In a mixture of 126 litres the ratio of water and milk is 2: 5 re spectively. How much water (in litres) must be added to make the ratio of water and milk as 2:3 respectively?

Options

A. 16
B. 24
C. 18
D. 20

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the current quantities of water and milk in the mixture.
Given the ratio of water to milk is 2:5, we can express the total mixture of 126 liters in terms of parts.

- Total parts = 2 (water) + 5 (milk) = 7 parts

Now, we can find the value of 1 part:
\[ \text{Value of 1 part} = \frac{126 \text{ liters}}{7} = 18 \text{ liters} \]

Now we can find the quantities of water and milk:
- Water = 2 parts = \(2 \times 18 = 36 \text{ liters}\)
- Milk = 5 parts = \(5 \times 18 = 90 \text{ liters}\)

### Step 2: Set up the new ratio after adding water.
We want to change the ratio of water to milk to 2:3. Let \(x\) be the amount of water we need to add.

After adding \(x\) liters of water, the new quantity of water will be:
\[ \text{New quantity of water} = 36 + x \text{ liters} \]
The quantity of milk remains the same at 90 liters.

### Step 3: Set up the equation based on the new ratio.
We want the new ratio of water to milk to be 2:3. Therefore, we can set up the following equation:
\[ \frac{36 + x}{90} = \frac{2}{3} \]

### Step 4: Cross-multiply to solve for \(x\).
Cross-multiplying gives us:
\[ 3(36 + x) = 2 \times 90 \]
\[ 108 + 3x = 180 \]

### Step 5: Solve for \(x\).
Now, we will isolate \(x\):
\[ 3x = 180 - 108 \]
\[ 3x = 72 \]
\[ x = \frac{72}{3} = 24 \]

### Conclusion:
The amount of water that must be added to achieve the desired ratio is \(24\) liters.

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