Profit and Loss
Welcome to the topic of Profit and Loss. This is a fundamental concept in arithmetic and is frequently tested in competitive exams. Understanding profit and loss is crucial for making informed financial decisions in everyday life as well. We will cover all aspects of this topic, from basic definitions to complex problems.
1. Basic Concepts and Definitions
Before diving into calculations, let's understand the key terms associated with profit and loss. These terms form the building blocks for solving any problem in this area.
1.1 Cost Price (CP)
The Cost Price is the price at which an article is purchased or manufactured. It's the amount of money spent to acquire an item.
1.2 Selling Price (SP)
The Selling Price is the price at which an article is sold. It's the amount of money received when an item is sold.
1.3 Profit
Profit occurs when the Selling Price (SP) is greater than the Cost Price (CP). The profit is the difference between the SP and CP.
Formula: Profit = Selling Price (SP) - Cost Price (CP)
1.4 Loss
Loss occurs when the Cost Price (CP) is greater than the Selling Price (SP). The loss is the difference between the CP and SP.
Formula: Loss = Cost Price (CP) - Selling Price (SP)
2. Calculating Profit and Loss Percentage
Profit and loss are often expressed as a percentage to give a relative measure of the gain or loss compared to the initial investment (Cost Price).
2.1 Profit Percentage
Profit Percentage is calculated on the Cost Price. If there is a profit, it is expressed as a percentage of the CP.
Formula: Profit Percentage = (Profit / Cost Price) × 100
Substituting the profit formula: Profit Percentage = ((SP - CP) / CP) × 100
2.2 Loss Percentage
Loss Percentage is also calculated on the Cost Price. If there is a loss, it is expressed as a percentage of the CP.
Formula: Loss Percentage = (Loss / Cost Price) × 100
Substituting the loss formula: Loss Percentage = ((CP - SP) / CP) × 100
3. Relationship between CP, SP, Profit, and Loss
We can derive formulas to find SP if CP and profit/loss percentage are known, or to find CP if SP and profit/loss percentage are known.
3.1 Finding Selling Price (SP)
Case 1: When there is a Profit If an article is sold at a profit of P%, then: SP = CP × (100 + P) / 100
Case 2: When there is a Loss If an article is sold at a loss of L%, then: SP = CP × (100 - L) / 100
3.2 Finding Cost Price (CP)
Case 1: When there is a Profit If an article is sold at a profit of P%, then: CP = SP × 100 / (100 + P)
Case 2: When there is a Loss If an article is sold at a loss of L%, then: CP = SP × 100 / (100 - L)
4. Examples and Applications
Let's work through some examples to solidify our understanding.
Example 1: Calculating Profit
A shopkeeper buys a toy for ₹200 and sells it for ₹250. Find the profit and profit percentage.
Given: CP = ₹200, SP = ₹250
Profit = SP - CP = ₹250 - ₹200 = ₹50
Profit Percentage = (Profit / CP) × 100 Profit Percentage = (50 / 200) × 100 Profit Percentage = (1 / 4) × 100 = 25%
So, the profit is ₹50 and the profit percentage is 25%.
Example 2: Calculating Loss
A vendor buys 10 kg of apples for ₹400. He sells 7 kg of apples for ₹350 and the remaining 3 kg for ₹150. Find his overall profit or loss percentage.
Total CP = ₹400
Sales from 7 kg = ₹350
Sales from 3 kg = ₹150
Total SP = ₹350 + ₹150 = ₹500
Since SP (₹500) > CP (₹400), there is a profit.
Profit = Total SP - Total CP = ₹500 - ₹400 = ₹100
Profit Percentage = (Profit / CP) × 100 Profit Percentage = (100 / 400) × 100 Profit Percentage = (1 / 4) × 100 = 25%
The overall profit percentage is 25%.
Example 3: Finding SP given CP and Profit %
A trader buys a table for ₹1500 and wants to make a profit of 20%. What should be the selling price?
Given: CP = ₹1500, Profit % = 20%
SP = CP × (100 + Profit %) / 100 SP = 1500 × (100 + 20) / 100 SP = 1500 × 120 / 100 SP = 1500 × 1.20 SP = ₹1800
The selling price should be ₹1800.
