This question asks about mixing two types of tea with different costs to achieve a specific target cost for the resulting mixture. We need to find the ratio in which the two types of tea should be mixed.

Understanding the Mixture Problem

We have two ingredients (tea types) with known costs per unit weight and a desired cost per unit weight for the final mixture. We need to determine the proportion or ratio of the weights of the two ingredients to be mixed.

Applying the Rule of Alligation

This type of problem can be efficiently solved using the Rule of Alligation. This rule helps in finding the ratio in which two ingredients at given prices should be mixed to produce a mixture of a desired price.

Let's define the terms:

• Cost of the cheaper ingredient (C)
• Cost of the dearer ingredient (D)
• Mean price of the mixture (M)

According to the Rule of Alligation, the ratio of the quantity of the cheaper ingredient to the quantity of the dearer ingredient is given by the difference between the dearer price and the mean price, and the difference between the mean price and the cheaper price.

Ratio of Quantity of Cheaper : Quantity of Dearer = (D − M) ∶ (M − C)

Solving the Tea Mixture Problem

In this problem:

• Cost of the cheaper tea (C) = Rs. 10 per kg
• Cost of the dearer tea (D) = Rs. 15 per kg
• Desired cost of the mixture (M) = Rs. 12 per kg

We can visualize this using the alligation diagram:

Now, calculate the differences diagonally:

• Difference between Dearer Price and Mean Price = D − M = 15 − 12 = 3
• Difference between Mean Price and Cheaper Price = M − C = 12 − 10 = 2

The ratio of the quantities of the cheaper tea to the dearer tea is the ratio of these differences, but in the opposite direction on the diagram.

Quantity of Cheaper Tea ∝ (D − M)

Quantity of Dearer Tea ∝ (M − C)

So, the ratio of Cheaper Tea : Dearer Tea = (D − M) ∶ (M − C) = (15 − 12) ∶ (12 − 10) = 3 ∶ 2.

The quantities are mixed in the ratio 3 ∶ 2.

• Quantity of tea at Rs. 10 per kg is proportional to 3.
• Quantity of tea at Rs. 15 per kg is proportional to 2.

Thus, the two varieties of tea must be mixed in the ratio 3 ∶ 2.

Revision Table: Key Concepts

Additional Information: Weighted Average Approach

The rule of alligation is a specific application of the concept of weighted average. Let's say we mix quantity \(Q_1\) of tea with price \(P_1\) and quantity \(Q_2\) of tea with price \(P_2\). The total cost of the mixture will be \(Q_1 P_1 + Q_2 P_2\). The total quantity will be \(Q_1 + Q_2\). The mean price of the mixture (M) is the total cost divided by the total quantity:

\(M = \frac{Q_1 P_1 + Q_2 P_2}{Q_1 + Q_2}\)

If we assume \(P_1\) is the cheaper price (Rs. 10) and \(P_2\) is the dearer price (Rs. 15), and the mean price is Rs. 12, then:

\(12 = \frac{Q_1 \times 10 + Q_2 \times 15}{Q_1 + Q_2}\)

\(12(Q_1 + Q_2) = 10 Q_1 + 15 Q_2\)

\(12 Q_1 + 12 Q_2 = 10 Q_1 + 15 Q_2\)

\(12 Q_1 - 10 Q_1 = 15 Q_2 - 12 Q_2\)

\(2 Q_1 = 3 Q_2\)

To find the ratio \(Q_1 : Q_2\), we rearrange the equation:

\(\frac{Q_1}{Q_2} = \frac{3}{2}\)

So, \(Q_1 : Q_2 = 3 : 2\). This confirms the result obtained using the Rule of Alligation. The ratio is Quantity of tea at Rs. 10 : Quantity of tea at Rs. 15, which is 3 : 2.