Calculating Cost Price Using Loss and Gain Percentages

This problem involves finding the original cost price of an article based on how changes in the selling price affect the percentage of loss or gain. We are given two scenarios for selling the same article and the difference in the selling prices.

Understanding the Problem Statement

We are told that a dealer initially sold an article at a 2% loss. If the selling price had been Rs. 44 higher, the dealer would have made a 20% gain. We need to determine the original cost price of the article.

Key Concepts: Cost Price, Selling Price, Loss, and Gain

• Cost Price (CP): The price at which the dealer bought the article. This is what we need to find.
• Selling Price (SP): The price at which the dealer sold the article.
• Loss: Occurs when SP < CP. Loss Percentage = $\left( \frac{\text{Loss}}{\text{CP}} \right) \times 100 = \left( \frac{\text{CP} - \text{SP}}{\text{CP}} \right) \times 100$.
• Gain (or Profit): Occurs when SP > CP. Gain Percentage = $\left( \frac{\text{Gain}}{\text{CP}} \right) \times 100 = \left( \frac{\text{SP} - \text{CP}}{\text{CP}} \right) \times 100$.

From the definitions, we can express SP in terms of CP and the percentage loss or gain:

• If there is a Loss of L%, SP = $\text{CP} \times \left( 1 - \frac{L}{100} \right)$.
• If there is a Gain of G%, SP = $\text{CP} \times \left( 1 + \frac{G}{100} \right)$.

Step-by-Step Solution to Find the Cost Price

Let's denote the Cost Price of the article as CP.

Scenario 1: Selling at a Loss of 2%

The initial sale was at a loss of 2%. Let the selling price in this scenario be SP$_1$.

Using the formula for SP with a loss:

$\text{SP}_1 = \text{CP} \times \left( 1 - \frac{2}{100} \right)$

$\text{SP}_1 = \text{CP} \times \left( 1 - 0.02 \right)$

$\text{SP}_1 = \text{CP} \times 0.98$

$\text{SP}_1 = 0.98 \times \text{CP}$

Scenario 2: Selling for Rs. 44 more and gaining 20%

If the selling price was Rs. 44 more than SP$_1$, the new selling price, let's call it SP$_2$, would result in a 20% gain on the Cost Price.

So, SP$_2 = \text{SP}_1 + 44$.

Also, we can calculate SP$_2$ based on the 20% gain on CP:

Using the formula for SP with a gain:

$\text{SP}_2 = \text{CP} \times \left( 1 + \frac{20}{100} \right)$

$\text{SP}_2 = \text{CP} \times \left( 1 + 0.20 \right)$

$\text{SP}_2 = \text{CP} \times 1.20$

$\text{SP}_2 = 1.20 \times \text{CP}$

Equating and Solving for Cost Price (CP)

We have two expressions for SP$_2$: SP$_2 = \text{SP}_1 + 44$ and SP$_2 = 1.20 \times \text{CP}$.

Substitute the expression for SP$_1$ ($0.98 \times \text{CP}$) into the first equation:

$\text{SP}_2 = (0.98 \times \text{CP}) + 44$

Now, equate this with the second expression for SP$_2$ ($1.20 \times \text{CP}$):

$0.98 \times \text{CP} + 44 = 1.20 \times \text{CP}$

To solve for CP, rearrange the equation:

$44 = 1.20 \times \text{CP} - 0.98 \times \text{CP}$

$44 = (1.20 - 0.98) \times \text{CP}$

$44 = 0.22 \times \text{CP}$

Now, isolate CP:

$\text{CP} = \frac{44}{0.22}$

To simplify the division, multiply the numerator and denominator by 100:

$\text{CP} = \frac{44 \times 100}{0.22 \times 100}$

$\text{CP} = \frac{4400}{22}$

$\text{CP} = 200$

So, the Cost Price of the article is Rs. 200.

Verification

Let's check if our answer makes sense:

• Cost Price = Rs. 200
• Scenario 1: 2% loss. Loss amount = 2% of 200 = $\frac{2}{100} \times 200 = 4$. SP$_1$ = $200 - 4 = 196$.
• Scenario 2: Sold for Rs. 44 more than SP$_1$. SP$_2 = 196 + 44 = 240$.
• Check Gain for SP$_2$: Gain = SP$_2$ - CP = $240 - 200 = 40$.
• Gain Percentage = $\left( \frac{\text{Gain}}{\text{CP}} \right) \times 100 = \left( \frac{40}{200} \right) \times 100 = \left( \frac{1}{5} \right) \times 100 = 20\%$.

The calculations match the problem statement. The cost price is indeed Rs. 200.

Revision Table: Profit and Loss Key Formulas

Additional Information: Solving Profit and Loss Problems

Profit and loss problems often involve relating cost price, selling price, profit/loss amount, and profit/loss percentage. Here are some tips for tackling such problems:

• Always identify whether the percentage is calculated on the cost price or the selling price (unless specified, it's usually on CP).
• Set up equations based on the information given. Represent the unknown (usually CP or SP) with a variable.
• Convert percentages to decimals or fractions for calculations.
• If there are multiple transactions or scenarios, analyse them one by one and connect them using the given relationships (like the difference in selling prices).
• Verify your final answer by plugging it back into the original problem conditions.
• Understanding the relationship between CP, SP, and percentage change is crucial. A loss of X% means SP is (100-X)% of CP, and a gain of Y% means SP is (100+Y)% of CP.