Example 4: Finding CP given SP and Loss %
A shopkeeper sold a watch for ₹720 at a loss of 10%. What was the cost price of the watch?
Given: SP = ₹720, Loss % = 10%
CP = SP × 100 / (100 - Loss %) CP = 720 × 100 / (100 - 10) CP = 720 × 100 / 90 CP = 720 × 10 / 9 CP = 80 × 10 CP = ₹800
The cost price of the watch was ₹800.
5. Problems involving Marked Price and Discount
In retail, items often have a Marked Price (MP) or List Price, which is higher than the CP. A discount is then offered on the MP to arrive at the SP.
5.1 Marked Price (MP)
The Marked Price is the price printed on the product or listed in the catalog. It's usually higher than the CP.
5.2 Discount
Discount is the reduction offered on the Marked Price. It is usually given as a percentage of the MP.
Formula: Discount = Marked Price (MP) - Selling Price (SP)
Formula: Discount Percentage = (Discount / Marked Price) × 100
5.3 Relationship between MP, SP, and Discount
Selling Price (SP) = Marked Price (MP) - Discount
SP = MP - (MP × Discount % / 100)
SP = MP × (100 - Discount %) / 100
5.4 Relationship between CP, SP, Profit/Loss, MP, and Discount
We can combine the concepts. A shopkeeper buys an item at CP, marks it at MP, offers a discount, and sells it at SP. This results in a profit or loss.
If there's a profit of P%, then SP = CP × (100 + P) / 100
If there's a loss of L%, then SP = CP × (100 - L) / 100
Also, SP = MP × (100 - Discount %) / 100
Equating these, we get: CP × (100 + P) / 100 = MP × (100 - Discount %) / 100 (For profit case)
Or, CP × (100 - L) / 100 = MP × (100 - Discount %) / 100 (For loss case)
This leads to a very important ratio: MP / CP = (100 + Profit %) / (100 - Discount %) or MP / CP = (100 - Loss %) / (100 - Discount %)
Example 5: Profit, MP, and Discount
A shopkeeper marks his goods 40% above his cost price and then offers a discount of 20% on the marked price. What is his profit percentage?
Let CP = ₹100
MP is 40% above CP: MP = 100 + (40% of 100) = 100 + 40 = ₹140
Discount is 20% on MP: Discount = 20% of 140 = (20 / 100) × 140 = 0.2 × 140 = ₹28
SP = MP - Discount = 140 - 28 = ₹112
Profit = SP - CP = 112 - 100 = ₹12
Profit Percentage = (Profit / CP) × 100 = (12 / 100) × 100 = 12%
Using the shortcut formula:
CP = 100, Profit % = P, Discount % = 20 MP = CP × (100 + P) / 100 SP = MP × (100 - 20) / 100 = MP × 80 / 100 MP = CP × 1.40 (Since MP is 40% above CP) SP = (CP × 1.40) × 0.80 = CP × 1.12
Since SP = CP × (100 + P) / 100, we have: CP × 1.12 = CP × (100 + P) / 100 1.12 = (100 + P) / 100 112 = 100 + P P = 12%
6. False Weights and Measures
Sometimes, problems involve a seller using faulty weights to cheat customers. This means the weight they claim to sell is different from the actual weight they give.
Example 6: Cheating Seller
A dishonest shopkeeper professes to sell his goods at cost price but uses a weight of 900 grams instead of 1 kilogram. What is his gain percentage?
The shopkeeper claims to sell at CP. This means if the CP of 1 kg is ₹100, he claims to sell for ₹100.
However, he uses a 900-gram weight instead of a 1000-gram weight.
Let the CP of 1 gram be ₹1.
Cost Price for 1000 grams = ₹1000
He sells 900 grams, so his actual cost incurred for the goods sold is the cost of 900 grams. Actual CP = Cost of 900 grams = ₹900
The selling price is what he would charge for 1000 grams if he were selling honestly at CP. So, he charges ₹1000 for the 900 grams he gives. Actual SP = ₹1000
Profit = Actual SP - Actual CP = ₹1000 - ₹900 = ₹100
Profit Percentage = (Profit / Actual CP) × 100 Profit Percentage = (100 / 900) × 100 Profit Percentage = (1 / 9) × 100 = 11.11% (approximately)
Similarly, if a seller claims to sell at a profit of P% but uses a false weight, the calculation becomes more complex.
