Ram sells a suitcase to Mohan at a 20% profit. Mohan sells it to Shyam at a 40% profit. If Shyam pays Rs. 1,430 for it, then the price at which Ram bought it is:
Rs. 851.19
Calculating Initial Price with Successive Profits
This problem involves calculating the original cost price of an item after it has been sold multiple times with different profit percentages applied in each transaction. We need to work backward from the final price paid by Shyam to find the initial price at which Ram bought the suitcase.
Understanding the Transactions
Let's break down the transactions:
- Ram sells the suitcase to Mohan at a 20% profit.
- Mohan sells the suitcase to Shyam at a 40% profit.
- Shyam pays Rs. 1,430 for the suitcase.
Step-by-Step Calculation
We can denote the prices at each stage:
- Let \(C_R\) be the cost price for Ram (the price Ram bought it for).
- The selling price for Ram (\(S_R\)) is the cost price for Mohan (\(C_M\)).
- The selling price for Mohan (\(S_M\)) is the cost price for Shyam (\(C_S\)).
Transaction 1: Ram to Mohan
Ram sells to Mohan at a 20% profit. This means Mohan's cost price is Ram's cost price plus 20% of Ram's cost price.
\(C_M = C_R + 20\% \text{ of } C_R\)
In terms of percentage, this is 100% (original price) + 20% (profit) = 120% of \(C_R\).
So, \(C_M = C_R \times \left(1 + \frac{20}{100}\right) = C_R \times 1.20\)
Transaction 2: Mohan to Shyam
Mohan sells to Shyam at a 40% profit. This means Shyam's cost price is Mohan's cost price plus 40% of Mohan's cost price.
\(C_S = C_M + 40\% \text{ of } C_M\)
In terms of percentage, this is 100% (Mohan's price) + 40% (profit) = 140% of \(C_M\).
So, \(C_S = C_M \times \left(1 + \frac{40}{100}\right) = C_M \times 1.40\)
Relating Shyam's Price to Ram's Price
We know \(C_S = 1.40 \times C_M\) and \(C_M = 1.20 \times C_R\).
Substitute the expression for \(C_M\) into the equation for \(C_S\):
\(C_S = 1.40 \times (1.20 \times C_R)\)
\(C_S = (1.40 \times 1.20) \times C_R\)
\(C_S = 1.68 \times C_R\)
Solving for Ram's Cost Price (\(C_R\))
We are given that Shyam pays Rs. 1,430, which means \(C_S = 1430\).
\(1430 = 1.68 \times C_R\)
To find \(C_R\), divide 1430 by 1.68:
\(C_R = \frac{1430}{1.68}\)
Let's perform the division:
\(C_R \approx 851.19047...\)
Rounding this to two decimal places gives approximately Rs. 851.19.
Summary of Calculation Steps
| Step | Description | Formula | Calculation |
|---|---|---|---|
| 1 | Mohan's cost price based on Ram's profit | \(C_M = C_R \times (1 + \text{Profit Rate 1})\) | \(C_M = C_R \times 1.20\) |
| 2 | Shyam's cost price based on Mohan's profit | \(C_S = C_M \times (1 + \text{Profit Rate 2})\) | \(C_S = C_M \times 1.40\) |
| 3 | Substitute \(C_M\) into the equation for \(C_S\) | \(C_S = (1 + \text{Profit Rate 2}) \times (1 + \text{Profit Rate 1}) \times C_R\) | \(C_S = 1.40 \times 1.20 \times C_R = 1.68 \times C_R\) |
| 4 | Solve for \(C_R\) using Shyam's price | \(C_R = \frac{C_S}{(1 + \text{Profit Rate 1}) \times (1 + \text{Profit Rate 2})}\) | \(C_R = \frac{1430}{1.68} \approx 851.19\) |
The price at which Ram bought the suitcase is approximately Rs. 851.19.
Revision Table: Profit Percentage Concepts
| Term | Definition | Formula (Profit) |
|---|---|---|
| Cost Price (CP) | The initial price at which an article is bought. | N/A |
| Selling Price (SP) | The price at which an article is sold. | \(SP = CP + \text{Profit}\) |
| Profit | The gain obtained by selling an article for more than its cost price. | \(SP - CP\) |
| Profit Percentage | Profit expressed as a percentage of the cost price. | \(\left(\frac{\text{Profit}}{CP}\right) \times 100\) |
| SP in terms of CP and Profit % | Selling Price directly calculated from Cost Price and Profit Percentage. | \(SP = CP \times \left(1 + \frac{\text{Profit \%}}{100}\right)\) |
Additional Information: Successive Profit Calculations
When an item is sold and resold with successive profit percentages, the final selling price is calculated by multiplying the original cost price by the successive profit factors. A profit factor for \(x\%\) profit is \(\left(1 + \frac{x}{100}\right)\). If an item is sold with a profit of \(p_1\%\) and then resold with a profit of \(p_2\%\), and the original cost price was CP, the final selling price (which is the final buyer's cost price) is given by:
Final Price \( = CP \times \left(1 + \frac{p_1}{100}\right) \times \left(1 + \frac{p_2}{100}\right)\)
In this problem, the final price (Shyam's cost price) is Rs. 1430, the first profit is 20%, and the second profit is 40%. The original cost price is Ram's cost price \(C_R\).
\(1430 = C_R \times \left(1 + \frac{20}{100}\right) \times \left(1 + \frac{40}{100}\right)\)
\(1430 = C_R \times (1.20) \times (1.40)\)
\(1430 = C_R \times 1.68\)
\(C_R = \frac{1430}{1.68}\)
This confirms our step-by-step calculation and provides a general formula for handling successive profits.
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