In 40 litres mixture of milk and water the ratio of milk to water is 7 : 1. In order to make the ratio of milk and water 3:1, the quantity of water (in litres) that should be added to the mixture will be

Options

A. 6
B. `6 "" (1)/(2)`
C. ` 6"" (2)/(3)`
D. `6"" (3)/(4)`

Answer

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

### Step 1: Understand the initial mixture
We have a total of 40 liters of a mixture of milk and water, where the ratio of milk to water is 7:1.

### Step 2: Calculate the quantities of milk and water
To find the quantity of milk and water in the mixture, we can use the ratio:
- The total parts in the ratio = 7 (milk) + 1 (water) = 8 parts.
- Each part = Total volume / Total parts = 40 liters / 8 = 5 liters.

Now, we can calculate the quantities:
- Quantity of milk = 7 parts = 7 * 5 = 35 liters.
- Quantity of water = 1 part = 1 * 5 = 5 liters.

### Step 3: Set up the equation for the new ratio
We want to change the ratio of milk to water to 3:1 by adding some quantity of water (let's denote this quantity as \( x \) liters).

After adding \( x \) liters of water, the new quantity of water will be:
- New quantity of water = Original water + Added water = 5 + \( x \) liters.

### Step 4: Write the equation based on the new ratio
We want the ratio of milk to water to be 3:1. Therefore, we can set up the equation:
\[ \frac{\text{Quantity of Milk}}{\text{Quantity of Water}} = \frac{3}{1} \]
Substituting the known quantities:
\[ \frac{35}{5 + x} = \frac{3}{1} \]

### Step 5: Cross-multiply to solve for \( x \)
Cross-multiplying gives us:
\[ 35 \cdot 1 = 3 \cdot (5 + x) \]
This simplifies to:
\[ 35 = 15 + 3x \]

### Step 6: Solve for \( x \)
Now, we can isolate \( x \):
\[ 35 - 15 = 3x \]
\[ 20 = 3x \]
\[ x = \frac{20}{3} \approx 6.67 \text{ liters} \]

### Final Answer
The quantity of water that should be added to the mixture to achieve the desired ratio of 3:1 is \( \frac{20}{3} \) liters or approximately 6.67 liters.

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