Example 7: Cheating Seller with Profit Margin
A dishonest dealer sells goods at a loss of 4% on the cost price but uses a weight of 700 grams instead of 1 kg. Find his actual gain percentage.
Let CP of 1000 grams = ₹1000. So, CP of 1 gram = ₹1. CP of 700 grams = ₹700.
The dealer claims to sell at a loss of 4%. So, the SP for 1000 grams would be ₹1000 - 4% of ₹1000 = ₹1000 - ₹40 = ₹960. This means he sells 700 grams for ₹960.
Actual CP for the goods sold = Cost of 700 grams = ₹700. Actual SP for the goods sold = ₹960.
Profit = Actual SP - Actual CP = ₹960 - ₹700 = ₹260.
Actual Gain Percentage = (Profit / Actual CP) × 100 Actual Gain Percentage = (260 / 700) × 100 Actual Gain Percentage = (26 / 70) × 100 = (13 / 35) × 100 Actual Gain Percentage ≈ 37.14%
7. Problems involving buying and selling multiple articles
These problems often involve scenarios where an individual buys two articles for the same price and sells them at different profit/loss percentages, or sells them for the same price but at different profit/loss percentages.
7.1 Case 1: Two articles bought at the same CP and sold at different profit/loss percentages.
If two articles are bought at the same Cost Price and sold at P% profit and L% loss respectively, the overall profit or loss percentage is: Overall Profit/Loss % = (P - L) / 2
If P = L, then there is no overall profit or loss.
7.2 Case 2: Two articles bought at the same CP and sold at the same profit/loss percentage.
If two articles are bought at the same Cost Price and sold at P% profit and P% loss, there is always a loss, and the loss percentage is: Loss % = P2 / 100
This is a very important and commonly tested scenario.
Example 8: Same CP, Different P/L %
A man buys two horses for ₹20,000 each. He sells the first horse at a profit of 10% and the second horse at a loss of 10%. Find his overall profit or loss percentage.
CP of each horse = ₹20,000. Profit on first horse = 10% Loss on second horse = 10%
Since the cost prices are the same and the profit % and loss % are the same (P=L=10%), we can use the shortcut.
Overall Loss % = P2 / 100 = 102 / 100 = 100 / 100 = 1%
There is a net loss of 1%.
Verification: SP of first horse = 20000 × (100 + 10) / 100 = 20000 × 1.1 = ₹22,000 SP of second horse = 20000 × (100 - 10) / 100 = 20000 × 0.9 = ₹18,000 Total CP = 20000 + 20000 = ₹40,000 Total SP = 22000 + 18000 = ₹40,000 Total Profit/Loss = Total SP - Total CP = 40000 - 40000 = ₹0. Wait, the calculation above is incorrect. The rule P2 / 100 applies when the SELLING PRICE is the same, not the Cost Price. Let's correct this.
7.3 Case 3: Two articles sold at the same SP and at different profit/loss percentages.
If two articles are sold at the same Selling Price and one is sold at P% profit and the other at L% loss: Overall Profit/Loss % = (P - L) / 2 (This is incorrect, this formula is for same CP)
If two articles are sold at the same Selling Price and one is sold at P% profit and the other at P% loss, there is ALWAYS a loss, and the loss percentage is: Loss % = P2 / 100
Example 9: Same SP, Same P/L %
A man sells two chairs for ₹1000 each. He sells one at a profit of 25% and the other at a loss of 25%. Find his overall profit or loss percentage.
SP of each chair = ₹1000 Profit on one chair = 25% Loss on other chair = 25%
Since the selling prices are the same and the profit % and loss % are the same (P=L=25%), we use the shortcut.
Overall Loss % = P2 / 100 = 252 / 100 = 625 / 100 = 6.25%
There is a net loss of 6.25%.
