The marked price of an article is Rs. 240. A shopkeeper sells it by allowing 18% discount on its marked price and still gains 23%. What is the cost price (in Rs.) of the article?
160
Calculating Cost Price from Marked Price, Discount, and Gain
This problem involves understanding the relationship between Marked Price (MP), Selling Price (SP), Cost Price (CP), discount percentage, and gain percentage. We are given the marked price, the discount percentage applied, and the profit percentage earned, and we need to find the cost price of the article.
Understanding the Terms
- Marked Price (MP): The price listed on the article.
- Discount: A reduction offered on the Marked Price.
- Selling Price (SP): The price at which the article is sold after the discount.
- Cost Price (CP): The price at which the shopkeeper bought the article.
- Gain (Profit): When SP is greater than CP. Gain % is calculated on CP.
Step-by-Step Solution
Step 1: Calculate the Selling Price (SP)
The shopkeeper allows an 18% discount on the Marked Price of Rs. 240.
The formula to calculate the selling price after a discount is:
$\text{SP} = \text{MP} - (\text{Discount \% of MP})$
Alternatively, the SP is $(100 - \text{Discount \%})\%$ of the MP.
$\text{SP} = \text{MP} \times \frac{100 - \text{Discount \%}}{100}$
Given:
- Marked Price (MP) = Rs. 240
- Discount Percentage = 18%
Calculation:
$\text{SP} = 240 \times \frac{100 - 18}{100}$
$\text{SP} = 240 \times \frac{82}{100}$
$\text{SP} = 240 \times 0.82$
$\text{SP} = 196.80$ Rs.
So, the selling price of the article is Rs. 196.80.
Step 2: Calculate the Cost Price (CP)
The shopkeeper gains 23% after selling the article for Rs. 196.80. The gain percentage is always calculated on the Cost Price.
The formula relating Selling Price, Cost Price, and Gain Percentage is:
$\text{SP} = \text{CP} + (\text{Gain \% of CP})$
Alternatively, the SP is $(100 + \text{Gain \%})\%$ of the CP.
$\text{SP} = \text{CP} \times \frac{100 + \text{Gain \%}}{100}$
We need to find CP. We can rearrange the formula:
$\text{CP} = \frac{\text{SP}}{\frac{100 + \text{Gain \%}}{100}} = \text{SP} \times \frac{100}{100 + \text{Gain \%}}$
Given:
- Selling Price (SP) = Rs. 196.80 (calculated in Step 1)
- Gain Percentage = 23%
Calculation:
$\text{CP} = 196.80 \times \frac{100}{100 + 23}$
$\text{CP} = 196.80 \times \frac{100}{123}$
$\text{CP} = \frac{19680}{123}$
To perform the division:
$19680 \div 123$
We can simplify this division. Let's perform long division or notice that $123 \times 100 = 12300$ and $123 \times 200 = 24600$. The value is closer to 12300, maybe around 160.
$123 \times 160 = 123 \times (100 + 60) = 12300 + 123 \times 60 = 12300 + 7380 = 19680$.
So, $\text{CP} = 160$ Rs.
The cost price of the article is Rs. 160.
Summary of Calculations
| Item | Value (Rs.) |
|---|---|
| Marked Price (MP) | 240 |
| Discount (%) | 18% |
| Selling Price (SP) | $240 \times (1 - 0.18) = 196.80$ |
| Gain (%) | 23% |
| Cost Price (CP) | $\frac{196.80}{1 + 0.23} = \frac{196.80}{1.23} = 160$ |
The calculated cost price is Rs. 160.
Revision Table: Profit, Loss, Discount Formulas
| Concept | Formula |
|---|---|
| Selling Price with Discount | $\text{SP} = \text{MP} \times \frac{100 - \text{Discount \%}}{100}$ |
| Selling Price with Gain | $\text{SP} = \text{CP} \times \frac{100 + \text{Gain \%}}{100}$ |
| Cost Price with Gain | $\text{CP} = \text{SP} \times \frac{100}{100 + \text{Gain \%}}$ |
| Discount Amount | $\text{Discount Amount} = \text{MP} - \text{SP}$ |
| Gain Amount | $\text{Gain Amount} = \text{SP} - \text{CP}$ (if SP > CP) |
| Loss Amount | $\text{Loss Amount} = \text{CP} - \text{SP}$ (if CP > SP) |
| Gain % | $\text{Gain \%} = \frac{\text{Gain Amount}}{\text{CP}} \times 100$ |
| Loss % | $\text{Loss \%} = \frac{\text{Loss Amount}}{\text{CP}} \times 100$ |
| Discount % | $\text{Discount \%} = \frac{\text{Discount Amount}}{\text{MP}} \times 100$ |
Additional Information on Pricing Concepts
Understanding the different price points and percentages is key to solving profit, loss, and discount problems. Here's a bit more detail:
- Marked Price vs. Selling Price: The marked price is often higher than the selling price, with the difference being the discount. Discounts are used to attract customers.
- Cost Price vs. Selling Price: The relationship between cost price and selling price determines whether there is a profit or a loss. If SP > CP, there is a profit (gain). If SP < CP, there is a loss.
- Profit/Loss Percentage Calculation: Profit or loss percentage is almost always calculated on the cost price. This is the standard convention unless otherwise specified.
- Discount Percentage Calculation: Discount percentage is always calculated on the marked price.
- Successive Discounts: Sometimes multiple discounts are applied one after another. The effective discount can be calculated, or each discount can be applied sequentially.
These concepts are fundamental in commercial arithmetic and are frequently tested in various exams.
Important Questions from Discount and MP
The marked price of an article is Rs. 1500. A shopkeeper sells it by giving 20% discount on its marked price. If the cost price of the article is Rs. 991, then his profit (in Rs.) is:
Surbhi sold an article for Rs. 176 after giving 12% discount on its marked price. Had she not given any discount; she would have earned a profit of 25%. What is the cost price (in Rs.) of the article?
Three shopkeepers A, B and C marked on an identical article at Rs. 4820. A, B and C sold their article on successive discounts of 20% and 20%; 25% and 15%; 30% and 10% respectively. Which shopkeeper gives the maximum discount and how much (in Rs.)?
Find a single discount percentage equivalent to successive discounts of 10%, 20% and 25%.
A shopkeeper marks the price of the article in such a way that after allowing 28% discount, he wants a gain of 12%. If the marked price is Rs. 224, then the cost price of the article is:
