Percentage; Profit, Loss and Discount

Percentage

8.1 Basic Concept

Percent (%) = per hundred = /100

x% = x/100

8.2 Conversion Table

Fraction	Percentage
1/2	50%
1/3	33.33%
1/4	25%
1/5	20%
1/6	16.66%
1/7	14.28%
1/8	12.5%
1/9	11.11%
1/10	10%
1/12	8.33%
1/20	5%
1/25	4%

8.3 Key Formulas

To Find	Formula
x% of A	(x/100) × A
A is what % of B	(A/B) × 100
A is x% more than B	A = B × (1 + x/100)
A is x% less than B	A = B × (1 - x/100)
If A is x% more than B, B is _% less than A	[x/(100+x)] × 100
If A is x% less than B, B is _% more than A	[x/(100-x)] × 100

8.4 Successive Percentage Change

If a value changes by x% first then by y%, the net percentage change:

**Net Change = x + y

(xy/100)** %

Example: Price increased by 20% then decreased by 20%:

Net change = 20 + (-20) + (20 × -20)/100 = 0 - 4 = -4% (net decrease of 4%)

8.5 Percentage Increase/Decrease

Percentage Increase = (Increase/Original) × 100
Percentage Decrease = (Decrease/Original) × 100

8.6 Population/Growth Formula

If a population P grows at r% per year for n years:

Final Population = P × (1 + r/100)ⁿ

If it decreases:

Final Population = P × (1 - r/100)ⁿ

8.7 Expenditure = Price × Consumption

If price increases by x% and consumption decreases by y%:

New expenditure = Original × (1

x/100)(1 - y/100)

To keep expenditure same: reduce consumption by [x/(100+x)] × 100 %

Profit, Loss and Discount

9.1 Basic Definitions

Term	Abbreviation	Meaning
Cost Price	CP	Price at which article is bought
Selling Price	SP	Price at which article is sold
Marked Price	MP	Price printed/marked on article
Discount	D	Reduction on Marked Price

9.2 Fundamental Formulas

Condition	Formula
Profit	SP > CP → Profit = SP - CP
Loss	SP < CP → Loss = CP - SP
Profit %	(Profit/CP) × 100
Loss %	(Loss/CP) × 100
SP from CP (Profit)	SP = CP × (100 + P%)/100
SP from CP (Loss)	SP = CP × (100 - L%)/100
CP from SP (Profit)	CP = SP × 100/(100 + P%)
CP from SP (Loss)	CP = SP × 100/(100 - L%)
Discount	Discount = MP - SP
Discount %	(Discount/MP) × 100
SP from MP	SP = MP × (100 - D%)/100

9.3 Important Cases

Case 1: SP = 2CP (SP is double the CP)

Profit = SP - CP = 2CP - CP = CP
Profit% = CP/CP × 100 = 100%

Dishonest Dealer Using False Weights

A dealer cheats using a faulty weight of (1000-error) gm instead of 1000 gm
Profit% = Error/(True weight - Error) × 100

Example: Dealer uses 800g instead of 1000g:

Profit% = 200/800 × 100 = 25%

Selling at Same Price — Two Articles

One at P% profit, other at P% loss
Net Result = P%² / 100 LOSS always (never gain)

Example: Two articles each sold at Rs. 9900, one with 10% profit, other with 10% loss:

Net Loss % = 10²/100 = 1% loss

9.4 Successive Discounts

If two discounts d₁% and d₂% are given on MP:
Net Discount = d₁ + d₂ - (d₁ × d₂/100)
SP = MP × (1 - d₁/100) × (1 - d₂/100)

Example: 20% and 30% successive discount is NOT 50% but:

Net = 20 + 30 - (20×30/100) = 50 - 6 = 44% discount

9.5 Profit When CP is Given as % of SP

If CP = x% of SP, then:
- CP/SP = x/100
- Profit = SP - CP
- Profit% = (SP - CP)/CP × 100 = (100-x)/x × 100