Quantitative Aptitude
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Module 3 · SSC CGL Quantitative Aptitude

Percentage

Percentage change, successive change, profit-loss.
Basic % · Successive Change · Profit-Loss %
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Learning Objectives

  • Understand the fundamental concepts of Percentage
  • Apply key formulas and techniques to solve problems
  • Practice with exam-level questions to build speed and accuracy

Key Concepts

What is Percentage?

Percentage means "per hundred" (parts out of 100). It forms the foundation for profit-loss, simple & compound interest, data interpretation, and many other SSC CGL topics. A percentage is a dimensionless ratio expressed as a fraction of 100.

Basic Formula

Percentage = (Part / Whole) x 100. To convert a fraction to a percentage, multiply by 100. To convert a percentage to a fraction, divide by 100. To express one quantity as a percentage of another: (Quantity / Base) x 100.

Tip 1: Fraction-Percents Are Faster
Avoid calculator decimals in SSC CGL. Memorise fraction equivalents: 12.5% = 1/8, 33.33% = 1/3, 6.25% = 1/16. Using fractions instead of decimals can save 10-15 seconds per question.
Tip 2: x% of y = y% of x
This property — x% of y is equal to y% of x — is an enormous shortcut. Example: 8% of 25 = 25% of 8 = 2. Always swap to the easier number when computing percentages mentally.

Fraction-to-Percent Conversion Table

Master this table cold. SSC CGL repeatedly tests these equivalences.

FractionPercentageFractionPercentage
1/1100%1/119.09%
1/250%1/128.33%
1/333.33%1/137.69%
1/425%1/147.14%
1/520%1/156.67%
1/616.67%1/166.25%
1/714.28%1/175.88%
1/812.5%1/185.56%
1/911.11%1/195.26%
1/1010%1/205%
Tip 3: Reciprocal Trick
To find 14.28% of a number, just divide by 7 (since 14.28% = 1/7). Similarly, 8.33% = divide by 12, 6.25% = divide by 16. This is the "multiply by reciprocal" shortcut — convert the percentage to its fraction form and divide.

Percentage Change

If a value changes from A to B: % increase = ((B - A) / A) x 100. % decrease = ((A - B) / A) x 100. For a decrease, the result is negative; the magnitude gives the decrease percentage.

Tip 4: Multiplier Method
Instead of percentage change formula, use multipliers: a 20% increase means multiply by 1.20. A 15% decrease means multiply by 0.85. Chain multipliers for successive changes: 10% increase then 10% decrease = 1.1 x 0.9 = 0.99 (1% net decrease).
Example 1 (Basic)
What is 12.5% of 640?
Solution: 12.5% = 1/8. So 640 / 8 = 80.
Example 2 (Percentage Increase)
The price of an item increased from Rs 800 to Rs 960. What is the percentage increase?
Solution: Increase = 160. % increase = (160/800) x 100 = 20%.
Example 3 (Percentage Decrease)
A number decreased from 250 to 200. Find the percentage decrease.
Solution: Decrease = 50. % decrease = (50/250) x 100 = 20%.

Successive Percentage Change

When a quantity undergoes two successive changes of x% and y%, the net percentage change is: x + y + (xy/100). Use the formula with signs: + for increase, - for decrease.

Tip 5: Net Change After Two Changes
For two successive discounts of x% and y%, the single equivalent discount is: x + y - (xy/100). Note the minus sign. For successive increases, it is x + y + (xy/100). For an increase followed by a decrease, use x + (-y) + (x*(-y)/100).
Tip 6: Three Successive Changes
For three successive changes x%, y%, z%, apply the formula pairwise: first find net of x and y, then apply with z. Example: 10%, 20%, 30% increases: first net = 10+20+2 = 32%. Then 32+30+(32x30/100) = 32+30+9.6 = 71.6%.
Example 4 (Successive Change)
A number is first increased by 10% and then decreased by 10%. What is the net percentage change?
Solution: Net = 10 + (-10) + (10 x -10 / 100) = 0 - 1 = -1% (1% decrease).
Example 5 (Successive Discounts)
A shopkeeper offers two successive discounts of 20% and 10%. What single discount is equivalent?
Solution: Single discount = 20 + 10 - (20x10/100) = 30 - 2 = 28%.

Population-Based Problems

When a population increases or decreases by a fixed percentage each year, use the compound multiplier method: Population after n years = P x (1 + r/100)^n. For decrease, use (1 - r/100)^n.

