Brief Summary
Alright guys, in this session, we're diving into Profit and Loss, a chapter that can be tricky with its concepts and language. We'll start from the basics, cover various types of questions, including those involving cost price (CP), selling price (SP), marked price (MP), and wrap it up with dishonest shopkeepers. The key takeaways are understanding how to calculate profit and loss, the relationship between CP, SP, and MP, and different methods to solve problems, like basic equations, assuming CP as 100%, and using fractions.
- Understanding of Profit and Loss
- CP, SP, and MP relationship
- Different methods to solve problems
Introduction to Profit and Loss
So, welcome to today's session where we're tackling Profit and Loss. This chapter can be a bit confusing, especially with the language and different concepts involved. But don't worry, by the end of this session, all your problems will be sorted. We'll start with the basics and go through everything in detail. Make sure you stick around till the end. Also, the PDF for this session will be available on the 'Quants by Sumit Sir' Telegram channel.
Topics to be Covered
Today, we're going to cover a few key areas. First, we'll look at the basics of cost price (CP) and selling price (SP). Then, we'll move on to miscellaneous types of profit and loss questions, focusing on the relationship between CP and SP. After that, we'll bring in the marked price (MP) and see how it relates to CP and SP. Finally, we'll wrap up with the concept of dishonest shopkeepers.
Understanding Profit and Loss Basics
Let's start with the basics of profit and loss. Imagine you go to the market and buy an article for ₹500. This ₹500 is your cost price (CP), the price you pay to acquire the article. Now, if you sell that article for ₹1000, that's your selling price (SP). Since you sold it for more than you bought it, you made a profit. Profit is calculated as SP - CP, so in this case, ₹1000 - ₹500 = ₹500 profit.
Calculating Profit Percentage
To find the profit percentage, you use the formula: (Profit / CP) * 100. This means profit percentage is always calculated on the cost price. Remember, even loss percentage is calculated on the CP. So, if you bought something for ₹500 and sold it for ₹1000, your profit percentage would be (₹500 / ₹500) * 100 = 100%.
Understanding Loss
Now, let's talk about loss. Suppose you bought an article for ₹500, but you had to sell it for ₹300. In this case, you've incurred a loss. Loss happens when your cost price (CP) is greater than your selling price (SP). The loss amount is calculated as CP - SP, so ₹500 - ₹300 = ₹200 loss.
Calculating Loss Percentage
To calculate the loss percentage, you use the formula: (Loss / CP) * 100. Just like profit percentage, loss percentage is always calculated on the cost price. So, if you bought something for ₹500 and sold it for ₹300, your loss percentage would be (₹200 / ₹500) * 100 = 40%.
Relationship Between CP, SP, and Profit/Loss
Here's how cost price (CP), selling price (SP), profit, and loss are related: SP = CP + Profit, and SP = CP - Loss. If you add profit to the cost price, you get the selling price. If you subtract loss from the cost price, you also get the selling price.
Understanding Marked Price (MP)
Now, let's introduce marked price (MP), also known as the listed price. This is the price a shopkeeper puts on an article. Imagine a shopkeeper marks an article at ₹500. When you bargain, you might get a discount.
Understanding Discount
Discount is the reduction in price offered by the shopkeeper on the marked price. If the marked price of an article is ₹500 and you get it for ₹150, the discount is ₹500 - ₹150 = ₹350. Discount is always calculated on the marked price.
Calculating Discount Percentage
To calculate the discount percentage, you use the formula: (Discount / MP) * 100. So, if the marked price is ₹500 and you get a discount of ₹350, the discount percentage would be (₹350 / ₹500) * 100 = 70%.
Relating CP, SP, MP, Profit, Loss, and Discount
Let's tie everything together. If you buy an article for ₹1000 (CP) and want to sell it at a profit of ₹500, your selling price (SP) would be ₹1500. Now, if you offer a discount of ₹500 on the marked price (MP), the MP would be ₹2000. So, CP + Profit = SP, and SP + Discount = MP.
Formulas for CP, SP, and MP Relationships
Here are a couple of important formulas to remember the relationship between CP, SP, and MP:
- CP * (100 + Profit%) = SP
- MP * (100 - Discount%) = SP
By equating these two, you can derive a relationship between CP and MP: CP / MP = (100 - Discount%) / (100 + Profit%). If there's a loss instead of profit, the formula becomes: CP / MP = (100 - Discount%) / (100 - Loss%).
Methods to Solve Profit and Loss Questions
There are three main methods to solve profit and loss questions:
- Basic Method: Using equations and variables (like assuming CP as 'x').
- 100% Assumption Method: Assuming the CP as 100%.
- Fraction Method: Using fractions to represent percentages.
The choice of method depends on the question. If the percentages are easy (like 10%, 20%), the 100% assumption method is best. If the percentages are complex (like 14.28%, 33.33%), the fraction method is more suitable.
Question 1: Calculating Profit on Two Items
Two bikes are purchased for ₹40,000 each. One is sold at a profit of ₹8,000. What should be the selling price of the second bike to make an overall profit of 25%?
The total CP of both bikes is ₹80,000. A 25% profit on ₹80,000 is ₹20,000. If the first bike already made a profit of ₹8,000, the second bike needs to make a profit of ₹12,000. So, the selling price of the second bike should be ₹40,000 + ₹12,000 = ₹52,000.
Question 2: Calculating Profit Percentage with Additional Expenses
A person buys a book for ₹1200 and spends ₹450 on binding. If he sells the book for ₹2100, what is his profit percentage?
The total cost price (CP) is ₹1200 (book) + ₹450 (binding) = ₹1650. The selling price (SP) is ₹2100. The profit is ₹2100 - ₹1650 = ₹450. The profit percentage is (₹450 / ₹1650) * 100 = 27.27%.
Question 3: Calculating Increase in Profit Percentage
A person buys an article for ₹1500 and sells it at a 10% profit. By what percentage would the profit increase if he sold it for ₹1950?
First, calculate the selling price at a 10% profit: ₹1500 + (10% of ₹1500) = ₹1650. Now, if he sells it for ₹1950, the profit is ₹1950 - ₹1500 = ₹450. The new profit percentage is (₹450 / ₹1500) * 100 = 30%. The increase in profit percentage is 30% - 10% = 20%.
Question 4: Ratio Based Profit and Loss
A person sells two articles, A and B. On article A, he makes a profit of ₹80, and on article B, he incurs a loss of ₹40. The ratio of the cost prices of A and B is 2:3. What is the loss percentage on article B?
Let the cost price of A be 2x and the cost price of B be 3x. Since the profit on A is ₹80, we can find the value of x. If we assume 2x is 100%, then to get a profit of ₹80, we can multiply by a factor to find the actual cost price. Once we find the cost price of B, we can calculate the loss percentage.
Question 5: Equal Profit and Loss Percentage
By selling an item for ₹2120, a profit is made. The percentage of profit earned is equal to the percentage of loss if it were sold for ₹1520. What price should it be sold at to make a 25% profit?
Let the cost price be 'x'. The profit is (2120 - x) and the loss is (x - 1520). Since the profit percentage and loss percentage are equal, we can equate (2120 - x) / x = (x - 1520) / x. Solving for x gives us the cost price. Then, to make a 25% profit, we need to sell it at 1.25 times the cost price.
Question 6: Article Based Profit and Loss
A person buys 5 lollipops for ₹3 each and another 5 lollipops for ₹4 each. He sells all of them for ₹4 each. Find the overall gain or loss percentage.
The total cost price is (5 * ₹3) + (5 * ₹4) = ₹35. He sells 10 lollipops for ₹4 each, so the total selling price is 10 * ₹4 = ₹40. The profit is ₹40 - ₹35 = ₹5. The profit percentage is (₹5 / ₹35) * 100 = 14.28%.
Question 7: Cycle Based Profit and Loss
A shopkeeper sells an article to P at a 30% profit. P sells it to Q at a 20% profit. If Q pays ₹3120, what was the cost price of the article for the shopkeeper? Also, find the profit amount received by the shopkeeper.
Let the cost price for the shopkeeper be 'x'. The shopkeeper sells it to P at 1.3x. P sells it to Q at 1.2 * 1.3x. We know that 1.2 * 1.3x = ₹3120. Solving for x gives us the cost price for the shopkeeper. The profit for the shopkeeper is 30% of x.
Question 8: Chain Based Profit and Loss
A sells a watch to B at a loss of 8.33%. B sells it to C at a loss of 12.5%. C sells it back to A at a profit of 16.66%. If A initially paid ₹2400 for the watch, how much does A pay to C?
This question involves a chain of transactions with different profit and loss percentages. We need to use fraction values to solve this question.
Question 9: Ratio and Percentage Based Profit and Loss
A person sells noodles at a 100% profit and sugar at a 20% profit. The selling price of sugar and noodles is the same. What is the profit percentage?
Let the cost price of noodles be 'x' and the cost price of sugar be 'y'. The selling price of noodles is 2x and the selling price of sugar is 1.2y. Since the selling prices are the same, 2x = 1.2y. From this, we can find the ratio of x to y. Then, we can calculate the overall profit percentage.
Question 10: Same Selling Price with Equal Profit and Loss Percentage
There are two articles, A and B. The cost price of article A is four times the cost price of article B. If both articles are sold at the same price, and the loss percentage on A is equal to the gain percentage on B, find the loss percentage on article A.
Let the cost price of A be 4x and the cost price of B be x. Let the selling price be 'a'. The loss on A is (4x - a) and the profit on B is (a - x). Since the loss percentage on A is equal to the profit percentage on B, we can equate (4x - a) / 4x = (a - x) / x. Solving for x gives us the cost price. Then, we can calculate the loss percentage.
