In festival season, a shopkeeper allows a discount of 10% on every item. Even after giving the discount, he makes a profit of 20%. If he does not give any discount, then what will be his profit percent? (correct to 2 decimal places)
33.33
Understanding the Profit and Discount Problem
This problem involves calculating the profit percentage a shopkeeper would make if they stopped offering a discount during a festival season. We are given the discount percentage offered and the profit percentage made even after offering that discount. We need to relate the Cost Price (CP), Selling Price (SP), and Marked Price (MP) to find the new profit percentage.
Setting up the Relationships for Discount and Profit
Let's assume the Marked Price (MP) of an item is $100 for easy calculation. Although not necessary, using a base value can sometimes clarify the steps. Alternatively, we can work directly with variables.
We are given a discount of 10% on the Marked Price (MP). This means the Selling Price (SP) is calculated as:
SP = MP - 10% of MP
SP = MP - $\frac{10}{100} \times$ MP
SP = MP - 0.10 $\times$ MP
SP = 0.90 $\times$ MP
So, the Selling Price is 90% of the Marked Price.
We are also given that the shopkeeper makes a profit of 20% on the Cost Price (CP) even after giving the discount. This means the Selling Price (SP) is calculated based on the Cost Price:
SP = CP + 20% of CP
SP = CP + $\frac{20}{100} \times$ CP
SP = CP + 0.20 $\times$ CP
SP = 1.20 $\times$ CP
So, the Selling Price is 120% of the Cost Price.
Relating Marked Price and Cost Price
Since the Selling Price is the same in both scenarios (with discount and profit), we can equate the two expressions for SP:
0.90 $\times$ MP = 1.20 $\times$ CP
Now, let's express the Marked Price (MP) in terms of the Cost Price (CP):
MP = $\frac{1.20}{0.90} \times$ CP
MP = $\frac{120}{90} \times$ CP
MP = $\frac{12}{9} \times$ CP
MP = $\frac{4}{3} \times$ CP
This relationship tells us that the shopkeeper marks up the price by a factor of $\frac{4}{3}$ from the cost price.
Calculating Profit Without Discount
The question asks for the profit percentage if the shopkeeper does not give any discount. If no discount is given, the customer buys the item at the Marked Price (MP). In this case, the Selling Price (SP) will be equal to the Marked Price (MP).
New Selling Price (SPnew) = MP
We know that MP = $\frac{4}{3} \times$ CP. So, the New Selling Price is:
SPnew = $\frac{4}{3} \times$ CP
Now, we can calculate the profit if the item is sold at this new selling price:
New Profit = SPnew - CP
New Profit = $(\frac{4}{3} \times \text{CP}) - \text{CP}
New Profit = $(\frac{4}{3} - 1) \times \text{CP}
New Profit = $(\frac{4-3}{3}) \times \text{CP}
New Profit = $\frac{1}{3} \times$ CP
Finally, to find the profit percentage without discount, we use the formula:
Profit Percentage = $\frac{\text{New Profit}}{\text{CP}} \times 100$
Profit Percentage = $\frac{\frac{1}{3} \times \text{CP}}{\text{CP}} \times 100$
Profit Percentage = $\frac{1}{3} \times 100$
Profit Percentage = $\frac{100}{3}$
Profit Percentage $\approx$ 33.3333...
Rounding the profit percentage to 2 decimal places, we get 33.33%.
Step-by-Step Calculation Summary
- Identify the given information: Discount = 10%, Profit (with discount) = 20%.
- Express SP in terms of MP using the discount: SP = 0.90 * MP.
- Express SP in terms of CP using the profit: SP = 1.20 * CP.
- Equate the expressions for SP to relate MP and CP: 0.90 * MP = 1.20 * CP.
- Solve for MP in terms of CP: MP = $\frac{1.20}{0.90}$ CP = $\frac{4}{3}$ CP.
- Determine the new SP if no discount is given: SPnew = MP = $\frac{4}{3}$ CP.
- Calculate the new profit: New Profit = SPnew - CP = $\frac{4}{3}$ CP - CP = $\frac{1}{3}$ CP.
- Calculate the new profit percentage: Profit % = $\frac{\text{New Profit}}{\text{CP}} \times 100 = \frac{1/3 \text{ CP}}{\text{CP}} \times 100 = \frac{1}{3} \times 100$.
- Calculate the final percentage: $\frac{100}{3} \approx$ 33.33%.
| Concept | Formula/Relationship | Value (in terms of CP) |
|---|---|---|
| Selling Price (with discount) | SP = MP - Discount | SP = 0.90 $\times$ MP |
| Selling Price (with profit) | SP = CP + Profit | SP = 1.20 $\times$ CP |
| Relationship between MP and CP | Equating SPs | MP = $\frac{4}{3}$ $\times$ CP |
| Selling Price (without discount) | SPnew = MP | SPnew = $\frac{4}{3}$ $\times$ CP |
| New Profit (without discount) | New Profit = SPnew - CP | New Profit = $\frac{1}{3}$ $\times$ CP |
| New Profit Percentage | $\frac{\text{New Profit}}{\text{CP}} \times 100$ | $\frac{1/3 \times \text{CP}}{\text{CP}} \times 100 = 33.33\%$ |
The profit percentage if the shopkeeper does not give any discount is approximately 33.33%.
Revision Table: Key Terms and Concepts
| Term | Definition | Relevance to Problem |
|---|---|---|
| Cost Price (CP) | The original price at which an item is bought or manufactured. | The base for calculating profit. |
| Marked Price (MP) | The price printed on the label or marked on the item. | The price before any discount is applied. |
| Selling Price (SP) | The price at which an item is sold. | Affected by discount and profit. |
| Discount | A reduction in price given on the Marked Price. | Given as 10% in the problem. |
| Profit | The amount gained when SP is greater than CP. | Occurs when SP > CP. |
| Profit Percentage | Profit expressed as a percentage of the Cost Price: $\frac{\text{Profit}}{\text{CP}} \times 100$. | The value we need to calculate without discount. |
Additional Information: Profit, Loss, and Discount Formulas
Understanding the relationship between CP, SP, MP, discount, and profit/loss is crucial for solving such problems. Here are some key formulas:
- Profit = SP - CP (when SP > CP)
- Loss = CP - SP (when CP > SP)
- Profit % = $\frac{\text{Profit}}{\text{CP}} \times 100$
- Loss % = $\frac{\text{Loss}}{\text{CP}} \times 100$
- Discount = MP - SP
- Discount % = $\frac{\text{Discount}}{\text{MP}} \times 100$
- SP = MP - (Discount % of MP)
- SP = MP $\times (1 - \frac{\text{Discount %}}{100})$
- SP = CP + (Profit % of CP)
- SP = CP $\times (1 + \frac{\text{Profit %}}{100})$
- CP = $\frac{100}{(100 + \text{Profit %})} \times$ SP
- CP = $\frac{100}{(100 - \text{Loss %})} \times$ SP
These formulas help link the different prices and percentages, allowing you to solve various problems involving profit, loss, and discount.
Similar Questions
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Important Questions from Discount and MP
If a trader marks the price of an article 50 percent more than the cost price and allows a discount of 30 percent, then what is his profit percentage?
Printed price of a mobile is Rs. 6,400. It is sold in Rs. 2,560 with two consecutive discount. If first discount is 20%, then what is the second discount?
A trader decided to mark his goods 15 percent above the cost price and then offered 40 percent discount. What will be the percentage of profit or loss?
On the sale of 20 percent discount, the value of an item is Rs. 760. What was the real value of the item?
What is the total discount applicable if 3 successive discounts of 20%, 10% and 12.5% are applied?
