On selling an article for Rs. 246.80, the gain is 20% more than the amount of loss incurred on selling it for Rs. 216. If the article is sold for Rs. 220.75, then what is the gain/loss percent (correct to nearest integer)?
Loss 4%
Understanding the Profit and Loss Scenario
The question describes a situation involving profit and loss when an article is sold at different prices. We are given two selling prices and the relationship between the gain and loss incurred at these prices. Our goal is to find the cost price (CP) of the article and then calculate the gain or loss percentage if it is sold at a third price.
Setting up the Equations for Cost Price
Let the Cost Price (CP) of the article be Rs. \(x\).
When the article is sold for Rs. 246.80, there is a gain. The amount of gain (G1) is calculated as:
Gain (G1) = Selling Price 1 - Cost Price
\[ G1 = 246.80 - x \]
When the article is sold for Rs. 216.00, there is a loss. The amount of loss (L2) is calculated as:
Loss (L2) = Cost Price - Selling Price 2
\[ L2 = x - 216.00 \]
We are given that the gain on selling the article for Rs. 246.80 is 20% more than the loss incurred on selling it for Rs. 216. This can be written as:
Gain (G1) = Loss (L2) + 20% of Loss (L2)
\[ G1 = L2 + 0.20 \times L2 \]
\[ G1 = 1.20 \times L2 \]
Now, substitute the expressions for G1 and L2 into this equation:
\[ 246.80 - x = 1.20 \times (x - 216.00) \]
Solving for the Cost Price
Let's solve the equation to find the cost price \(x\):
\[ 246.80 - x = 1.20x - 1.20 \times 216.00 \]
\[ 246.80 - x = 1.20x - 259.20 \]
Now, rearrange the terms to isolate \(x\):
\[ 246.80 + 259.20 = 1.20x + x \]
\[ 506.00 = 2.20x \]
\[ x = \frac{506.00}{2.20} \]
\[ x = \frac{5060}{22} \]
\[ x = 230 \]
So, the Cost Price (CP) of the article is Rs. 230.00.
Calculating Gain or Loss at the New Selling Price
The article is now sold for a third price, Rs. 220.75. To determine if there is a gain or loss, we compare this selling price (SP3) with the cost price (CP).
Selling Price 3 (SP3) = Rs. 220.75
Cost Price (CP) = Rs. 230.00
Since SP3 < CP, there is a loss.
Amount of Loss = Cost Price - Selling Price 3
\[ \text{Loss} = 230.00 - 220.75 \]
\[ \text{Loss} = 9.25 \]
The loss incurred is Rs. 9.25.
Calculating the Loss Percentage
To find the loss percentage, we use the formula:
\[ \text{Loss Percentage} = \frac{\text{Loss}}{\text{Cost Price}} \times 100 \]
Substitute the values:
\[ \text{Loss Percentage} = \frac{9.25}{230.00} \times 100 \]
\[ \text{Loss Percentage} = \frac{925}{230} \]
\[ \text{Loss Percentage} \approx 4.0217 \dots \]
Rounding to the Nearest Integer
The question asks for the gain/loss percentage correct to the nearest integer. Rounding 4.0217... to the nearest integer gives 4.
Since we calculated a loss, the result is approximately 4% loss.
Comparing with Options
Based on our calculation, there is a loss of approximately 4%.
- Option 1: Profit 7% (Incorrect)
- Option 2: Loss 4% (Correct)
- Option 3: Loss 5% (Incorrect)
- Option 4: Profit 3% (Incorrect)
The calculated loss percentage of approximately 4% matches Option 2.
| Scenario | Selling Price (Rs.) | Outcome | Calculation |
|---|---|---|---|
| Scenario 1 | 246.80 | Gain | Gain = 246.80 - CP |
| Scenario 2 | 216.00 | Loss | Loss = CP - 216.00 |
| Scenario 3 | 220.75 | Loss | Loss = 230.00 - 220.75 = 9.25 |
Revision Table: Key Concepts in Profit and Loss
| Concept | Formula | Description |
|---|---|---|
| Cost Price (CP) | - | The original price at which an article is bought. |
| Selling Price (SP) | - | The price at which an article is sold. |
| Gain or Profit | Gain = SP - CP (when SP > CP) | The amount by which selling price exceeds cost price. |
| Loss | Loss = CP - SP (when CP > SP) | The amount by which cost price exceeds selling price. |
| Gain Percentage | \(\frac{\text{Gain}}{\text{CP}} \times 100\) | Gain expressed as a percentage of the cost price. |
| Loss Percentage | \(\frac{\text{Loss}}{\text{CP}} \times 100\) | Loss expressed as a percentage of the cost price. |
Additional Information: Percentage Calculations
Understanding percentage increase and decrease is crucial in profit and loss problems.
- If quantity A is P% more than quantity B, then A = B + P% of B = \(B \times (1 + \frac{P}{100})\). In our problem, Gain is 20% more than Loss, so Gain = Loss \(\times (1 + \frac{20}{100}) = \text{Loss} \times 1.20\).
- Calculating percentages requires the base value, which is typically the Cost Price (CP) in profit and loss scenarios. The gain or loss amount is divided by the CP, and then multiplied by 100 to get the percentage.
These fundamental concepts help in setting up the correct equations to solve problems involving varying selling prices and their resulting gains or losses.
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