Profit and Loss

ARITHMETIC HANDBOOK — PROFIT & LOSS

1. Basic Concept

Profit and Loss problems deal with the difference between the Cost Price (CP) and the Selling Price (SP) of an article. These questions are very common in competitive examinations.

Cost Price (CP)

The price at which an article is purchased is called its Cost Price.

Selling Price (SP)

The price at which an article is sold is called its Selling Price.

Profit

When the selling price is greater than the cost price, there is a profit.

Profit = SP − CP
Profit% = (Profit / CP) × 100

Loss

When the selling price is less than the cost price, there is a loss.

Loss = CP − SP
Loss% = (Loss / CP) × 100

2. Important Basic Formulas

If Profit = p%:
SP = CP × (100 + p)/100

If Loss = l%:
SP = CP × (100 − l)/100

If profit is p%:
CP = SP × 100/(100 + p)

If loss is l%:
CP = SP × 100/(100 − l)

3. Solved Examples

Example 1: An article is bought for ₹800 and sold for ₹960. Find the profit and profit percentage.

Profit = 960 − 800 = ₹160
Profit% = (160/800) × 100 = 20%

Example 2: An article costing ₹1,500 is sold at a loss of 12%. Find the selling price.

SP = 1500 × 88/100
SP = ₹1,320

Example 3: An article is sold for ₹1,380 at a profit of 15%. Find its cost price.

CP = 1380 × 100/115
CP = ₹1,200

Exam Tip: Profit or loss percentage is always calculated on the Cost Price, unless the question specifically states otherwise.

PROFIT & LOSS — ADVANCED CONCEPTS

4. Marked Price and Discount

The price written or printed on an article is called the Marked Price (MP) or List Price.

A reduction offered on the marked price is called a Discount.

Discount = MP − SP
Discount% = (Discount / MP) × 100
SP = MP × (100 − Discount%)/100

Example: A shirt is marked at ₹2,000 and a discount of 15% is offered. Find the selling price.

SP = 2000 × 85/100
= ₹1,700

5. Profit After Discount

An important type of question gives both the marked price and discount, while asking for the profit percentage.

Example: An article costs ₹800. It is marked at ₹1,200 and sold after a discount of 10%. Find the profit percentage.

SP = 1200 × 90/100 = ₹1,080
Profit = 1080 − 800 = ₹280
Profit% = (280/800) × 100
= 35%

6. Relation Between CP, SP and MP

Situation	Relation
Profit	SP > CP
Loss	SP < CP
No Profit, No Loss	SP = CP
Discount	SP < MP

7. Successive Discounts

When two or more discounts are given one after another, they are called successive discounts.

Equivalent Discount = a + b − (ab/100)

For three discounts:
Equivalent factor = (1 − a/100)(1 − b/100)(1 − c/100)

Example: Find the equivalent discount of 20% and 10%.

Equivalent Discount = 20 + 10 − (20×10)/100
= 30 − 2
= 28%

8. Important Percentage Relations

If SP is p% more than CP:
SP : CP = (100+p) : 100

If SP is l% less than CP:
SP : CP = (100−l) : 100

Remember: A 20% increase followed by a 20% decrease does NOT bring the value back to the original value.

PROFIT & LOSS — SPECIAL CASES

9. Same Selling Price: Profit and Loss

If two articles are sold at the same selling price, one at a profit of x% and the other at a loss of x%, the overall result is always a loss.

Overall Loss% = x²/100

Example: Two articles are sold for ₹1,000 each. One gives a profit of 20% and the other gives a loss of 20%. Find the overall loss percentage.

Loss% = 20²/100
= 4%

10. Same Cost Price

If two articles have the same cost price and one is sold at a profit of x% and the other at a loss of y%, the net percentage result can be found by comparing the total CP and total SP.

Example: Two articles each cost ₹500. One is sold at 20% profit and the other at 10% loss.

