How much quantity (in kg.) of wheat costing Rs. 84 per kg. must be mixed with 81 kg. of wheat costing Rs. 60 per kg, so that on selling the mixture at Rs.75.9 per kg. there is a gain of 15%?

Options

A. 27
B. 20.5
C. 22.75
D. 24

Answer

To solve the problem step by step, we will determine how much quantity of wheat costing Rs. 84 per kg must be mixed with 81 kg of wheat costing Rs. 60 per kg, so that on selling the mixture at Rs. 75.9 per kg there is a gain of 15%.

### Step 1: Determine the Cost Price (CP) of the Mixture
We know the Selling Price (SP) and the gain percentage. We can find the Cost Price (CP) using the formula:

\[ \text{CP} = \frac{\text{SP} \times 100}{100 + \text{Gain \%}} \]

Given:
- SP = Rs. 75.9 per kg
- Gain % = 15%

Substituting the values:

\[ \text{CP} = \frac{75.9 \times 100}{100 + 15} = \frac{75.9 \times 100}{115} = \frac{7590}{115} = 66 \text{ Rs. per kg} \]

### Step 2: Set Up the Alligation
Now we will use the method of alligation to find the ratio in which the two types of wheat should be mixed.

- Cost of wheat A (Rs. 84 per kg)
- Cost of wheat B (Rs. 60 per kg)
- Cost of mixture (Rs. 66 per kg)

We will find the differences:

\[ \text{Difference between A and mixture} = 84 - 66 = 18 \]
\[ \text{Difference between B and mixture} = 66 - 60 = 6 \]

### Step 3: Calculate the Ratio
Using the differences, we can find the ratio of the quantities of wheat A and B:

\[ \text{Ratio} = \frac{6}{18} = \frac{1}{3} \]

This means for every 1 part of wheat costing Rs. 84, we need 3 parts of wheat costing Rs. 60.

### Step 4: Determine the Quantity of Wheat A
Let the quantity of wheat A (costing Rs. 84 per kg) be \( x \) kg. The quantity of wheat B is given as 81 kg.

Using the ratio:

\[ \frac{x}{81} = \frac{1}{3} \]

Cross-multiplying gives:

\[ 3x = 81 \implies x = \frac{81}{3} = 27 \text{ kg} \]

### Conclusion
The quantity of wheat costing Rs. 84 per kg that must be mixed is **27 kg**.

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