The marked price of an article is 40% above its cost price. If its selling price is \(73 \frac{1}{2}\%\) of the marked price, then the profit percentage is:
2.9%
Calculating Profit Percentage
The problem asks us to find the profit percentage on an article given its marked price relative to the cost price and its selling price relative to the marked price. Let's break down the steps to solve this.
Defining the Terms
- Cost Price (CP): The original price at which the article was bought.
- Marked Price (MP): The price listed on the article, usually higher than the cost price.
- Selling Price (SP): The price at which the article is sold.
- Profit: When Selling Price is greater than Cost Price (SP > CP). Profit = SP - CP.
- Profit Percentage: \(\frac{\text{Profit}}{\text{CP}} \times 100\).
Step-by-Step Calculation
To make the calculation easy, let's assume a value for the Cost Price.
Step 1: Assume a Cost Price (CP)
Let the Cost Price (CP) of the article be \(₹100\).
Step 2: Calculate the Marked Price (MP)
The marked price is 40% above its cost price.
MP = CP + 40% of CP
MP = \(100 + \frac{40}{100} \times 100\)
MP = \(100 + 40\)
MP = \(₹140\)
Step 3: Calculate the Selling Price (SP)
The selling price is \(73 \frac{1}{2}\%\) of the marked price.
\(73 \frac{1}{2}\%\) can be written as \(73.5\%\).
SP = \(73.5\%\) of MP
SP = \(\frac{73.5}{100} \times 140\)
SP = \(\frac{73.5 \times 140}{100}\)
SP = \(\frac{73.5 \times 14}{10}\)
SP = \(73.5 \times 1.4\)
Let's perform the multiplication:
\(73.5 \times 1.4\)
\(73.5 \times (1 + 0.4) = 73.5 \times 1 + 73.5 \times 0.4\)
\(73.5 + 29.4\)
\(102.9\)
So, SP = \(₹102.9\)
Step 4: Calculate the Profit
Profit = Selling Price (SP) - Cost Price (CP)
Profit = \(102.9 - 100\)
Profit = \(₹2.9\)
Step 5: Calculate the Profit Percentage
Profit Percentage = \(\frac{\text{Profit}}{\text{CP}} \times 100\)
Profit Percentage = \(\frac{2.9}{100} \times 100\)
Profit Percentage = \(2.9\%\)
Summary of Calculations
| Item | Value (assuming CP = ₹100) |
|---|---|
| Cost Price (CP) | ₹100 |
| Marked Price (MP) | ₹140 (40% above CP) |
| Selling Price (SP) | ₹102.9 (\(73.5\%\) of MP) |
| Profit (SP - CP) | ₹2.9 |
| Profit Percentage | 2.9% |
The profit percentage is 2.9%.
Revision Table: Profit and Loss Concepts
| Concept | Formula | Description |
|---|---|---|
| Profit | SP - CP (if SP > CP) | Gain from selling an article. |
| Loss | CP - SP (if CP > SP) | Loss incurred from selling an article. |
| Profit % | \(\frac{\text{Profit}}{\text{CP}} \times 100\) | Profit expressed as a percentage of the cost price. |
| Loss % | \(\frac{\text{Loss}}{\text{CP}} \times 100\) | Loss expressed as a percentage of the cost price. |
| Marked Price (MP) | CP + Markup | Price set by the seller, often higher than CP. |
| Selling Price (SP) | MP - Discount | Actual price at which the article is sold. |
Additional Information: Percentage Calculations
Understanding percentages is crucial in profit and loss problems. Here are a few key ideas:
- Percentage Increase: To increase a quantity by x%, multiply the quantity by \((1 + \frac{x}{100})\). For example, increasing CP by 40% gives MP = CP \(\times (1 + \frac{40}{100}) = \text{CP} \times 1.4\).
- Percentage of a Quantity: To find y% of a quantity, multiply the quantity by \(\frac{y}{100}\). For example, \(73.5\%\) of MP is MP \(\times \frac{73.5}{100} = \text{MP} \times 0.735\).
- Converting Mixed Number to Decimal Percentage: \(73 \frac{1}{2}\%\) means \(73 + \frac{1}{2}\) percent. \(\frac{1}{2} = 0.5\), so \(73 \frac{1}{2}\%\) is \(73.5\%\).
These concepts allow us to directly calculate MP from CP and SP from MP using multiplication factors, which can sometimes simplify calculations.
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