Tip 7: Population Reverse Calculation
If the current population is P and it increases by r% annually, the population n years ago was P / (1 + r/100)^n. This is frequently asked in SSC CGL. Example: if present population is 1,21,000 and growth rate is 10%, population 2 years ago = 1,21,000 / (1.21) = 1,00,000.
Tip 8: Net Population After Migration
For problems involving birth rate, death rate, or migration: Net change = (birth rate - death rate + immigration - emigration)%. Apply as a single multiplier.
Example 6 (Population)
The population of a town increases by 10% annually. If the present population is 1,21,000, what was it two years ago?
Solution: Let P be the population two years ago. P x 1.1 x 1.1 = 121000. So P = 121000 / 1.21 = 1,00,000.

Profit & Loss in Percentage Terms

Profit % = (Profit / CP) x 100. Loss % = (Loss / CP) x 100. SP = CP x (1 + Profit%/100) for profit, SP = CP x (1 - Loss%/100) for loss.

Tip 9: CP vs SP Based Percentages
A common SSC CGL trap: "If SP is Rs 120 and profit is 20%, what is CP?" Do NOT do 120 - 20% of 120. Instead, CP = SP / (1 + profit%/100) = 120 / 1.2 = 100. The same applies for loss: CP = SP / (1 - loss%/100).
Tip 10: Profit % on SP vs CP
If profit is given on SP, convert: Profit % on CP = (Profit % on SP) / (100 - Profit % on SP) x 100. For example, 20% profit on SP = 25% profit on CP. This is a very common SSC CGL twist.
Example 7 (Profit %)
A man buys an item for Rs 500 and sells it for Rs 600. What is his profit percentage?
Solution: Profit = 100. Profit % = (100/500) x 100 = 20%.
Example 8 (Profit on SP)
If an article is sold at Rs 120 with a profit of 20% on the selling price, find the cost price.
Solution: Profit = 20% of 120 = 24. CP = SP - Profit = 120 - 24 = 96. Check: Profit % on CP = (24/96) x 100 = 25%.

Election Problems

In elections, percentages are used to distribute votes among candidates. Common scenario: Total votes = Valid votes + Invalid votes. A candidate's vote share is calculated on valid votes unless stated otherwise. Votes can also be "not polled" or "absent."

Tip 11: Winner's Margin
If candidate A gets a% and B gets b% of valid votes, and A wins by M votes, then M = (a - b)% of valid votes. Total valid votes = M / ((a-b)/100). If there are more than 2 candidates, take the difference between top two.
Tip 12: Invalid Votes Adjustment
When a% of total votes are invalid and the winner gets b% of valid votes, first find valid votes = Total x (100-a)/100. Then winner's votes = Valid x b/100.
Example 9 (Election)
In an election, candidate A got 60% of the total valid votes. Total votes were 50,000 and 10% were invalid. How many valid votes did A get?
Solution: Valid votes = 50,000 x 90% = 45,000. A's votes = 60% of 45,000 = 27,000.
Example 10 (Election Margin)
In an election, the winning candidate got 55% of the valid votes and won by 2000 votes. Only two candidates contested and 10% of total votes were invalid. Find total votes.
Solution: Loser got 45% of valid votes. Margin = 10% of valid votes = 2000. So valid votes = 2000 / 0.1 = 20,000. Total votes = 20,000 / 0.9 = 22,222 (approx).

Marks / Score Problems

In exam score problems, percentage = (Marks obtained / Maximum marks) x 100. Passing percentage is calculated on maximum marks. If a student gets a% and fails by b marks, while another gets c% and exceeds passing marks by d marks, set up equations.

Tip 13: Pass Marks Equation
If student 1 gets a% and fails by p marks, and student 2 gets b% and passes by q marks, then: Pass marks = (a% of M) + p = (b% of M) - q. Solve for M (max marks), then find pass marks.
Example 11 (Score)
A student scored 40% marks and failed by 30 marks. Another student scored 50% and got 20 marks more than the passing marks. Find the maximum marks.
Solution: Let max = M. 0.4M = P - 30, 0.5M = P + 20. Subtract: 0.1M = 50 => M = 500.

Mixture Problems in Percentage

When two or more substances are mixed, the percentage concentration of a component in the mixture is (Quantity of component / Total mixture quantity) x 100. Use the alligation method for mixing problems.

