Percentage, Profit & Loss, Discount and Marks Problems — Complete TNPSC Aptitude Guide

Percentage-based questions form one of the most heavily weighted topics in TNPSC Aptitude and Mental Ability sections across Group 1, Group 2, Group 2A, Group 4, and VAO examinations. Nearly every other topic in Arithmetic — Profit & Loss, Discount, Simple & Compound Interest, Ratio & Proportion, Data Interpretation, and even Averages — is built on a strong foundation of percentage calculation. This chapter covers the complete syllabus from basic percentage concepts to advanced successive percentage change, profit-loss-discount applications, income-expenditure problems, and marks-based percentage questions, exactly as tested in TNPSC examinations.

Why this topic matters: On average, TNPSC papers carry 3 to 6 questions directly from Percentage and its applications (Profit/Loss, Discount, Marks problems) in every Group exam. Mastering the shortcut methods in this chapter can save you 30-40 seconds per question — a huge advantage in a time-bound exam.

1. Basic Concepts of Percentage

The word "percent" comes from the Latin phrase "per centum," meaning "out of hundred." A percentage is simply a fraction whose denominator is fixed at 100. When we say 40%, we mean 40 out of every 100 parts, or the fraction 40/100.

Definition: Percentage is a way of expressing a number as a fraction of 100. It is denoted by the symbol %.

Form	Conversion Rule	Example
Fraction to Percentage	Multiply the fraction by 100	3/4 × 100 = 75%
Percentage to Fraction	Divide by 100	65% = 65/100 = 13/20
Decimal to Percentage	Multiply by 100	0.45 × 100 = 45%
Percentage to Decimal	Divide by 100	28% = 0.28

1.1 Important Percentage-Fraction Conversion Table (Must Memorise)

This is the single most important table for TNPSC aptitude. Memorising it converts multiplication/division problems into simple mental arithmetic, saving huge amounts of time.

Fraction	Percentage	Fraction	Percentage
1/2	50%	1/9	11.11%
1/3	33.33%	1/10	10%
2/3	66.67%	1/11	9.09%
1/4	25%	1/12	8.33%
3/4	75%	1/13	7.69%
1/5	20%	1/15	6.67%
1/6	16.67%	1/16	6.25%
1/7	14.28%	1/20	5%
1/8	12.5%	1/25	4%

Shortcut Tip: To calculate x% of a number quickly, use the fraction equivalent instead of long multiplication. For example, to find 12.5% of 640, instead of multiplying by 0.125, simply divide 640 by 8 (since 12.5% = 1/8) = 80. This is much faster under exam pressure.

2. Basic Formulas of Percentage

Concept	Formula
x% of a number N	(x/100) × N
What percent is A of B	(A/B) × 100
Finding the number when percentage is known	If x% of N = A, then N = (A × 100)/x
Percentage change	[(New Value − Old Value)/Old Value] × 100

2.1 Beginner-Level Explanation

Suppose a question asks: "What is 20% of 350?" A beginner should think of it step by step: 20% means 20/100. So the answer is (20/100) × 350 = 70. As you practice more, you should be able to see immediately that 20% = 1/5, so 350/5 = 70, without writing a single extra step.

Similarly, if a question asks "45 is what percent of 180?", set up the formula (A/B) × 100 where A = 45 (the part) and B = 180 (the whole): (45/180) × 100 = 25%.

3. Percentage Increase and Decrease

This is one of the most frequently tested applications of percentage in TNPSC exams — questions about population growth, price rise, salary increments, decrease in production, and so on.

Situation	Formula
New value after x% increase	New Value = Old Value × (100 + x)/100
New value after x% decrease	New Value = Old Value × (100 − x)/100
Percentage increase required to restore original value after x% decrease	[x/(100 − x)] × 100
Percentage decrease required to restore original value after x% increase	[x/(100 + x)] × 100

Important Rule (Very Frequently Tested): If the price of an item increases by x%, the consumption must be decreased by [x/(100+x)] × 100 percent to keep expenditure the same. Conversely, if price decreases by x%, consumption can be increased by [x/(100−x)] × 100 percent to keep expenditure constant. This exact formula appears repeatedly in TNPSC Group 2 and Group 4 papers under "Income/Expenditure" type questions.

3.1 Worked Example

Example 1: The population of a town increases from 25,000 to 27,500. Find the percentage increase.

Solution: Percentage increase = [(27500 − 25000)/25000] × 100 = (2500/25000) × 100 = 10%.

Example 2: The price of sugar increases by 25%. By what percent should a family reduce its consumption so that the expenditure remains unchanged?

