The ratio in which the Darjeeling tea at 32 per kg is mixed with the Assam tea at 25 per kg so as to gain 20% by selling the mixture at 32.40 per kg is

Options

A. a)`4 :3`
B. b)`3 :4`
C. c)`5 :2`
D. d)`2 :5`

Answer

To solve the problem of finding the ratio in which Darjeeling tea at ₹32 per kg is mixed with Assam tea at ₹25 per kg to achieve a 20% profit by selling the mixture at ₹32.40 per kg, we can follow these steps:

### Step 1: Calculate the Cost Price (CP)
Given that the selling price (SP) is ₹32.40 and the profit is 20%, we can find the cost price using the formula:

\[ \text{SP} = \text{CP} + \text{Profit} \]

Since the profit is 20% of the cost price, we can express this as:

\[ \text{SP} = \text{CP} + 0.20 \times \text{CP} = 1.20 \times \text{CP} \]

Now, substituting the selling price into the equation:

\[ 32.40 = 1.20 \times \text{CP} \]

To find CP, we rearrange the equation:

\[ \text{CP} = \frac{32.40}{1.20} = 27 \text{ per kg} \]

### Step 2: Set Up the Allegation Method
Now we will use the allegation method to find the ratio of the two types of tea. We have:

- Cost price of Darjeeling tea (C1) = ₹32 per kg
- Cost price of Assam tea (C2) = ₹25 per kg
- Average cost price of the mixture (Cavg) = ₹27 per kg

### Step 3: Apply the Allegation Formula
Using the allegation method, we can find the ratio as follows:

1. Calculate the difference between Cavg and C1:
\[ Cavg - C1 = 27 - 32 = -5 \]

2. Calculate the difference between C2 and Cavg:
\[ C2 - Cavg = 25 - 27 = -2 \]

3. The ratio of the two teas is given by the absolute values of these differences:
\[ \text{Ratio} = \frac{Cavg - C1}{C2 - Cavg} = \frac{5}{2} \]

### Step 4: Final Ratio
Thus, the ratio in which Darjeeling tea is mixed with Assam tea is:

\[ \text{Ratio} = 5:2 \]

### Conclusion
The final answer is that the ratio in which Darjeeling tea at ₹32 per kg is mixed with Assam tea at ₹25 per kg to gain a 20% profit by selling the mixture at ₹32.40 per kg is **5:2**.

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