Ajar contains a mixture of milk and water in the respective ratio of 5:1.18 litres of mixture is taken out from the jar and 6 litres of pure water is added in the jar. If the resultant ratio between milk and water in the jar is 3 : 1, what was the initial quantity of mixture in the jar before the replacements ? (inlitres)

Options

A. 72
B. 84
C. 60
D. 68

Answer

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

### Step 1: Understand the initial ratio of milk and water
The initial ratio of milk to water in the jar is given as 5:1. This means for every 5 parts of milk, there is 1 part of water.

### Step 2: Calculate the total parts in the mixture
The total parts in the mixture = 5 (milk) + 1 (water) = 6 parts.

### Step 3: Let the initial quantity of the mixture be 'x' liters
Since the total parts are 6, we can express the quantities of milk and water in terms of 'x':
- Quantity of milk = (5/6)x
- Quantity of water = (1/6)x

### Step 4: Calculate the quantities after taking out 18 liters
When 18 liters of the mixture is taken out, the quantity of milk and water taken out will also maintain the same ratio of 5:1.
- Quantity of milk taken out = (5/6) * 18 = 15 liters
- Quantity of water taken out = (1/6) * 18 = 3 liters

### Step 5: Calculate the remaining quantities of milk and water
After removing 18 liters:
- Remaining quantity of milk = (5/6)x - 15
- Remaining quantity of water = (1/6)x - 3

### Step 6: Add 6 liters of pure water
After adding 6 liters of pure water, the new quantity of water becomes:
- New quantity of water = (1/6)x - 3 + 6 = (1/6)x + 3

### Step 7: Set up the equation for the new ratio
The problem states that the new ratio of milk to water is 3:1. Therefore, we can set up the equation:
\[ \frac{\text{Remaining milk}}{\text{New water}} = \frac{3}{1} \]
Substituting the values we have:
\[ \frac{(5/6)x - 15}{(1/6)x + 3} = 3 \]

### Step 8: Cross-multiply to solve the equation
Cross-multiplying gives:
\[ (5/6)x - 15 = 3 \left((1/6)x + 3\right) \]
Expanding the right side:
\[ (5/6)x - 15 = (1/2)x + 9 \]

### Step 9: Rearrange the equation
Rearranging the equation to isolate 'x':
\[ (5/6)x - (1/2)x = 15 + 9 \]
Finding a common denominator (which is 3):
\[ \frac{5}{6}x - \frac{3}{6}x = 24 \]
\[ \frac{2}{6}x = 24 \]
\[ \frac{1}{3}x = 24 \]

### Step 10: Solve for 'x'
Multiplying both sides by 3:
\[ x = 72 \]

### Conclusion
The initial quantity of the mixture in the jar before the replacements was **72 liters**.

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