Understanding the Grain Bag Profit Problem

The question asks us to determine the selling price per bag for the remaining grain bags so that the trader achieves a specific overall profit percentage on the entire lot of 60 bags. We are given the initial cost, the number of bags sold at a certain profit, and the target overall profit percentage.

Calculating the Total Cost Price

The trader buys 60 bags of grain at Rs. 400 each. The total cost price is calculated by multiplying the number of bags by the cost per bag.

Total Cost Price = Number of Bags \(\times\) Cost Price per Bag

Total Cost Price = \(60 \times 400 = \text{Rs. } 24000\)

Calculating the Target Overall Selling Price

The trader wants to make a 16.4% overall profit on selling all 60 bags. This profit is calculated on the total cost price.

Overall Profit Amount = 16.4% of Total Cost Price

Overall Profit Amount = \(\frac{16.4}{100} \times 24000\)

Overall Profit Amount = \(16.4 \times 240 = \text{Rs. } 3936\)

The target total selling price for all 60 bags is the sum of the total cost price and the overall profit amount.

Target Total Selling Price = Total Cost Price + Overall Profit Amount

Target Total Selling Price = \(24000 + 3936 = \text{Rs. } 27936\)

Calculating Revenue from the First 18 Bags

The trader sells 18 bags at an 8% profit. First, let's find the selling price per bag for these 18 bags.

Profit per Bag (on first 18) = 8% of Cost Price per Bag

Profit per Bag = \(\frac{8}{100} \times 400 = 8 \times 4 = \text{Rs. } 32\)

Selling Price per Bag (for first 18) = Cost Price per Bag + Profit per Bag

Selling Price per Bag = \(400 + 32 = \text{Rs. } 432\)

The total revenue generated from selling these 18 bags is the number of bags multiplied by their selling price per bag.

Total Revenue from 18 Bags = Number of Bags \(\times\) Selling Price per Bag

Total Revenue from 18 Bags = \(18 \times 432 = \text{Rs. } 7776\)

Calculating Required Revenue from Remaining Bags

The total number of bags is 60. After selling 18 bags, the number of remaining bags is:

Remaining Bags = Total Bags - Bags Sold

Remaining Bags = \(60 - 18 = 42\)

The total revenue needed from all 60 bags is Rs. 27936. The revenue already earned from the first 18 bags is Rs. 7776. The revenue that must be generated from the remaining 42 bags is the difference.

Required Revenue from Remaining Bags = Target Total Selling Price - Total Revenue from 18 Bags

Required Revenue from Remaining Bags = \(27936 - 7776 = \text{Rs. } 20160\)

Calculating Selling Price per Bag for Remaining Bags

To find the selling price per bag for the remaining 42 bags, we divide the required revenue from these bags by the number of remaining bags.

Selling Price per Bag (for remaining bags) = Required Revenue from Remaining Bags / Number of Remaining Bags

Selling Price per Bag = \(\frac{20160}{42}\)

Let's perform the division:

\(\frac{20160}{42} = \frac{10080}{21} = \frac{3360}{7} = 480\)

So, the selling price per bag for the remaining 42 bags should be Rs. 480.

Summary of Calculations

The trader should sell the remaining 42 bags at Rs. 480 each to achieve an overall profit of 16.4% on the 60 bags.

Revision Table: Key Profit Concepts

Additional Information: Calculating Profit & Selling Price

Understanding how profit percentages relate to cost price and selling price is crucial in business math problems. Here's a bit more detail:

• When an item is sold at a profit, the selling price is the cost price plus the profit amount. If the profit percentage is P%, the selling price is \(CP + P\% \text{ of } CP = CP \times (1 + P/100)\).
• If you know the selling price and the profit percentage, you can find the cost price using the formula: \(CP = \frac{SP}{1 + P/100}\).
• In this problem, we calculated the selling price of the first 18 bags using the formula: \(SP = 400 \times (1 + 8/100) = 400 \times (1.08) = 432\).
• For the remaining bags, we worked backward. We found the required total selling price for the remaining bags and then divided by the number of bags to get the price per bag. This price is designed to ensure the overall profit target is met.
• It's important to distinguish between profit on a portion of goods and the overall profit on the entire transaction. The overall profit percentage is always calculated on the total initial cost of all items considered in the transaction.