Tip 14: Alligation for Percentage Mixtures
To find the ratio in which two mixtures with different concentrations should be mixed to achieve a target concentration: draw the alligation cross. Place the target in the middle, subtract diagonally, and read the ratio.
Tip 15: Repeated Dilution
If a container has V litres of pure liquid and x litres are replaced with water n times, the remaining pure liquid = V x (1 - x/V)^n. The percentage of pure liquid = (1 - x/V)^n x 100%. This formula appears regularly in SSC CGL.
Example 12 (Mixture)
In a 60-litre mixture of milk and water, 20% is water. How much milk should be added so that water becomes 10% of the new mixture?
Solution: Water = 12L. After adding milk, water should be 10% of new total. So 12 = 10% of (60 + x). 12 = 0.1(60+x). 120 = 60+x. So x = 60L of milk.
Example 13 (Alligation)
In what ratio must a 30% sugar solution be mixed with a 50% sugar solution to get a 42% sugar solution?
Solution: Alligation: 30% target 42% with 50%. Differences: 50-42=8, 42-30=12. Ratio = 8:12 = 2:3 (30% : 50%).

SSC CGL Shortcut: Multiply by Reciprocal

The "multiply by reciprocal" trick is a powerful SSC CGL technique. Instead of computing percentages step by step, convert the percentage to its reciprocal fraction and multiply. This works because x% of a number = number x (x/100) = number / (100/x).

Tip 16: Shortcut Table for Speed
Memorise these: 20% = 1/5 = multiply by 0.2 or divide by 5. 25% = 1/4 = divide by 4. 33.33% = 1/3 = divide by 3. 12.5% = 1/8 = divide by 8. 6.25% = 1/16 = divide by 16. 8.33% = 1/12 = divide by 12.
Tip 17: A is what % of B
A is what % of B = (A/B) x 100. A is what % more than B = ((A-B)/B) x 100. A is what % less than B = ((B-A)/B) x 100. The base is always the one after "than". Example: If A = 80 and B = 100, A is 20% less than B.
Tip 18: Value after Multiple Changes
Instead of applying each percentage separately, chain all multipliers: For a 20% increase followed by a 30% increase followed by a 10% decrease: multiplier = 1.2 x 1.3 x 0.9 = 1.404. That is a 40.4% net increase. This approach is much faster than stepwise calculation.
Tip 19: Salary Comparison
If A's salary is a% more than B's, then B's salary is [a/(100+a)] x 100% less than A's. If A's salary is a% less than B's, then B's salary is [a/(100-a)] x 100% more than A's. Example: A is 20% less than B means B is 25% more than A.
Tip 20: Fraction of a Fraction
a% of b% of c% of a number = (a x b x c) / (100 x 100) % of the number. Example: 20% of 30% of 40% of 500 = 0.2 x 0.3 x 0.4 x 500 = 12.
Tip 21: If x% of y% of z is given
If a% of b% of c = d, then c = d x 10000 / (a x b). This type of problem appears often in SSC CGL Tier 1. Solve by treating it as a chain of multiplications.
Tip 22: Compound Growth Shortcut
For a quantity growing at r% per period, the approximate doubling time = 72/r years (Rule of 72). Not directly tested but useful for estimation in DI questions.
Tip 23: Expenditure = Price x Consumption
If price increases by a% and expenditure remains constant, consumption decreases by [a/(100+a)] x 100%. If price decreases by a%, consumption increases by [a/(100-a)] x 100%. This is a classic SSC CGL question type.
Tip 24: Tax / GST Problems
If an item is marked at M, and a GST of r% is applied, the final price = M x (1 + r/100). If the GST-inclusive price is given, the base price = Inclusive price / (1 + r/100).

Solved Examples — Advanced

Example 14 (Population Decrease)
The population of a town decreases by 5% annually. If the current population is 1,80,500, what will it be after 2 years?
Solution: After 2 years = 1,80,500 x 0.95 x 0.95 = 1,80,500 x 0.9025 = 1,62,901 (approx).
Example 15 (Successive Increases)
The salary of a person increased by 10% in the first year, 15% in the second year, and 20% in the third year. What was the net increase after 3 years?
Solution: Net multiplier = 1.1 x 1.15 x 1.20 = 1.518. Net increase = 51.8%. Alternatively, pairwise: 10+15+1.5 = 26.5%. Then 26.5+20+(26.5x20/100) = 26.5+20+5.3 = 51.8%.

