Let's break down this profit and loss problem step by step to find the selling price needed to achieve a 10% gain.

The problem states that an article was initially sold at a loss of 15%. If the selling price were Rs. 30.60 more, it would have resulted in a gain of 9%. We need to find the selling price required to make a gain of 10%.

Understanding Profit and Loss Concepts

In profit and loss calculations, we usually work with the Cost Price (CP) and the Selling Price (SP). Percentage gain or loss is always calculated on the Cost Price unless stated otherwise.

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

Step-by-Step Solution for Finding the Cost Price

Let the Cost Price of the article be $CP$.

Scenario 1: Sold at a loss of 15%

The selling price (SP1) in this case is:

$SP1 = CP \times \frac{(100 - 15)}{100} = CP \times \frac{85}{100} = 0.85 \times CP$

Scenario 2: Sold with a gain of 9%

The selling price (SP2) in this case is:

$SP2 = CP \times \frac{(100 + 9)}{100} = CP \times \frac{109}{100} = 1.09 \times CP$

According to the problem, if the article was sold for Rs. 30.60 more than the first selling price (SP1), it would be equal to the second selling price (SP2). So, we can write the equation:

$SP2 = SP1 + 30.60$

Substitute the expressions for SP1 and SP2 in terms of CP:

$1.09 \times CP = 0.85 \times CP + 30.60$

Now, let's solve for CP:

$1.09 \times CP - 0.85 \times CP = 30.60$

$(1.09 - 0.85) \times CP = 30.60$

$0.24 \times CP = 30.60$

$CP = \frac{30.60}{0.24}$

$CP = \frac{3060}{240}$ (Multiplying numerator and denominator by 100)

$CP = \frac{306}{24}$ (Dividing numerator and denominator by 10)

$CP = \frac{102}{8}$ (Dividing numerator and denominator by 3)

$CP = \frac{51}{4}$ (Dividing numerator and denominator by 2)

$CP = 12.75 \times 10$ (Since 51/4 = 12.75 and we had 306/24 = 102/8. Let's recheck: 30.60/0.24 = 3060/24 = 127.5)

Let's calculate $3060 \div 240$: $3060/240 = 306/24$. $306 \div 24 = 12.75$. So $CP = 127.50$.

The Cost Price (CP) of the article is Rs. 127.50.

Calculating Selling Price for 10% Gain

Now we need to find the selling price (SP3) at which the person would gain 10%. The gain is calculated on the Cost Price (CP).

Gain required = 10% of CP

Selling Price (SP3) = $CP + (10\% \text{ of } CP)$

$SP3 = CP \times \frac{(100 + 10)}{100} = CP \times \frac{110}{100} = 1.10 \times CP$

Substitute the value of CP we found:

$SP3 = 1.10 \times 127.50$

$SP3 = 1.1 \times 127.5 = 140.25$

So, to gain 10%, the person should have sold the article for Rs. 140.25.

Let's verify the steps:

• CP = 127.50
• 15% Loss: $127.50 \times 0.85 = 108.375$ (approx Rs. 108.38)
• 9% Gain: $127.50 \times 1.09 = 139.975$ (approx Rs. 139.98)
• Difference: $139.975 - 108.375 = 31.6$ (The difference in the problem is 30.60. Let's recheck calculation $30.60 / 0.24$). $30.60 / 0.24 = (3060/100) / (24/100) = 3060/24$. $3060 \div 24$. $3060/24 = 127.5$. Okay, CP is indeed 127.50. SP1 = $127.50 \times 0.85 = 108.375$. SP2 = $127.50 \times 1.09 = 138.975$. Difference = $138.975 - 108.375 = 30.60$. This matches the problem statement exactly.
• 10% Gain SP: $127.50 \times 1.10 = 140.25$.

The calculation is correct.

The selling price required to gain 10% is Rs. 140.25.

Revision Table: Profit and Loss Basics

Additional Information on Profit and Loss Calculations

Profit and loss problems often involve converting between cost price, selling price, profit percentage, and loss percentage. A key point is always identifying the base for the percentage calculation, which is usually the cost price.

In this problem, the difference in selling prices corresponds to the combined percentage change from 15% loss to 9% gain. The total percentage change is $15\% (\text{loss}) + 9\% (\text{gain}) = 24\%$. This 24% corresponds to the difference of Rs. 30.60. So, $24\%$ of $CP = 30.60$. This confirms our equation $0.24 \times CP = 30.60$. Understanding this relationship can sometimes provide a quicker way to set up the problem.

Once the Cost Price is known, calculating the selling price for any desired profit or loss percentage is straightforward using the formulas $SP = CP \times (1 + \text{gain %}/100)$ or $SP = CP \times (1 - \text{loss %}/100)$.