First SP = 500 × 120/100 = ₹600
Second SP = 500 × 90/100 = ₹450
Total CP = ₹1,000
Total SP = ₹1,050
Profit = ₹50
Profit% = 50/1000 × 100 = 5%

11. False Weight and False Measure

Sometimes a seller sells less quantity while claiming to sell the full quantity. This creates an additional gain.

If an article is sold at cost price but only x units are given for every y units charged:

Gain% = [(y − x)/x] × 100

Example: A shopkeeper claims to sell 1 kg but actually gives only 800 g at the cost price. Find the gain percentage.

Gain% = (1000 − 800)/800 × 100
= 200/800 × 100
= 25%

12. Finding Discount for Desired Profit

If the cost price, marked price and desired profit are known, first calculate the required selling price and then find the discount.

Example: An article costs ₹800. It is marked at ₹1,200. What discount should be offered to gain 20%?

Required SP = 800 × 120/100 = ₹960
Discount = 1200 − 960 = ₹240
Discount% = 240/1200 × 100
= 20%

13. Ratio Method

Profit p%:
CP : SP = 100 : (100+p)

Loss l%:
CP : SP = 100 : (100−l)

Example: An article is sold at 25% profit. Find the ratio of CP to SP.

CP : SP = 100 : 125
= 4 : 5

Common Mistake: Do not calculate profit percentage by dividing profit by SP. Standard profit percentage is calculated using CP as the base.

PROFIT & LOSS — FORMULA & PRACTICE

14. Complete Formula Sheet

Concept	Formula
Profit	SP − CP
Loss	CP − SP
Profit%	Profit/CP × 100
Loss%	Loss/CP × 100
SP at p% profit	CP(100+p)/100
SP at l% loss	CP(100−l)/100
CP at p% profit	SP×100/(100+p)
CP at l% loss	SP×100/(100−l)
Discount	MP − SP
Discount%	Discount/MP × 100
Successive discount	a+b−ab/100
Equal SP, ±x%	Loss% = x²/100

15. Solved Revision Examples

Example 1: CP = ₹2,400 and profit = 25%.
SP = 2400 × 125/100 = ₹3,000

Example 2: SP = ₹1,700 and loss = 15%.
CP = 1700 × 100/85 = ₹2,000

Example 3: MP = ₹5,000 with discounts of 20% and 10%.
Equivalent discount = 20 + 10 − 2 = 28%
SP = 5000 × 72/100 = ₹3,600

16. Practice Questions

An article costs ₹600 and is sold for ₹750. Find the profit percentage.
An article is sold for ₹1,440 at a loss of 10%. Find its CP.
An article costing ₹2,500 is sold at a profit of 16%. Find the SP.
A shirt marked at ₹2,000 is sold at a discount of 15%. Find the SP.
Find the equivalent discount of 25% and 20%.
An article is sold for ₹900 at a profit of 20%. Find the CP.
An article costs ₹1,200 and is sold at a loss of 15%. Find the SP.
Two articles are sold at the same price, one at 10% profit and the other at 10% loss. Find the overall loss percentage.
A shopkeeper gives 900 g instead of 1 kg while charging the price of 1 kg at cost price. Find his gain percentage.
An article costs ₹1,000. It is marked at ₹1,500. What discount should be given to earn a profit of 20%?

17. Answers

1. 25%

2. ₹1,600

3. ₹2,900

4. ₹1,700

5. 40%

6. ₹750

7. ₹1,020

8. 1%

9. 11 1/9%

10. 20%

18. Quick Exam Shortcuts

Shortcut 1: If CP = 100 and profit = 20%, take SP = 120. This makes ratio questions very easy.

``` Shortcut 2: If CP = 100 and loss = 20%, take SP = 80.

Shortcut 3: For successive discounts of a% and b%:
Equivalent discount = a + b − ab/100.

Shortcut 4: In equal-SP questions with equal profit and loss percentages, the result is always a loss:
Loss% = x²/100.

Shortcut 5: Always identify the base before calculating a percentage. Profit/Loss uses CP; Discount uses MP. ```

Final Rule: CP is the base for Profit/Loss percentage, while MP is the base for Discount percentage. Remembering this single rule prevents many examination mistakes.