Finding the Cost Price of an Article in a Trader Problem

This problem involves calculating the original cost price of an article based on two different selling scenarios: one resulting in a loss and the other in a gain, with a given difference in their selling prices.

Understanding the Problem Scenarios

• Initially, the trader sells the article at a loss of 16% on the cost price.
• In a second scenario, if the selling price were Rs. 192.20 higher than the initial selling price, the trader would have made a gain of 15% on the cost price.
• We need to find the cost price (CP) of the article.

Setting up the Equations

Let the Cost Price of the article be CP (in Rs.).

Scenario 1: Selling at a 16% Loss

The selling price in the first scenario (SP1) is the Cost Price minus the loss amount.

Loss amount = 16% of CP

Loss amount = $\frac{16}{100}\times \text{CP}$

SP1 = CP − Loss amount

SP1 = $\text{CP}-\frac{16}{100}\text{CP}$

SP1 = $\text{CP}\times \left(1-\frac{16}{100}\right)$

SP1 = $\text{CP}\times \left(\frac{100-16}{100}\right)$

SP1 = $\text{CP}\times \frac{84}{100}$

SP1 = $0.84\times \text{CP}$

Scenario 2: Selling for Rs. 192.20 More, Resulting in a 15% Gain

The selling price in the second scenario (SP2) is SP1 plus Rs. 192.20.

SP2 = SP1 + 192.20

Also, SP2 represents a 15% gain on the Cost Price.

Gain amount = 15% of CP

Gain amount = $\frac{15}{100}\times \text{CP}$

SP2 = CP + Gain amount

SP2 = $\text{CP}+\frac{15}{100}\text{CP}$

SP2 = $\text{CP}\times \left(1+\frac{15}{100}\right)$

SP2 = $\text{CP}\times \left(\frac{100+15}{100}\right)$

SP2 = $\text{CP}\times \frac{115}{100}$

SP2 = $1.15\times \text{CP}$

Solving for the Cost Price (CP)

We have two expressions for SP2:

• SP2 = SP1 + 192.20
• SP2 = $1.15\times \text{CP}$

Substitute the expression for SP1 ($0.84\times \text{CP}$) into the first equation:

$$1.15\times \text{CP}=0.84\times \text{CP}+192.20$$

Now, rearrange the equation to group the CP terms:

$$1.15\times \text{CP}-0.84\times \text{CP}=192.20$$

Subtract the CP terms:

$$\left(1.15-0.84\right)\times \text{CP}=192.20$$

$$0.31\times \text{CP}=192.20$$

Now, solve for CP by dividing 192.20 by 0.31:

$$\text{CP}=\frac{192.20}{0.31}$$

To simplify the division, multiply both the numerator and denominator by 100 to remove the decimals:

$$\text{CP}=\frac{192.20\times 100}{0.31\times 100}$$

CP=1922031

Performing the division:

$$19220÷31=620$$

So, the Cost Price (CP) of the article is Rs. 620.

Summary of Calculation

The cost price of the article is Rs. 620.

Revision Table: Profit and Loss Concepts

Additional Information: Percentage Difference in Selling Prices

In this problem, the difference of Rs. 192.20 in selling prices corresponds to the combined effect of changing from a 16% loss to a 15% gain. The total percentage change relative to the cost price is the sum of the loss percentage and the gain percentage because one is below CP and the other is above CP.

• Going from a 16% loss means the price moved 16% UP towards the Cost Price (100% of CP).
• Going from the Cost Price to a 15% gain means the price moved another 15% UP from the Cost Price.
• Total percentage difference = 16% (to reach CP) + 15% (to go above CP)
• Total percentage difference = $16\%+15\%=31\%$

This 31% difference is on the Cost Price. So, 31% of CP is equal to the difference in selling prices, which is Rs. 192.20.

$$31\%\times \text{CP}=192.20$$

$$\frac{31}{100}\times \text{CP}=192.20$$

$$\text{CP}=\frac{192.20\times 100}{31}$$

$$\text{CP}=\frac{19220}{31}$$

$$\text{CP}=620$$

This confirms the result obtained through the previous method and highlights the direct relationship between the price difference and the total percentage change relative to the cost price when switching from a loss to a gain scenario (or vice versa).