What will be the cost price of the goods if a shopkeeper allows a discount of 10% on the marked price of Rs. 100 and gains 8%(rounded off to the nearest integer)?
Rs. 83
Understanding Profit and Loss Calculations
This problem involves concepts of marked price, discount, selling price, cost price, and gain percentage. We are given the marked price of an item, the discount percentage allowed, and the gain percentage made by the shopkeeper. Our goal is to find the cost price of the goods.
Step-by-Step Calculation of Cost Price
1. Calculate the Selling Price (SP)
The shopkeeper allows a discount on the marked price. The selling price is the price after the discount is applied.
- Marked Price (MP) = Rs. 100
- Discount Percentage = 10%
- Discount Amount = Discount % of MP
- Discount Amount = $\frac{10}{100} \times 100 = 10$ Rupees
- Selling Price (SP) = Marked Price - Discount Amount
- SP = $100 - 10 = 90$ Rupees
So, the selling price of the goods is Rs. 90.
2. Calculate the Cost Price (CP) using Selling Price and Gain Percentage
The shopkeeper gains 8% on the transaction. The gain is calculated on the cost price. The relationship between Selling Price (SP), Cost Price (CP), and Gain Percentage is given by the formula:
$\text{SP} = \text{CP} \times \left(1 + \frac{\text{Gain \%}}{100}\right)$
We know SP = 90 and Gain % = 8. We need to find CP.
Substitute the values into the formula:
$90 = \text{CP} \times \left(1 + \frac{8}{100}\right)$
$90 = \text{CP} \times \left(1 + 0.08\right)$
$90 = \text{CP} \times 1.08$
Now, solve for CP:
$\text{CP} = \frac{90}{1.08}$
Let's perform the division:
$\text{CP} = 83.333...$
3. Round off to the Nearest Integer
The question asks to round off the cost price to the nearest integer.
CP $\approx$ 83 Rupees
Therefore, the cost price of the goods is approximately Rs. 83.
Summary of Calculations
| Parameter | Value |
|---|---|
| Marked Price (MP) | Rs. 100 |
| Discount % | 10% |
| Discount Amount | Rs. 10 |
| Selling Price (SP) | Rs. 90 |
| Gain % | 8% |
| Cost Price (CP) (Calculated) | Rs. 83.33... |
| Cost Price (CP) (Rounded) | Rs. 83 |
The calculated cost price, rounded to the nearest integer, is Rs. 83.
Revision Table: Key Concepts in Profit and Loss
| Concept | Definition/Formula |
|---|---|
| Marked Price (MP) | The price listed on the tag. |
| Discount | Reduction in Marked Price. Discount = Discount % of MP. |
| Selling Price (SP) | Price at which an item is sold. SP = MP - Discount. Also, SP = CP + Gain or SP = CP - Loss. |
| Cost Price (CP) | The original price at which the shopkeeper bought the goods. |
| Gain or Profit | When SP > CP. Gain = SP - CP. |
| Loss | When SP < CP. Loss = CP - SP. |
| Gain % | $\frac{\text{Gain}}{\text{CP}} \times 100$ |
| Loss % | $\frac{\text{Loss}}{\text{CP}} \times 100$ |
Additional Information on Discount and Gain Calculations
Discounts are always calculated on the Marked Price (MP), unless stated otherwise. Gain or Loss is always calculated on the Cost Price (CP), unless stated otherwise.
The formula $\text{SP} = \text{CP} \times \left(1 + \frac{\text{Gain \%}}{100}\right)$ is a quick way to find the Selling Price if the Cost Price and Gain Percentage are known. Conversely, it can be rearranged to find the Cost Price if the Selling Price and Gain Percentage are known, as demonstrated in this problem: $\text{CP} = \frac{\text{SP}}{\left(1 + \frac{\text{Gain \%}}{100}\right)}$.
Similarly, if there is a loss percentage, the formula is $\text{SP} = \text{CP} \times \left(1 - \frac{\text{Loss \%}}{100}\right)$.
These formulas are very useful for quickly solving profit and loss problems involving percentages.
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