Question 11: Mixture Based Profit and Loss
Some sarees are sold at a 10% profit, and the remaining sarees are sold at a 20% profit. If the overall profit is 16%, what is the ratio of the sarees sold at 10% profit to those sold at 20% profit?
This is a mixture based question. We can use the allegation method to solve this question.
Question 12: Profit and Discount Based Question
A person buys two items, P and Q, with the same cost price. He labels both items 40% above the cost price. He sells item P at a 30% discount and item Q at ₹35 more than the selling price of item P. In the whole deal, he earns a 5% profit. What was the cost price of item P?
Let the cost price of each item be 'x'. The marked price of each item is 1.4x. The selling price of P is 0.7 * 1.4x = 0.98x. The selling price of Q is 0.98x + ₹35. The total selling price is 0.98x + 0.98x + ₹35 = 1.96x + ₹35. The total cost price is 2x. Since the overall profit is 5%, we can equate 1.05 * 2x = 1.96x + ₹35. Solving for x gives us the cost price of item P.
Question 13: Profit and Loss with Different Rates
A person sells some items at an 18% profit and other items at a 25% profit. If the total profit is 20%, what is the ratio of the quantities of the two types of items?
This is a mixture based question. We can use the allegation method to solve this question.
Question 14: Pencil and Pen Profit and Loss
A shopkeeper sells pencils and pens. If he sells pencils at a profit of 16% and pens at a loss of 20%, he gains ₹12. However, if he sells pencils at a loss of 20% and pens at a profit of 16%, he loses ₹6. Find the cost price of the pencils and pens.
Let the cost price of pencils be '100A' and the cost price of pens be '100B'. In the first case, the profit is 16A - 20B = ₹12. In the second case, the loss is 20A - 16B = ₹6. We have two equations and two variables. Solving these equations will give us the values of A and B, and hence the cost prices of pencils and pens.
Question 15: Jeans Based Profit and Loss
Rajbir purchases a pair of jeans for ₹3200 and sells it at a 10% loss. From that money, he purchases another pair of jeans and sells it at a 10% profit. What is his overall profit or loss?
Rajbir buys jeans for ₹3200. He sells it at a 10% loss, so he gets ₹3200 - (10% of ₹3200) = ₹2880. He buys another pair of jeans for ₹2880 and sells it at a 10% profit, so he gets ₹2880 + (10% of ₹2880) = ₹3168. His overall loss is ₹3200 - ₹3168 = ₹32. The loss percentage is (₹32 / ₹3200) * 100 = 1%.
Understanding Discount Concepts
Let's recap the discount concepts. Marked Price (MP) = Selling Price (SP) + Discount. Discount = MP - SP. SP = MP - Discount. These are just different ways of expressing the same relationship.
Question 16: Shopkeeper Marking Up and Giving Discounts
A shopkeeper marks an item 60% above the cost price. He gives successive discounts of 10% and 20%. What is the profit percentage for the shopkeeper?
Let the cost price be ₹100. The marked price is ₹160. After a 10% discount, the price becomes ₹160 - (10% of ₹160) = ₹144. After a 20% discount, the price becomes ₹144 - (20% of ₹144) = ₹115.2. The profit is ₹115.2 - ₹100 = ₹15.2. The profit percentage is (₹15.2 / ₹100) * 100 = 15.2%.
Question 17: Discount and Profit Percentage
A saree is bought for ₹7200. After giving a discount of 33%, a profit of 34% is still earned. What is the marked price of the saree?
We can use the relationship between CP and MP: CP / MP = (100 - Discount%) / (100 + Profit%). Plugging in the values, we get ₹7200 / MP = (100 - 33) / (100 + 34) = 67 / 134 = 1/2. So, MP = 2 * ₹7200 = ₹14400.
Question 18: Discount and Gift Based Question
A shopkeeper offers a 15% discount and a gift worth ₹600. Overall, he gives a 20% discount and still makes a profit of 20%. What is the cost price if the marked price is ₹12000?
Let the cost price be 'x'. The marked price is ₹12000. The overall discount is 20%, so the selling price is ₹12000 - (20% of ₹12000) = ₹9600. The shopkeeper also gives a gift worth ₹600, so the effective selling price is ₹9600 - ₹600 = ₹9000. Since he makes a 20% profit, we can equate 1.2x = ₹9000. Solving for x gives us the cost price.
Question 19: Free Article Based Question
A shopkeeper marks an article 36% above the cost price. He offers a scheme: on the purchase of every 28 articles at the marked price, he gives 6 articles free. If a customer takes a total of 68 articles (including free articles), find the profit or loss percentage for the shopkeeper.
The shopkeeper marks the price 36% above the cost price. He gives 6 articles free on every 28 articles purchased. This means the customer pays for 28 articles but gets 34 articles. The profit or loss percentage can be calculated by comparing the cost price and the revenue generated.
Homework Question
Here's a homework question for you all. Make sure you comment with your answers. We'll discuss it in the next one-shot session.
