Calculating Rice Quantity Sold at Different Profits

This problem involves a shopkeeper selling a total quantity of rice in two parts, each at a different profit percentage, resulting in an overall profit percentage. We need to find the specific quantity sold at one of the given profit rates.

This type of problem can be efficiently solved using the concept of Alligation or weighted averages. The Alligation method helps us find the ratio in which two quantities with different characteristics (here, profit percentages) are mixed to get a mixture with an average characteristic.

Understanding the Alligation Method for Profit

In this scenario, the two characteristics are the profit percentages on the two parts of the rice, and the average characteristic is the overall profit percentage on the total quantity of rice. The method works as follows:

1. Write down the two individual profit percentages.
2. Write down the overall average profit percentage in the middle.
3. Find the difference between the overall profit and each individual profit. These differences give the ratio of the quantities sold at the respective profit percentages.

Applying Alligation to the Shopkeeper's Rice Sale

• Profit percentage on Part 1: 20%
• Profit percentage on Part 2: 12%
• Overall average profit percentage: 18%

Let's set up the alligation cross:

The ratio of the quantity sold at the lower profit (12%) to the quantity sold at the higher profit (20%) is the inverse of the differences calculated. That is:

\(\text{Quantity at 12% Profit} : \text{Quantity at 20% Profit} = (20 - 18) : (18 - 12)\)

\(\text{Ratio} = 2 : 6\)

This ratio can be simplified by dividing both parts by their greatest common divisor, which is 2:

\(\text{Ratio} = \frac{2}{2} : \frac{6}{2} = 1 : 3\)

So, for every 1 part of rice sold at 12% profit, 3 parts were sold at 20% profit.

Calculating the Quantities Sold

The total ratio parts are \(1 + 3 = 4\). This means the total quantity of rice (2220 kg) is divided into 4 equal parts according to this ratio.

The quantity sold at 12% profit corresponds to 1 part of the ratio.

The quantity sold at 20% profit corresponds to 3 parts of the ratio.

Total quantity of rice = 2220 kg

Quantity sold at 12% profit \( = \frac{\text{Ratio part for 12%}}{\text{Total ratio parts}} \times \text{Total quantity} \)

Quantity sold at 12% profit \( = \frac{1}{4} \times 2220 \text{ kg} \)

Let's perform the calculation:

So, the quantity of rice sold at a 12% profit is 555 kg.

For completeness, the quantity sold at 20% profit would be:

Quantity sold at 20% profit \( = \frac{3}{4} \times 2220 \text{ kg} = 3 \times 555 \text{ kg} = 1665 \text{ kg} \)

Check: \(555 \text{ kg} + 1665 \text{ kg} = 2220 \text{ kg}\), which is the total quantity.

The question asks for the quantity (in kg) sold at a 12% profit, which we calculated to be 555 kg.

Final Answer Summary

By using the alligation method, we found the ratio of quantities sold at 12% and 20% profit to be 1:3. Applying this ratio to the total quantity of 2220 kg, we determined the quantity sold at a 12% profit.

Quantity sold at 12% profit = 555 kg.

Revision Table: Key Concepts Revisited

Additional Information: Why Alligation Works

The alligation method is derived from the weighted average concept. Let \(Q_1\) be the quantity sold at profit \(P_1\) and \(Q_2\) be the quantity sold at profit \(P_2\). The total profit is \(Q_1 \times P_1 + Q_2 \times P_2\). The total quantity is \(Q_1 + Q_2\). The overall average profit \(P_{avg}\) is given by:

Rearranging this equation gives:

Assuming \(P_1 > P_{avg} > P_2\), this becomes:

This shows the inverse relationship between the quantities and the differences in profit percentages from the average. The ratio \(Q_1 : Q_2\) is indeed \((P_{avg} - P_2) : (P_1 - P_{avg})\), which is exactly what the alligation diagram represents.

In our problem, \(P_1 = 20\%\), \(P_2 = 12\%\), and \(P_{avg} = 18\%\). Note that we used \(P_1 = 20\%\) and \(P_2 = 12\%\) labels slightly differently in the alligation table for clarity with "Higher" and "Lower", but the calculation of differences remains consistent. The difference \(P_{avg} - P_2 = 18 - 12 = 6\) corresponds to the quantity \(Q_1\) (at 20%), and the difference \(P_1 - P_{avg} = 20 - 18 = 2\) corresponds to the quantity \(Q_2\) (at 12%). So the ratio \(Q_1 : Q_2\) is \(6 : 2\), or \(3 : 1\). This means Quantity at 20% Profit : Quantity at 12% Profit = 3 : 1. This matches our earlier finding using the inverse ratio \(1:3\) for \(Q_{12} : Q_{20}\). The method is consistent and provides a quick way to solve such mixture problems involving profit percentages.