Understanding the Problem: Selling Toffees for Profit

This question asks us to figure out the selling price strategy required for a vendor to achieve a specific profit percentage when selling toffees. We are given the cost price of the toffees and the desired profit margin. We need to determine how many toffees the vendor must sell for one rupee to make a 25% profit.

Calculating the Cost Price of Toffees

The vendor bought toffees at a rate of 10 toffees for a rupee. This means the cost price (CP) for 10 toffees is 1 rupee.

Let's denote the Cost Price (CP) of 10 toffees:

CP of 10 toffees = 1 Rupee

Determining the Desired Selling Price for Toffees

The vendor wants to gain 25% profit. The profit percentage is calculated on the cost price. To find the selling price (SP) for 10 toffees that yields a 25% profit, we can use the formula:

\( \text{SP} = \text{CP} \times \left(1 + \frac{\text{Profit Percentage}}{100}\right) \)

In this case, CP is 1 Rupee and the Profit Percentage is 25%.

\( \text{SP of 10 toffees} = 1 \text{ Rupee} \times \left(1 + \frac{25}{100}\right) \)

\( \text{SP of 10 toffees} = 1 \times \left(1 + 0.25\right) \)

\( \text{SP of 10 toffees} = 1 \times 1.25 \)

\( \text{SP of 10 toffees} = 1.25 \text{ Rupees} \)

So, to make a 25% profit, the vendor must sell 10 toffees for 1.25 Rupees.

Finding How Many Toffees to Sell for One Rupee

We now know that 10 toffees must be sold for 1.25 Rupees. The question asks how many toffees must be sold for one rupee. We can set up a simple ratio or use a unitary method:

• If 1.25 Rupees buys 10 toffees,
• Then 1 Rupee buys \( \frac{10}{1.25} \) toffees.

Let's calculate the number of toffees for 1 Rupee:

\( \text{Number of toffees for 1 Rupee} = \frac{10}{1.25} \)

To simplify the division, we can write 1.25 as a fraction \( \frac{125}{100} \) or \( \frac{5}{4} \):

\( \text{Number of toffees for 1 Rupee} = \frac{10}{\frac{5}{4}} \)

\( \text{Number of toffees for 1 Rupee} = 10 \times \frac{4}{5} \)

\( \text{Number of toffees for 1 Rupee} = \frac{40}{5} \)

\( \text{Number of toffees for 1 Rupee} = 8 \)

Therefore, the vendor must sell 8 toffees for one rupee to gain 25%.

Summary of Steps

Here is a quick summary of the steps taken to solve this toffee selling problem:

1. Identify the Cost Price (CP) for a known quantity of toffees (10 toffees for 1 rupee).
2. Calculate the desired Selling Price (SP) for the same quantity of toffees (10 toffees) to achieve the target profit percentage (25%).
3. Determine how many toffees can be sold for one rupee based on the calculated Selling Price per quantity.

Final Answer

Based on our calculations, the vendor must sell 8 toffees for a rupee to achieve a 25% profit.

Revision Table: Profit and Loss Concepts

Additional Information: Solving Price-Quantity Problems

Problems involving buying a certain number of items for a price and selling a different number for another price (often the same price) are common in profit and loss. The key is often to find the price per item or, as we did here, the price for a convenient quantity (like the original quantity bought) under the new selling condition.

Alternatively, you could find the CP per toffee first:

• CP of 10 toffees = 1 Rupee
• CP of 1 toffee = \( \frac{1}{10} \) Rupees

Then calculate the desired SP per toffee for a 25% profit:

\( \text{SP of 1 toffee} = \text{CP of 1 toffee} \times \left(1 + \frac{25}{100}\right) \)

\( \text{SP of 1 toffee} = \frac{1}{10} \times 1.25 \)

\( \text{SP of 1 toffee} = \frac{1}{10} \times \frac{5}{4} = \frac{5}{40} = \frac{1}{8} \text{ Rupees} \)

This means the vendor must sell each toffee for \( \frac{1}{8} \) of a rupee.

The question asks how many toffees are sold for 1 Rupee. If 1 toffee is sold for \( \frac{1}{8} \) Rupee, then for 1 Rupee, the number of toffees sold will be:

\( \frac{1 \text{ Rupee}}{\text{SP per toffee}} = \frac{1}{\frac{1}{8}} = 1 \times 8 = 8 \text{ toffees} \)

Both methods yield the same result, confirming the answer. Working with a group (like 10 toffees) can sometimes be less prone to calculation errors with fractions or decimals early on.