A jar contains a mixture of mango juice and water in the ratio of `22:3`.Now `50` litres of the mixture was taken out and `25` litres of water was added to it. If water was 34% in the resultant mixture, what was the initial quantity of the mixture in the jar? (in litres)

Options

A. 175
B. 125
C. 150
D. 75

Answer

To solve the problem step-by-step, we will follow the given information and perform the necessary calculations.

### Step 1: Define the initial quantities
Given the ratio of mango juice to water is \(22:3\), we can express the quantities in terms of a variable \(x\):
- Mango juice = \(22x\)
- Water = \(3x\)

Thus, the total initial quantity of the mixture is:
\[ \text{Total mixture} = 22x + 3x = 25x \]

### Step 2: Calculate the quantities taken out
When \(50\) liters of the mixture is taken out, we need to determine how much mango juice and water is in that \(50\) liters. The proportions of mango juice and water in the mixture are:
- Proportion of mango juice = \(\frac{22}{25}\)
- Proportion of water = \(\frac{3}{25}\)

Calculating the quantities taken out:
- Mango juice taken out = \(50 \times \frac{22}{25} = 44\) liters
- Water taken out = \(50 \times \frac{3}{25} = 6\) liters

### Step 3: Calculate the remaining quantities
After taking out \(50\) liters:
- Remaining mango juice = \(22x - 44\)
- Remaining water = \(3x - 6\)

### Step 4: Add the water
Next, \(25\) liters of water is added to the remaining mixture:
- New quantity of water = \((3x - 6) + 25 = 3x + 19\)

### Step 5: Calculate the total mixture after the changes
The total quantity of the mixture after taking out and adding water is:
\[ \text{Total mixture} = (22x - 44) + (3x + 19) = 25x - 25 \]

### Step 6: Set up the equation for the percentage of water
We know that in the resultant mixture, water constitutes \(34\%\). Therefore, we can set up the equation:
\[ \frac{3x + 19}{25x - 25} = \frac{34}{100} \]

### Step 7: Cross-multiply to solve for \(x\)
Cross-multiplying gives:
\[ 100(3x + 19) = 34(25x - 25) \]
Expanding both sides:
\[ 300x + 1900 = 850x - 850 \]

### Step 8: Rearranging the equation
Rearranging gives:
\[ 300x - 850x = -850 - 1900 \]
\[ -550x = -2750 \]
Dividing by \(-550\):
\[ x = 5 \]

### Step 9: Calculate the initial quantity of the mixture
Now that we have \(x\), we can find the initial quantity of the mixture:
\[ \text{Initial quantity} = 25x = 25 \times 5 = 125 \text{ liters} \]

### Final Answer
The initial quantity of the mixture in the jar is \(125\) liters.

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