The percentages form an essential part of the competitive examinations and are used extensively in problems related to profit & loss, discount, interest, marks and data interpretation. Rather than remembering formulas, first understand the basic logic and then work on solving complex questions using percentages.

The term “percentage” stands for “out of 100.” Thus, 25% basically implies 25 out of 100, or .

Use of  percentages

To compare quantities with different bases. For example, comparing and is easier if we convert both to percentages: 60% and 70%.

For e.g.
If a shirt costs Rs.800 and a shoe costs Rs.1,200, saying the shirt is 66.66% of the shoe’s price gives an instant sense of proportion.

Fraction 

memorise this fractions it makes calculation faster.

Remember this method :

1. Complement method: If a fraction is close to 1, subtract from 1.This method is use in more/less percentage comparision 
Example: .

2. Splitting method: Break into known parts .use for make calculation easier
Example: .

3. Scaling method: Multiply or divide both fraction and percentage by powers of 10.use to simplify calculations
Example: Since , then and .

Type 1: Find x% of y : 
Formula:

e.g.: 35% of 240 = .

Type 2: a is what % of b : 
Formula:
e.g.: 54 is what % of 90 =  .

Percentage Change and Comparison

Increase by r%: Multiply by .
Decrease by r%: Multiply by .

Example: A phone costing Rs.800 is marked up by 25%, i.e., to Rs.1,000. Then reduced by 20% → Rs.800. It is back to its original cost!

Comparison formulas:

• A is what % more than B  =
• A is what % less than B = 

Example: 90 is what % more than 75 =  .
75 is what % less than 90 =  .

Note: The base is always the number after “than” or “of.”

Successive Percentage Change

When two changes happen one after another, don’t just add or subtract. Use to get net chnages.

AB Method:

Example: +20% then –10%
net increase.

Ratio method: Convert each change to an Old : New ratio, then multiply corresponding terms.

Example: +40% then –20%
+40% → Old : New = 5 : 7
–20% → Old : New = 5 : 4
Old total = 5 × 5 = 25, New total = 7 × 4 = 28 → net change +8 therefore ,net percentage change is 12%

For more than two changes: Multiply the factors.
Example: +10%, –20%, +25% → → 10% net increase.

Real-World Applications

A. Salary, Expenditure, and Savings

When a person spends a percentage of salary, then a percentage of the remaining tems also change.

E.g.
Salary = Rs.8,000. Spends 20% on rent, 25% of the remaining on food, and 10% of the remaining on travel.

B. Election Problems

Elections involve stages: Total voters ⟩ Cast votes ⟩ Valid votes ⟩ Candidate votes.

Formula:

For e.g:
Total voters = 20,000. 80% cast votes, 5% of cast votes are invalid then valid votes are

C. Set Theory and Venn Diagrams

Two subjects:

E.g.
60 students passed Math, 50 passed Science, 20 passed both. Total students = 120.

Total Students:  Math Only:  Science Only: Both Subjects: Passed at least one subject (Total Union): Neither Subject Passed(Fail):  (this is outside the intersecting circles)

Three subjects:

E.g.

Failed in A = 40%

Failed in B = 50%

Failed in C = 60%

Failed in both A & B = 20%

Failed in both B & C = 25%

Failed in both A & C = 30%

Failed in all three (A, B, C) = 10%

Total Fail = (A + B + C) – (A∩B + B∩C + A∩C) + (A∩B∩C)

Calculation:
Total Fail = (40 + 50 + 60) – (20 + 25 + 30) + 10
Total Fail = 150 – 75 + 10 = 85%

Passed All Three:
100% – 85% = 15%

Commission, Income Tax, and Population

Commission:
A salesman gets 5% up to Rs.20,000 and 3% above that what is total commision.

Calculation:

1. First Slab: 5% of 20,000 = 1,000

2. Second Slab: 3% of (40,000 – 20,000) = 3% of 20,000 = 600

Total Commission:
1,000 + 600 = Rs. 1,600

Income Tax:
(Gross – Tax = Net.)
If tax rate increases by 20% and net income decreases by 2%, find original rate.

Population (weighted change):

E.g. Total number of males and females is 10,000. males increases by 15%, and females increases by 5%. and total number increases by 8%. Find the original number of males.

Total = 10,000

Males received a 15% increase, while females received a 5% increase.

If everyone had received only a 5% increase, the total increase would have been:

However, the actual total increase was 8%:

Therefore, the difference between the actual increase and the normal 5% increase is:

This extra 300 came from the males’ additional 10% increase, because males received a 15% increase instead of 5%:

Therefore:

Therefore, the number of males is 3,000.

E. Miscellaneous Problems

Passing marks:
A student got 25% and failed by 20 marks. Another got 40% and passed by 10 marks.

Difference in marks =

That 15% corresponds to marks

Total marks

Pass marks

Answer: 70 marks.

Advanced Variable-Based Problems

Sometimes percentages are given as , , or . Treat them as algebraic expressions and form equations.

e.g 1:
A number increased by becomes 330 from 300.

→ → → .

e.g. 2:
Population 5,000. Increased by then , final 6,006.

so we can Solve the quadratic to find x, or by options method.