Profit and Loss

CP, SP, discount, marked price

Profit and Loss is a fundamental commercial mathematics topic that deals with calculating gains or losses incurred in business transactions. It involves understanding the relationship between Cost Price (CP), Selling Price (SP), Marked Price (MP), and various types of discounts. Mastery of this topic requires familiarity with percentage calculations and the ability to solve problems involving successive profits, losses, and combined transactions.

Key Formulas

\text{Profit = Selling Price (SP) - Cost Price (CP)}

\text{Loss = Cost Price (CP) - Selling Price (SP)}

\text{Profit \% = (Profit / CP) x 100}

\text{Loss \% = (Loss / CP) x 100}

\text{Selling Price = CP x (100 + Profit\%) / 100}

\text{Selling Price = CP x (100 - Loss\%) / 100}

\text{Cost Price = (SP x 100) / (100 + Profit\%)}

\text{Cost Price = (SP x 100) / (100 - Loss\%)}

\text{Discount = Marked Price - Selling Price}

\text{Discount \% = (Discount / MP) x 100}

\text{Selling Price = MP x (100 - Discount\%) / 100}

Key Concepts

Cost Price (CP) and Selling Price (SP)

Cost Price is the amount paid to purchase or manufacture an article. It includes all expenses incurred before the item is ready for sale. Selling Price is the amount at which an article is sold to the customer. The difference between SP and CP determines whether a transaction results in profit or loss. When SP > CP, there is a profit; when SP < CP, there is a loss.

Marked Price (MP) and Discount

Marked Price, also called List Price, is the price tagged on an article by the seller. It is usually higher than the expected selling price to allow room for discounts. Discount is the reduction given on the Marked Price to attract customers. The actual selling price is calculated after deducting the discount from the marked price: SP = MP - Discount.

Successive Profits and Losses

When multiple profits or losses occur in sequence, they are compounded. For successive profits of a% and b%, the overall profit percentage is calculated as: (a + b + ab/100)%. Similarly, for successive discounts of a% and b%, the equivalent single discount is: (a + b - ab/100)%. When there is a profit followed by a loss (or vice versa), the formula becomes (a - b - ab/100)% or (-a + b - ab/100)% depending on which is larger.

False Weight Problems

In false weight problems, a merchant claims to sell goods at a certain weight but actually uses a defective scale. If a seller claims to sell at cost price but uses a weight that is x% less than the actual weight, their profit percentage is [x / (100 - x)] x 100%. This concept tests understanding of how measurement errors translate into hidden profits.

Overall Profit/Loss in Multiple Transactions

When dealing with multiple articles bought or sold together, calculate the total cost price and total selling price separately. The overall profit or loss is found by comparing these totals. For transactions where some items are sold at profit and others at loss, the net result depends on the quantities and individual percentages. Always verify whether the final outcome is profit or loss by comparing total CP and total SP.

Solved Examples

Problem 1:

A merchant bought 50 kg of rice at Rs. 40 per kg. He sold 30 kg at Rs. 45 per kg and the remaining 20 kg at Rs. 38 per kg. Find his overall profit or loss percentage.

Solution:

Step 1: Calculate Total Cost Price
CP = 50 kg x Rs. 40/kg = Rs. 2000

Step 2: Calculate Total Selling Price
SP of first batch = 30 kg x Rs. 45/kg = Rs. 1350
SP of second batch = 20 kg x Rs. 38/kg = Rs. 760
Total SP = Rs. 1350 + Rs. 760 = Rs. 2110

Step 3: Determine Profit/Loss
Since SP > CP, there is a profit.
Profit = Rs. 2110 - Rs. 2000 = Rs. 110

Step 4: Calculate Profit Percentage
Profit % = (110 / 2000) x 100 = 5.5%

Answer: The merchant made an overall profit of 5.5%.

Problem 2:

A shopkeeper marks his goods 25% above the cost price and then offers a discount of 10%. What is his actual profit percentage?

Solution:

Step 1: Assume Cost Price
Let CP = Rs. 100 (for easy calculation)

Step 2: Calculate Marked Price
MP = CP x (100 + 25%) / 100 = 100 x 1.25 = Rs. 125

Step 3: Calculate Selling Price after Discount
Discount = 10% of MP = 0.10 x 125 = Rs. 12.50
SP = MP - Discount = 125 - 12.50 = Rs. 112.50

Step 4: Calculate Profit
Profit = SP - CP = 112.50 - 100 = Rs. 12.50

Step 5: Calculate Profit Percentage
Profit % = (12.50 / 100) x 100 = 12.5%

Alternative method using successive percentages:
Profit % = 25 - 10 - (25 x 10)/100 = 15 - 2.5 = 12.5%

Answer: The shopkeeper's actual profit percentage is 12.5%.

Problem 3:

A trader sells two articles at the same selling price of Rs. 1200 each. On one he gains 20% and on the other he loses 20%. Find his overall profit or loss percentage.

Solution:

Step 1: Calculate Cost Price of First Article (Profit 20%)
CP = (SP x 100) / (100 + Profit%) = (1200 x 100) / 120 = Rs. 1000

Step 2: Calculate Cost Price of Second Article (Loss 20%)
CP = (SP x 100) / (100 - Loss%) = (1200 x 100) / 80 = Rs. 1500

Step 3: Calculate Total CP and Total SP
Total CP = 1000 + 1500 = Rs. 2500
Total SP = 1200 + 1200 = Rs. 2400

Step 4: Determine Overall Result
Since Total CP > Total SP, there is a loss.
Loss = 2500 - 2400 = Rs. 100

Step 5: Calculate Loss Percentage
Loss % = (100 / 2500) x 100 = 4%

Note: When SP is same and profit% = loss%, the overall result is always a loss of (p^2/100)% where p is the profit/loss percentage.
Here: Loss % = (20 x 20) / 100 = 4%

Answer: The trader suffers an overall loss of 4%.

Problem 4:

A dishonest shopkeeper claims to sell rice at cost price but uses a weight of 900 grams instead of 1 kg. What is his profit percentage?

Solution:

Step 1: Understand the Problem
The shopkeeper gives only 900g but charges for 1000g (1 kg).

Step 2: Assume Cost Price per gram
Let CP of 1 gram = Re. 1
Then CP of 900 grams = Rs. 900
CP of 1000 grams = Rs. 1000

Step 3: Determine Selling Price
Since he claims to sell at cost price,
SP of 1000g (which he claims to sell) = Rs. 1000
But he actually gives only 900g, so his actual CP = Rs. 900

Step 4: Calculate Profit
Profit = SP - Actual CP = 1000 - 900 = Rs. 100

Step 5: Calculate Profit Percentage
Profit % = (Profit / Actual CP) x 100 = (100 / 900) x 100 = 100/9 % = 11.11% (approx)

Alternative formula:
Profit % = [Error / (True Value - Error)] x 100
= [100 / (1000 - 100)] x 100 = (100 / 900) x 100 = 11.11%

Answer: The shopkeeper's profit percentage is 11.11% or 100/9 %.

Tips & Tricks

Always identify and note down CP, SP, MP, and any discounts given in the problem before starting calculations.
When no specific values are given, assume CP = 100 for easier percentage calculations.
For successive profit/loss problems, use the formula (a + b + ab/100) or its variations rather than calculating step by step.
In false weight problems, remember the shortcut: Profit% = [Error / (True - Error)] x 100, where Error = True Weight - False Weight.
When two items are sold at the same price with equal profit and loss percentages, there is always an overall loss of (p^2/100)%.
For complex transactions involving multiple articles, calculate total CP and total SP separately rather than working with individual profit/loss percentages.

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