Verification: For the first chair (25% profit): CP = SP × 100 / (100 + P%) = 1000 × 100 / 125 = 1000 × 0.8 = ₹800 For the second chair (25% loss): CP = SP × 100 / (100 - L%) = 1000 × 100 / 75 = 1000 × 4 / 3 = ₹4000/3 ≈ ₹1333.33 Total CP = 800 + 4000/3 = (2400 + 4000) / 3 = 6400 / 3 ≈ ₹2133.33 Total SP = 1000 + 1000 = ₹2000 Total Loss = Total CP - Total SP = 6400/3 - 2000 = (6400 - 6000) / 3 = 400 / 3 ≈ ₹133.33 Loss Percentage = (Loss / Total CP) × 100 Loss Percentage = (400/3) / (6400/3) × 100 = 400 / 6400 × 100 = 1 / 16 × 100 = 6.25% The verification confirms the shortcut.
7.4 Case 4: Two articles bought at the same CP and sold at different SPs.
In this case, we need to calculate the individual SPs, sum them up, and compare with the total CP. There's no single shortcut formula.
7.5 Case 5: Two articles sold at the same SP and at different SPs.
In this case, we need to calculate the individual CPs, sum them up, and compare with the total SP. There's no single shortcut formula.
8. Miscellaneous Concepts
8.1 Overhead Charges
Sometimes, in addition to the purchase price, there are other expenses incurred to make the article ready for sale, such as transportation, repairs, taxes, etc. These are called overhead charges.
Formula: Total Cost Price = Purchase Price + Overhead Charges
All profit and loss calculations are then based on this Total Cost Price.
Example 10: Overhead Charges
A dealer buys an article for ₹400 and spends ₹50 on its transportation. He then sells it for ₹550. Find his profit percentage.
Purchase Price = ₹400 Overhead Charges = ₹50 Total CP = 400 + 50 = ₹450 SP = ₹550
Profit = SP - Total CP = 550 - 450 = ₹100
Profit Percentage = (Profit / Total CP) × 100 Profit Percentage = (100 / 450) × 100 Profit Percentage = (10 / 45) × 100 = (2 / 9) × 100 = 22.22%
8.2 Average Cost Price
When items are bought at different prices and sold at a single price, or vice-versa, we might need to calculate an average cost price or average selling price.
Example 11: Average CP
A person buys 3 articles for ₹500 each. He sells the first article at a 10% profit, the second at a 20% profit, and the third at a 30% profit. What is the average profit percentage?
CP of each article = ₹500 Total CP = 3 × 500 = ₹1500
SP of 1st article = 500 × (100 + 10) / 100 = 500 × 1.1 = ₹550 SP of 2nd article = 500 × (100 + 20) / 100 = 500 × 1.2 = ₹600 SP of 3rd article = 500 × (100 + 30) / 100 = 500 × 1.3 = ₹650
Total SP = 550 + 600 + 650 = ₹1800
Total Profit = Total SP - Total CP = 1800 - 1500 = ₹300
Average Profit Percentage = (Total Profit / Total CP) × 100 Average Profit Percentage = (300 / 1500) × 100 = (1 / 5) × 100 = 20%
Note: The average profit percentage is not the simple average of 10%, 20%, and 30% (which is 20%). This is because the profit percentages are applied to the same base cost price. If the selling prices were the same, the calculation would be different.
9. Common Mistakes to Avoid
1. Calculating Percentage on SP: Always calculate profit and loss percentages based on the Cost Price (CP) unless specifically mentioned otherwise. 2. Confusing MP and CP: Remember that MP is the marked price, often higher than CP, used for offering discounts. CP is the actual cost incurred. 3. Forgetting the P2 / 100 Rule: This rule for net loss applies ONLY when two items are sold at the SAME SELLING PRICE and at the SAME PROFIT and LOSS percentages. It does not apply if the cost prices are the same. 4. Ignoring Overhead Charges: If overheads are mentioned, they must be added to the purchase price to get the true Cost Price. 5. Simple Averaging: Don't simply average percentages when dealing with multiple items unless the base values (CP or SP) are identical for all items and the operation (profit/loss) is consistent.