Understanding Profit and Loss Percentages

This problem involves finding the cost price (CP) of an article using information about gain and loss at different selling prices (SP), and then calculating the gain or loss percentage at a third selling price.

Defining Variables

• Let CP be the Cost Price of the article.
• When sold for Rs 123.40, there is a Gain (G1).
• When sold for Rs 108, there is a Loss (L2).
• When sold for Rs 120.75, we need to find the gain or loss percent.

Setting Up the Equations

We know the formulas for Gain and Loss:

• Gain = Selling Price - Cost Price (if SP > CP)
• Loss = Cost Price - Selling Price (if CP > SP)

From the first sale (SP1 = Rs 123.40), the gain G1 is:

\(G1 = 123.40 - CP\)

From the second sale (SP2 = Rs 108), the loss L2 is:

\(L2 = CP - 108\)

The question states that the gain is 20% more than the loss. This can be written as:

\(G1 = L2 + 20\% \text{ of } L2\)

\(G1 = L2 + \frac{20}{100} \times L2\)

\(G1 = L2 + 0.20 \times L2\)

\(G1 = 1.20 \times L2\)

Solving for the Cost Price (CP)

Now substitute the expressions for G1 and L2 into the equation \(G1 = 1.20 \times L2\):

\(123.40 - CP = 1.20 \times (CP - 108)\)

Distribute the 1.20 on the right side:

\(123.40 - CP = 1.20 \times CP - 1.20 \times 108\)

\(123.40 - CP = 1.20 CP - 129.60\)

Now, rearrange the equation to group the CP terms on one side and the constant terms on the other:

\(123.40 + 129.60 = 1.20 CP + CP\)

\(253 = 2.20 CP\)

Solve for CP:

\(CP = \frac{253}{2.20}\)

\(CP = \frac{2530}{22}\)

Performing the division:

\(CP = 115\)

So, the Cost Price of the article is Rs 115.

Calculating Gain/Loss at the New Selling Price

The article is now sold for Rs 120.75 (SP3).

Cost Price (CP) = Rs 115

Selling Price (SP3) = Rs 120.75

Since SP3 > CP, there is a Gain.

Gain = SP3 - CP

Gain = \(120.75 - 115\)

Gain = \(5.75\)

The gain is Rs 5.75.

Calculating the Gain Percent

The Gain Percent is calculated with respect to the Cost Price:

Gain % = \(\frac{\text{Gain}}{\text{CP}} \times 100\)

Gain % = \(\frac{5.75}{115} \times 100\)

To make the division easier, multiply the numerator and denominator by 100:

Gain % = \(\frac{575}{115} \times \frac{1}{100} \times 100\)

Gain % = \(\frac{575}{115}\)

Now, divide 575 by 115. We can see that \(115 \times 5 = 575\).

Gain % = 5

So, the gain percent is 5%.

Summary of Calculations

When the article is sold for Rs. 120.75, there is a Gain of 5%.

Revision Table: Key Profit and Loss Concepts

Additional Information on Percentage Calculations

Percentage calculations are fundamental in profit and loss problems. A percentage represents a fraction of 100. For example, 20% means \(\frac{20}{100}\) or 0.20.

To find a value that is 'X% more than Y', you calculate \(Y + X\%\) of \(Y\), which is \(Y + \frac{X}{100} \times Y = Y \times (1 + \frac{X}{100})\).

In our problem, the gain (G1) was 20% more than the loss (L2), so \(G1 = L2 \times (1 + \frac{20}{100}) = L2 \times 1.20\).

Always remember that gain and loss percentages are calculated on the Cost Price unless specified otherwise.