Question

A vessel of 160 L is filled with milk and water.70% of milk and 30% of water is taken out of the vessel.It is found out that the vessel is vacated by 55%,then the quantity of milk and water in the original mixture is

• A. Milk 100 L; Water = 60 L
• B. Milk 60 L; Water = 100 L
• C. Milk 90 L: Water = 70 L
• D. Milk 70 L; Water = 90 L

Answer 1

The Correct Option is A

To solve this problem, let's begin by understanding the process step by step. We are given that a vessel with a total volume of 160 liters is filled with milk and water.

Let's assume the quantity of milk in the original mixture is \(M\) liters, and the quantity of water is \(W\) liters. According to the problem, the total volume is given as:

\(M + W = 160\) 

Next, it is stated that 70% of the milk and 30% of the water is taken out of the vessel.

The volume of milk removed is \(0.7M\) liters, and the volume of water removed is \(0.3W\) liters.

The total volume of liquid removed from the vessel is \(0.7M + 0.3W\).

According to the problem, the vessel is vacated by 55%. Hence, 55% of the original volume (160 liters) has been removed:

\((M \times 0.7 + W \times 0.3) = 0.55 \times 160\)

Simplifying, we get:

\(0.7M + 0.3W = 88\)

We now have a system of linear equations:

1. \(M + W = 160\)
2. \(0.7M + 0.3W = 88\)

Let's solve these equations to find the values of \(M\) and \(W\).

From the first equation, express \(W\) in terms of \(M\):

\(W = 160 - M\)

Substitute this into the second equation:

\(0.7M + 0.3(160 - M) = 88\)

Simplifying, we get:

\(0.7M + 48 - 0.3M = 88\)

\(0.4M = 40\)

\(M = 100\)

Now, substitute \(M = 100\) back into the equation for \(W\):

\(W = 160 - 100 = 60\)

Thus, the quantities of milk and water in the original mixture are 100 liters and 60 liters respectively.

Therefore, the correct answer is:

Milk 100 L; Water = 60 L

Answer 2

Let the quantity of milk and water in the vessel initially be \( M \) and \( W \), respectively. We know:

\[ M + W = 160 \]

Out of the total milk, 70% is retained, and after removing 55% of the mixture:

\[ 0.3M + 0.7W = 0.45 \times 160 = 72 \]

Now, we have the system of equations:

\[ M + W = 160 \quad \text{(1)} \]

\[ 0.3M + 0.7W = 72 \quad \text{(2)} \]

Multiply equation (1) by 0.3:

\[ 0.3M + 0.3W = 48 \quad \text{(3)} \]

Subtract equation (3) from equation (2):

\[ (0.3M + 0.7W) - (0.3M + 0.3W) = 72 - 48 \]

\[ 0.4W = 24 \quad \Rightarrow \quad W = 60 \]

Substitute \( W = 60 \) into equation (1):

\[ M + 60 = 160 \quad \Rightarrow \quad M = 100 \]

Thus, the quantity of milk is 100 L and the quantity of water is 60 L.