How many kilograms of tea worth Rs 25 per kg must be blended with 30kg of tea worth Rs 30 per kg, so that by selling the blended variety at Rs 30 per kg, there should be a gain of 10%?

Options

A. 36 kg
B. 40 kg
C. 32 kg
D. 42 kg

Answer

To solve the problem step by step, we need to determine how many kilograms of tea worth Rs 25 per kg must be blended with 30 kg of tea worth Rs 30 per kg, such that when sold at Rs 30 per kg, there is a gain of 10%.

### Step 1: Determine the Selling Price (SP) and Cost Price (CP) of the Mixture
To achieve a gain of 10%, we first need to find out the selling price of the mixture.

The selling price (SP) of the mixture is given as Rs 30 per kg.

To find the cost price (CP) of the mixture that would yield a 10% profit when sold at Rs 30, we can use the formula:
\[ SP = CP + \text{Gain} \]
Where Gain = 10% of CP.

Thus, we can express this as:
\[ SP = CP + 0.1 \times CP = 1.1 \times CP \]
Setting SP to 30, we have:
\[ 30 = 1.1 \times CP \]
Now, solving for CP:
\[ CP = \frac{30}{1.1} = \frac{300}{11} \text{ Rs per kg} \]

### Step 2: Set Up the Allegation Method
Now we will use the allegation method to find out how many kg of tea worth Rs 25 per kg must be blended with 30 kg of tea worth Rs 30 per kg.

Let \( X \) be the kg of tea worth Rs 25 per kg that we need to blend.

- Cost Price of Type 1 tea (X kg): Rs 25 per kg
- Cost Price of Type 2 tea (30 kg): Rs 30 per kg
- Cost Price of the mixture: Rs \( \frac{300}{11} \) per kg

### Step 3: Apply the Allegation Formula
Using the allegation method, we set up the following:

\[ \text{Type 1 (Rs 25)} \quad \text{Type 2 (Rs 30)} \quad \text{Mixture (Rs } \frac{300}{11}\text{)} \]
The differences are calculated as follows:

- Difference between Type 1 and Mixture:
\[ \frac{300}{11} - 25 = \frac{300}{11} - \frac{275}{11} = \frac{25}{11} \]

- Difference between Type 2 and Mixture:
\[ 30 - \frac{300}{11} = \frac{330}{11} - \frac{300}{11} = \frac{30}{11} \]

### Step 4: Determine the Ratio of Quantities
The ratio of the quantities of Type 1 tea (X kg) to Type 2 tea (30 kg) is given by the differences calculated above:

\[ \text{Ratio} = \frac{\frac{25}{11}}{\frac{30}{11}} = \frac{25}{30} = \frac{5}{6} \]

This means for every 5 parts of Type 1 tea, there are 6 parts of Type 2 tea.

### Step 5: Calculate the Value of X
Let \( X \) be the amount of Type 1 tea. Then, we can set up the equation based on the ratio:
\[ \frac{X}{30} = \frac{5}{6} \]

Cross-multiplying gives:
\[ 6X = 5 \times 30 \]
\[ 6X = 150 \]
\[ X = \frac{150}{6} = 25 \text{ kg} \]

### Conclusion
Thus, the amount of tea worth Rs 25 per kg that must be blended with 30 kg of tea worth Rs 30 per kg is **25 kg**.

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