80 litres of a mixture contains milk and water in the ratio of 27: 5. How much more water is to be added to get a mixture con taining milk and water in the ratio of 3:1?

Options

A. 5litres
B. 10litres
C. 15litres
D. 20litres

Answer

To solve the problem step by step, we will first determine the quantities of milk and water in the initial mixture and then find out how much more water needs to be added to achieve the desired ratio.

### Step 1: Determine the initial quantities of milk and water

We know that the total volume of the mixture is 80 liters, and the ratio of milk to water is 27:5.

To find the total parts in the mixture:
- Total parts = 27 (milk) + 5 (water) = 32 parts

Now, we can find the quantity of milk and water in the mixture:
- Quantity of milk = (27/32) * 80 liters
- Quantity of water = (5/32) * 80 liters

Calculating these:
- Quantity of milk = (27/32) * 80 = 67.5 liters
- Quantity of water = (5/32) * 80 = 12.5 liters

### Step 2: Set up the desired ratio

We want the new ratio of milk to water to be 3:1. This means for every 3 parts of milk, there should be 1 part of water.

Let’s denote the amount of water to be added as \( x \) liters. After adding \( x \) liters of water, the new quantities will be:
- Milk = 67.5 liters (remains the same)
- Water = 12.5 + \( x \) liters

### Step 3: Set up the equation based on the desired ratio

According to the desired ratio of milk to water (3:1), we can set up the equation:
\[ \frac{\text{Milk}}{\text{Water}} = \frac{3}{1} \]
Substituting the known quantities:
\[ \frac{67.5}{12.5 + x} = \frac{3}{1} \]

### Step 4: Cross-multiply to solve for \( x \)

Cross-multiplying gives us:
\[ 67.5 = 3(12.5 + x) \]
Expanding the right side:
\[ 67.5 = 37.5 + 3x \]

### Step 5: Isolate \( x \)

Now, we can isolate \( x \):
\[ 67.5 - 37.5 = 3x \]
\[ 30 = 3x \]
\[ x = \frac{30}{3} = 10 \]

### Conclusion

Thus, the amount of water that needs to be added to achieve the desired ratio of milk to water (3:1) is **10 liters**.