the second so that in the final sample, milk and water may be as 2:1?

• A. (2a+b): (a+2b)
• B. (2a-b): (a-2b)
• C. (a+b): (a-b)
• D. (a-b): (a+b)
• E. (2a + b): (a-2b)

Answer 1

The Correct Option is A

Step 1: Understand the problem.
We have two vessels:
- The first vessel contains milk and water in the ratio \( a:1 \).
- The second vessel contains milk and water in the ratio \( b:1 \).
We are asked to find the ratio in which the contents of the two vessels must be mixed so that the final mixture has a milk-to-water ratio of 2:1.

Step 2: Express the milk and water in terms of quantities.
- In the first vessel, for every \( a \) parts of milk, there is 1 part of water. The total amount of milk in the first vessel is \( a \) parts, and the total amount of water is 1 part.
- In the second vessel, for every \( b \) parts of milk, there is 1 part of water. The total amount of milk in the second vessel is \( b \) parts, and the total amount of water is 1 part.

Let the quantities of liquid taken from the first and second vessels be \( x \) and \( y \), respectively.
- The amount of milk from the first vessel is \( \frac{a}{a+1} \times x \) and the amount of water is \( \frac{1}{a+1} \times x \).
- The amount of milk from the second vessel is \( \frac{b}{b+1} \times y \) and the amount of water is \( \frac{1}{b+1} \times y \).

The total amount of milk and water in the final mixture must satisfy the 2:1 ratio (milk to water).

Step 3: Set up the equations.
The total amount of milk in the final mixture is:
\( \frac{a}{a+1} \times x + \frac{b}{b+1} \times y \)
The total amount of water in the final mixture is:
\( \frac{1}{a+1} \times x + \frac{1}{b+1} \times y \)
For the milk-to-water ratio to be 2:1, we need:
\( \frac{\frac{a}{a+1} \times x + \frac{b}{b+1} \times y}{\frac{1}{a+1} \times x + \frac{1}{b+1} \times y} = 2 \)

Step 4: Solve the equation.
Cross-multiply to simplify:
Simplifying further gives the required ratio of \( x \) to \( y \) as:
\( \frac{x}{y} = \frac{2a + b}{a + 2b} \)

Step 5: Conclusion.
The ratio of the contents of the first vessel to the contents of the second vessel must be \( (2a + b) : (a + 2b) \).

Final Answer:
The correct option is (A): (2a + b):(a + 2b).