In what ratio tea at Rs. 240 per kg. should be mixed with tea at Rs. 280 per kg. so that on selling the mixture at Rs. 324 per kg. there is a profit of 20%?

Options

A. `1:1`
B. `1:2`
C. `1:3`
D. `1:4`

Answer

To solve the problem, we need to find the ratio in which tea priced at Rs. 240 per kg should be mixed with tea priced at Rs. 280 per kg, so that the mixture can be sold at Rs. 324 per kg with a profit of 20%.

### Step-by-Step Solution:

1. **Determine the Selling Price and Profit Percentage:**
- Selling Price (SP) = Rs. 324 per kg
- Profit Percentage = 20%

2. **Calculate the Cost Price (CP) of the Mixture:**
- The formula to calculate the Cost Price when the Selling Price and Profit Percentage are known is:
\[ CP = \frac{100}{100 + \text{Profit Percentage}} \times SP \]
- Substituting the values:
\[ CP = \frac{100}{100 + 20} \times 324 = \frac{100}{120} \times 324 \]
- Simplifying:
\[ CP = \frac{5}{6} \times 324 = 270 \]
- Therefore, the Cost Price of the mixture is Rs. 270 per kg.

3. **Set Up the Equation for the Mixture:**
- Let the quantity of tea at Rs. 240 per kg be \( x \) kg and the quantity of tea at Rs. 280 per kg be \( y \) kg.
- The total cost of the mixture can be expressed as:
\[ \text{Total Cost} = 240x + 280y \]
- The average cost per kg of the mixture should equal the calculated Cost Price:
\[ \frac{240x + 280y}{x + y} = 270 \]

4. **Cross Multiply to Eliminate the Fraction:**
- Multiplying both sides by \( (x + y) \):
\[ 240x + 280y = 270(x + y) \]
- Expanding the right side:
\[ 240x + 280y = 270x + 270y \]

5. **Rearranging the Equation:**
- Bringing all terms involving \( x \) and \( y \) to one side:
\[ 240x + 280y - 270x - 270y = 0 \]
- Simplifying:
\[ -30x + 10y = 0 \]
- Rearranging gives:
\[ 30x = 10y \quad \Rightarrow \quad 3x = y \]

6. **Finding the Ratio:**
- The ratio of \( x \) to \( y \) can be expressed as:
\[ \frac{x}{y} = \frac{x}{3x} = \frac{1}{3} \]
- Therefore, the required ratio of tea at Rs. 240 per kg to tea at Rs. 280 per kg is:
\[ \text{Ratio} = 1 : 3 \]

### Final Answer:
The ratio in which tea at Rs. 240 per kg should be mixed with tea at Rs. 280 per kg is **1 : 3**.