Solving Sugar Mixing Ratio Problems

This problem asks us to find the specific ratio in which two types of sugar, with different costs per kilogram, must be combined to create a mixture with a desired intermediate cost per kilogram. These types of problems can be efficiently solved using the Rule of Alligation, which is a visual or mathematical method to determine the ratio of two ingredients needed to achieve a specific price for the mixture.

Understanding the Rule of Alligation for Mixture Problems

The Rule of Alligation helps us find the ratio of quantities of two ingredients when their individual prices and the price of the final mixture are known. It's based on the principle that the quantity of the cheaper ingredient used is proportional to the difference between the price of the dearer ingredient and the mean price of the mixture, and the quantity of the dearer ingredient used is proportional to the difference between the mean price of the mixture and the price of the cheaper ingredient.

Let's apply this rule to our sugar mixing problem:

• Price of Cheaper Sugar (C): Rs. 7 per kg
• Price of Dearer Sugar (D): Rs. 12 per kg
• Mean Price of Mixture (M): Rs. 8 per kg

Applying Alligation to Find the Mixing Ratio

We can set up the alligation cross:

The differences obtained are the quantities of the ingredients needed, but in reverse order. Specifically:

• The quantity of the Cheaper Sugar (at Rs. 7/kg) is proportional to the difference (Dearer Price - Mean Price) = \(12 - 8 = 4\).
• The quantity of the Dearer Sugar (at Rs. 12/kg) is proportional to the difference (Mean Price - Cheaper Price) = \(8 - 7 = 1\).

So, the ratio of the quantity of Cheaper Sugar to the quantity of Dearer Sugar is \(4 : 1\).

The question asks for the proportion in which the grocer must mix sugar at Rs. 12 a kg (Dearer) and Rs. 7 a kg (Cheaper). This means the ratio of Dearer Sugar : Cheaper Sugar.

Based on our calculation, the ratio is:

Quantity of Dearer Sugar : Quantity of Cheaper Sugar = \(1 : 4\)

Step-by-Step Solution Summary

1. Identify the prices of the two types of sugar and the desired price of the mixture.
2. Use the Rule of Alligation: subtract the mean price from the dearer price to find the quantity proportional to the cheaper ingredient, and subtract the cheaper price from the mean price to find the quantity proportional to the dearer ingredient.
3. The differences give the inverse ratio of quantities.
4. Ratio of Cheaper Sugar (Rs. 7/kg) : Dearer Sugar (Rs. 12/kg) is \((12 - 8) : (8 - 7) = 4 : 1\).
5. The question asks for the ratio of sugar at Rs. 12/kg : sugar at Rs. 7/kg, which is Dearer Sugar : Cheaper Sugar.
6. Therefore, the required mixing ratio is \(1 : 4\).

Thus, the grocer must mix the sugar at Rs. 12 a kg and Rs. 7 a kg in the proportion \(1 : 4\) to obtain a mixture worth Rs. 8 a kg.

Revision Table: Key Concepts in Mixture Problems

Additional Information: Weighted Average in Mixtures

Mixture problems can also be solved using the concept of weighted average. If you mix quantity \(Q_1\) of an ingredient with price \(P_1\) and quantity \(Q_2\) of an ingredient with price \(P_2\), the mean price of the mixture (\(M\)) is given by the formula:

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

In our problem, let \(Q_{12}\) be the quantity of sugar at Rs. 12/kg (\(P_1 = 12\)) and \(Q_7\) be the quantity of sugar at Rs. 7/kg (\(P_2 = 7\)). The mean price is \(M = 8\).

Substituting the values into the formula:

\[8 = \frac{(Q_{12} \times 12) + (Q_7 \times 7)}{Q_{12} + Q_7}\]

Now, we solve for the ratio \(Q_{12} : Q_7\):

\[8 \times (Q_{12} + Q_7) = (Q_{12} \times 12) + (Q_7 \times 7)\]

\[8 Q_{12} + 8 Q_7 = 12 Q_{12} + 7 Q_7\]

Rearrange the terms to group \(Q_{12}\) and \(Q_7\):

\[8 Q_7 - 7 Q_7 = 12 Q_{12} - 8 Q_{12}\]

\[1 Q_7 = 4 Q_{12}\]

To find the ratio \(Q_{12} : Q_7\), we can write this as:

\[\frac{Q_{12}}{Q_7} = \frac{1}{4}\]

So, the ratio \(Q_{12} : Q_7\) is \(1 : 4\). This confirms the result obtained using the Rule of Alligation.

Both methods, Alligation and Weighted Average, are useful for solving mixture problems, but Alligation is often faster for finding ratios when only two ingredients are involved.