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 Ittres) that should be added to the mixture will be

Options

A. 6
B. 6 `/2`
C. 6 `2/3`
D. 6 `3/4`

Answer

To solve the problem step by step, we will determine how much water needs to be added to a 40-liter mixture of milk and water to change the ratio of milk to water from 7:1 to 3:1.

### Step 1: Determine the initial quantities of milk and water in the mixture.
The ratio of milk to water is given as 7:1. This means that for every 8 parts of the mixture, 7 parts are milk and 1 part is water.

Total parts = 7 (milk) + 1 (water) = 8 parts.

Now, we can find the quantity of milk and water in the 40 liters mixture.

- Quantity of milk = (7/8) * 40 = 35 liters
- Quantity of water = (1/8) * 40 = 5 liters

### Step 2: Set up the equation for the new ratio.
We want to change the ratio of milk to water to 3:1. Let \( x \) be the amount of water we need to add.

After adding \( x \) liters of water, the new quantity of water will be \( 5 + x \) liters. The quantity of milk remains the same at 35 liters.

The new ratio of milk to water can be expressed as:
\[ \frac{35}{5 + x} = \frac{3}{1} \]

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

### Step 4: Solve for \( x \).
Now, isolate \( x \):
\[ 35 - 15 = 3x \]
\[ 20 = 3x \]
\[ x = \frac{20}{3} \text{ liters} \]

### Step 5: Conclusion.
The quantity of water that should be added to the mixture is \( \frac{20}{3} \) liters, which is approximately 6.67 liters.

### Summary of the Solution:
- Initial quantities: 35 liters of milk and 5 liters of water.
- Required water to achieve a 3:1 ratio: \( \frac{20}{3} \) liters or approximately 6.67 liters.