In jar A, 180 litres milk was mix with 36 litres water. Some of the mixture was taken out from Jar A and put in jar B. If after adding 6 litres of water in the mixture, the ratio of milk to water in Jar B was 5:2, then what was the amount of mixture that was taken out from Jar A (in litres)?

Options

A. 24
B. 54
C. 30
D. 36

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the initial mixture in Jar A
In Jar A, we have:
- Milk = 180 liters
- Water = 36 liters

Total mixture in Jar A = 180 liters (milk) + 36 liters (water) = 216 liters.

### Step 2: Define the amount of mixture taken out
Let the amount of mixture taken out from Jar A be \( x \) liters.

### Step 3: Calculate the proportions of milk and water in the mixture taken out
The proportion of milk in the mixture is:
\[ \text{Proportion of milk} = \frac{180}{216} = \frac{5}{6} \]
The proportion of water in the mixture is:
\[ \text{Proportion of water} = \frac{36}{216} = \frac{1}{6} \]

### Step 4: Calculate the amount of milk and water in the mixture taken out
From the \( x \) liters taken out:
- Amount of milk taken out = \( \frac{5}{6}x \)
- Amount of water taken out = \( \frac{1}{6}x \)

### Step 5: Add water to the mixture in Jar B
After transferring the mixture to Jar B, 6 liters of water is added. Therefore, the total amount of water in Jar B becomes:
\[ \text{Total water in Jar B} = \frac{1}{6}x + 6 \]

### Step 6: Set up the ratio of milk to water in Jar B
According to the problem, the ratio of milk to water in Jar B is \( 5:2 \). Therefore, we can set up the equation:
\[ \frac{\frac{5}{6}x}{\frac{1}{6}x + 6} = \frac{5}{2} \]

### Step 7: Cross-multiply to solve for \( x \)
Cross-multiplying gives:
\[ 5 \left( \frac{1}{6}x + 6 \right) = 2 \left( \frac{5}{6}x \right) \]
Expanding both sides:
\[ \frac{5}{6}x + 30 = \frac{10}{6}x \]

### Step 8: Simplify the equation
Rearranging gives:
\[ 30 = \frac{10}{6}x - \frac{5}{6}x \]
\[ 30 = \frac{5}{6}x \]

### Step 9: Solve for \( x \)
Multiplying both sides by \( \frac{6}{5} \):
\[ x = 30 \times \frac{6}{5} = 36 \]

### Conclusion
The amount of mixture taken out from Jar A is \( \boxed{36} \) liters.

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