a dealer mixes tea costing ₹6.92 per kg. with tea costing ₹7.77 per kg and sells the mixture at ₹8.80 per kg and earns a profit of 17.5% on his sale price. in what proportion does he 2 mix them?

To determine the proportion in which the dealer mixes the two types of tea, we need to use the concept of weighted averages and profit calculation.

### Step 1: Calculate the cost price of the mixture
The dealer sells the mixture at ₹8.80 per kg and earns a profit of 17.5% on the sale price.

Let's denote the cost price of the mixture as \( C \).

The relationship between the selling price (SP), cost price (CP), and profit percentage is given by:

\[
\text{Profit Percentage} = \frac{\text{SP} - \text{CP}}{\text{CP}} \times 100
\]

Given:
- Selling Price (SP) = ₹8.80 per kg
- Profit Percentage = 17.5%

Using the profit formula:

\[
17.5 = \frac{8.80 - C}{C} \times 100
\]

Solving for \( C \):

\[
0.175 = \frac{8.80 - C}{C}
\]

\[
0.175C = 8.80 - C
\]

\[
C + 0.175C = 8.80
\]

\[
1.175C = 8.80
\]

\[
C = \frac{8.80}{1.175} \approx 7.4894 \, \text{per kg}
\]

So, the cost price of the mixture is approximately ₹7.4894 per kg.

### Step 2: Use the rule of alligation to find the ratio
Now, we use the rule of alligation to find the ratio in which the two types of tea should be mixed.

Let:
- Cost of the first type of tea = ₹6.92 per kg
- Cost of the second type of tea = ₹7.77 per kg
- Mean price (Cost price of the mixture) = ₹7.4894 per kg

The formula for alligation is:

\[
\text{Ratio} = \frac{\text{Cost of second type} - \text{Mean Price}}{\text{Mean Price} - \text{Cost of first type}}
\]

Substituting the values:

\[
\text{Ratio} = \frac{7.77 - 7.4894}{7.4894 - 6.92}
\]

\[
\text{Ratio} = \frac{0.2806}{0.5694} \approx \frac{1}{2.0295} \approx 1 : 2.03
\]

### Conclusion
The dealer should mix the two types of tea in approximately a **1:2** ratio (first type to second type).

---

Would you like further details or explanations? Here are some related questions to consider:

1. How would the ratio change if the cost of one type of tea increased?
2. What if the selling price was ₹9.00 per kg instead?
3. How does the profit percentage affect the mixing ratio?
4. What if the dealer wanted a different profit percentage, say 20%?
5. How could the dealer calculate the maximum profit margin given the same costs?

**Tip:** Understanding the rule of alligation can be particularly useful when dealing with problems involving mixtures and averages.

Learn how to calculate the proportion in which a dealer mixes two types of tea, costing ₹6.92 per kg and ₹7.77 per kg, to sell at ₹8.80 per kg with a profit of 17.5%. Understand the use of weighted averages and the rule of alligation in solving mixture problems. Suitable for advanced high school students and those preparing for competitive exams.

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