Solving the Rice Trader Profit and Loss Problem

This problem involves a trader selling a total quantity of rice in two parts, one at a profit and the other at a loss, resulting in an overall profit percentage. We need to determine the specific quantity sold at a loss.

Here's the information given:

• Total quantity of rice bought = 640 kg
• First part sold at 20% profit
• Second part sold at 5% loss
• Overall profit on the entire transaction = 15%

We want to find the quantity of rice sold at a 5% loss.

Using the Alligation Method

The Alligation rule is a useful technique for solving problems involving the mixing of two ingredients or entities with different characteristics (like profit/loss percentages) to produce a mixture with an average characteristic.

In this case, we are mixing two quantities of rice sold at different profit/loss percentages to get an average profit percentage. The rule states that the ratio of the quantities mixed is inversely proportional to the difference between their individual percentages and the average percentage.

Let \(P_1\) be the profit percentage on the first part (20%).

Let \(L_2\) be the loss percentage on the second part (5%). Note that loss is treated as negative profit, so \(L_2 = -5\%\).

Let \(P_{avg}\) be the average profit percentage on the entire transaction (15%).

According to the Alligation rule, the ratio of the quantity sold at profit \(Q_p\) to the quantity sold at loss \(Q_l\) is given by the ratio of the differences:

The ratio of the quantity sold at profit (\(Q_p\)) to the quantity sold at loss (\(Q_l\)) is:

Ratio \(Q_p : Q_l\) = (Difference corresponding to Loss) : (Difference corresponding to Profit)

Ratio \(Q_p : Q_l\) = \( |P_1 - P_{avg}| : |P_{avg} - L_2| \) - This is the inverse relationship for quantities.

Ratio \(Q_p : Q_l\) = \( |20 - 15| : |15 - (-5)| \)

Ratio \(Q_p : Q_l\) = \( |5| : |20| \)

Ratio \(Q_p : Q_l\) = \( 5 : 20 \)

Simplifying the ratio, we get \( 1 : 4 \). However, the alligation diagram is usually set up as:

The ratio of Quantity 1 (sold at 20% profit, \(Q_p\)) to Quantity 2 (sold at 5% loss, \(Q_l\)) is the *inverse* ratio of the differences shown above the quantities.

So, the ratio of quantities is \(Q_p : Q_l\) = (Difference opposite 20%) : (Difference opposite -5%)

Ratio \(Q_p : Q_l\) = \(|15 - (-5)| : |20 - 15|\)

Ratio \(Q_p : Q_l\) = \(|20| : |5|\)

Ratio \(Q_p : Q_l\) = \(20 : 5\)

Simplifying the ratio, \(Q_p : Q_l = 4 : 1\).

The total ratio parts are \(4 + 1 = 5\).

The total quantity of rice is 640 kg.

The quantity sold at 5% loss (\(Q_l\)) corresponds to 1 part out of the total 5 parts.

Therefore, the quantity sold at 5% loss is:

\( Q_l = \frac{\text{Ratio part for loss}}{\text{Total ratio parts}} \times \text{Total Quantity} \)

\( Q_l = \frac{1}{5} \times 640 \text{ kg} \)

\( Q_l = 128 \text{ kg} \)

Verification

Quantity sold at 5% loss = 128 kg

Quantity sold at 20% profit = Total quantity - Quantity at loss = 640 kg - 128 kg = 512 kg

Let the cost price per kg be $1.00 for easy calculation.

• Cost of 512 kg = $512
• Selling price of 512 kg at 20% profit = $512 \times (1 + 0.20) = $512 \times 1.20 = $614.40
• Cost of 128 kg = $128
• Selling price of 128 kg at 5% loss = $128 \times (1 - 0.05) = $128 \times 0.95 = $121.60
• Total cost price = $512 + $128 = $640
• Total selling price = $614.40 + $121.60 = $736.00
• Overall profit = Total selling price - Total cost price = $736.00 - $640.00 = $96.00
• Overall profit percentage = \( \frac{\text{Overall Profit}}{\text{Total Cost Price}} \times 100\% = \frac{96}{640} \times 100\% \)
• Overall profit percentage = \( \frac{960}{64} \% = 15\% \)

The calculated overall profit percentage matches the given information, confirming our result.

Revision Table: Rice Trader Profit Loss Analysis

Additional Information: Profit and Loss Concepts

Understanding basic profit and loss concepts is crucial for solving such problems.

• Cost Price (CP): The price at which an article is bought.
• Selling Price (SP): The price at which an article is sold.
• Profit: When SP > CP. Profit = SP - CP.
• Loss: When SP < CP. Loss = CP - SP.
• Profit Percentage: \( \frac{\text{Profit}}{\text{CP}} \times 100\% \)
• Loss Percentage: \( \frac{\text{Loss}}{\text{CP}} \times 100\% \)
• Relationship between SP and CP:
  • If there is a profit of P%, SP = \( CP \times (1 + \frac{P}{100}) \)
  • If there is a loss of L%, SP = \( CP \times (1 - \frac{L}{100}) \)
• Alligation Rule: Applicable when two varieties of items with certain characteristics are mixed to produce a mixture with a mean characteristic. The ratio of the quantities of the two varieties is inversely proportional to the differences between their individual characteristic values and the mean characteristic value.

This problem demonstrates how different profit/loss margins on parts of a transaction contribute to the overall profit or loss. The Alligation method provides an efficient way to determine the proportions of these parts.