8 Advanced Problems

Advanced Problem 1
A's salary is 20% less than B's salary. B's salary is 25% more than C's salary. By what percent is A's salary less than C's salary?
Solution: Let C = 100. Then B = 125. A = 125 x 0.8 = 100. A is 0% less than C. So no difference. This shows how comparative percentages can cancel out.
Advanced Problem 2
The price of sugar increases by 25%. By what percent must a family reduce its consumption so that expenditure remains unchanged?
Solution: Let original price = P, consumption = C, expenditure = PC. New price = 1.25P. For expenditure to remain PC: 1.25P x newC = PC => newC = C/1.25 = 0.8C. Reduction = 20%. Formula: a/(100+a) = 25/125 = 20%.
Advanced Problem 3
If 60% of (x - y) = 30% of (x + y), then what percent of x is y?
Solution: 0.6(x - y) = 0.3(x + y) => 0.6x - 0.6y = 0.3x + 0.3y => 0.3x = 0.9y => x = 3y => y = (1/3)x = 33.33% of x.
Advanced Problem 4
In a mixture of 80 litres, milk and water are in the ratio 3:1. How much water must be added to make the water 25% of the new mixture?
Solution: Milk = 60L, Water = 20L. After adding x L water, water becomes 20+x. Total becomes 80+x. (20+x)/(80+x) = 25/100 = 1/4. Cross-multiply: 80+4x = 80+x => 3x = 0 => x = 0. Wait — the mixture already has 25% water! So no water needs to be added.
Advanced Problem 5
A number is first increased by 25% and then the increased number is decreased by 20%. What is the net change?
Solution: Net = 25 + (-20) + (25 x -20 / 100) = 5 - 5 = 0%. No net change.
Advanced Problem 6
In an examination, 30% of the students failed in English, 25% failed in Maths, and 10% failed in both. What percentage passed in both?
Solution: Failed in at least one = 30 + 25 - 10 = 45%. Passed in both = 100 - 45 = 55%. This uses the inclusion-exclusion principle.
Advanced Problem 7
If the price of an item is first increased by 20% and then decreased by 20%, the final price is Rs 96. What was the original price?
Solution: Let original = P. Final = P x 1.2 x 0.8 = 0.96P = 96 => P = 100.
Advanced Problem 8
A shopkeeper gives a 10% discount on the marked price and still makes a 20% profit. If the marked price is Rs 600, find the cost price.
Solution: SP = 600 x 0.9 = 540. CP = SP / 1.2 = 540 / 1.2 = 450.

10 Common Mistakes

  • Mistake 1: Assuming "20% profit on SP" means the same as "20% profit on CP." They are different. Always check the base.
  • Mistake 2: Applying successive percentage changes additively instead of using the formula x + y + xy/100.
  • Mistake 3: Using the wrong base for percentage difference. "A is what % more than B" uses B as the base. "A is what % less than B" also uses B as the base.
  • Mistake 4: Forgetting to deduct invalid votes before calculating a candidate's vote share in election problems.
  • Mistake 5: Taking percentage of the wrong quantity in mixture problems. Water added to a mixture changes the total quantity.
  • Mistake 6: Confusing percentage change with percentage point change. If a rate increases from 5% to 8%, it is a 3 percentage point increase but a 60% increase.
  • Mistake 7: Using (new - old)/new instead of (new - old)/old for percentage change.
  • Mistake 8: Forgetting to convert between fractions and percentages correctly (e.g., 0.33 does not equal 33.33%).
  • Mistake 9: Not checking for "none of these" as an option when the answer does not match the other choices.
  • Mistake 10: Assuming that a% increase followed by a% decrease brings you back to the original (it does not, unless a = 0).

Pro Tips for SSC CGL

Percentage questions in SSC CGL are designed to test speed as much as accuracy. Here are the most important strategies:

  • Memorise the fraction table from 1/2 to 1/20 cold. This alone can save 30-40 seconds per question.
  • Use the multiplier method for all chain calculations. Never apply percentage changes step-by-step.
  • Start with 100 as the assumed value for unknown quantities. This avoids fractions and reduces calculation errors.
  • Cross-check with reverse calculation when time permits: if you got 80 as 12.5% of 640, verify that 80/640 = 0.125 = 12.5%.
  • Learn the "divides by" trick: To find what percent one number is of another, simplify the fraction first. If A = 125 and B = 500, then A/B = 125/500 = 1/4 = 25%.
Pro Tip: Exam Strategy
Attempt percentage questions first in the quant section — they are among the quickest to solve. If a percentage question takes more than 45 seconds, you are probably missing a shortcut. Move on and come back if needed.

Practice Questions