Solution: Using the formula [x/(100+x)] × 100 = [25/125] × 100 = 20%.

4. Successive Percentage Change

Successive percentage problems involve two or more percentage changes applied one after another — for example, a price increased by 10% and then further increased by 20%, or a value increased by 20% and then decreased by 10%. A very common student mistake is simply adding these percentages together, which gives the wrong answer. This section explains the correct method.

Master Formula for Successive Percentage Change:
If a value undergoes two successive changes of x% and y% (where increase is taken as positive and decrease as negative), the net effective percentage change is:
Net % Change = x + y + (xy/100)

This single formula handles all four combinations:

Combination	Sign Convention	Result Nature
Increase then Increase	Both x and y positive	Net effect always greater than simple sum (compounding effect)
Decrease then Decrease	Both x and y negative	Net effect a larger decrease than simple sum
Increase then Decrease	x positive, y negative	Net effect is always a decrease (never zero, since xy/100 is subtracted from x+y)
Decrease then Increase	x negative, y positive	Same numeric result as increase-then-decrease with same values

4.1 Worked Examples

Example 3: A number is first increased by 20% and then decreased by 10%. Find the net percentage change.

Solution: Net change = 20 + (−10) + [20 × (−10)]/100 = 20 − 10 − 2 = +8% (net increase of 8%).

Example 4: The price of an article is increased by 10% and then again increased by 10%. Find the total percentage increase.

Solution: Net change = 10 + 10 + (10×10)/100 = 10 + 10 + 1 = 21% (NOT 20%, this is the classic trap TNPSC sets).

Common Trap Alert: Students very often assume that two successive 10% increases equal a 20% increase. This is WRONG. The correct answer is always 21% because the second 10% is calculated on the already-increased value, not the original value. TNPSC frequently tests exactly this misconception.

Example 5 (Three successive changes): A value is increased by 10%, then decreased by 20%, then increased by 25%. Find the net effect.

Solution: Apply the formula twice.
Step 1: Combine 10% and −20%: 10 − 20 + (10×−20)/100 = 10 − 20 − 2 = −12%.
Step 2: Combine −12% with +25%: −12 + 25 + (−12×25)/100 = 13 − 3 = +10% net increase.

4.2 Shortcut Using Multiplying Factors (Advanced Method)

For students comfortable with decimals, the multiplying factor method is faster for successive changes involving more than two steps. Convert each percentage change into a multiplying factor: increase of x% → (1 + x/100); decrease of x% → (1 − x/100). Multiply all factors together, then convert back to percentage.

Applying to Example 5: Factors = 1.10 × 0.80 × 1.25 = 1.10 × 1.00 = 1.10, i.e., a net increase of 10%. Note how 0.80 × 1.25 = 1.00 exactly — recognising such combinations instantly (this is a very common TNPSC pairing) saves valuable time.

5. Percentage Comparison

These questions ask you to compare two quantities using percentages — for example, "A's salary is what percent more/less than B's salary?" This is different from a simple percentage-of question because the comparison is relative to one specific quantity (the base), not the total.

Question Type	Formula
A is what % more than B	[(A − B)/B] × 100
A is what % less than B	[(B − A)/B] × 100
If A is x% more than B, then B is less than A by	[x/(100+x)] × 100
If A is x% less than B, then B is more than A by	[x/(100−x)] × 100

Critical Rule to Remember: The base of comparison always changes depending on the direction of the question. "A is x% more than B" uses B as the base. But the reverse statement "B is how much % less than A" uses A as the new base, giving a DIFFERENT numeric answer even though it describes the same real-world situation. This asymmetry is one of the most tested traps in TNPSC exams.

5.1 Worked Example

Example 6: A's income is 25% more than B's income. By what percent is B's income less than A's income?

Solution: Using the shortcut formula: [25/(100+25)] × 100 = [25/125] × 100 = 20%. (Note: it is NOT 25%.)

Example 7: The salary of Raja is Rs. 24,000 and that of Kumar is Rs. 30,000. Kumar's salary is what percent more than Raja's?

Solution: [(30000 − 24000)/24000] × 100 = (6000/24000) × 100 = 25%.

6. Profit and Loss

Profit and Loss is a direct real-world application of percentage concepts to buying and selling transactions. This is one of the highest-weightage topics across all TNPSC Group exams.

6.1 Basic Definitions

Term	Meaning
Cost Price (CP)	The price at which an article is purchased
Selling Price (SP)	The price at which an article is sold
Profit (Gain)	When SP is greater than CP
Loss	When SP is less than CP
Marked Price (MP)	The price labelled on the article before any discount
Overhead Charges	Additional expenses (transport, repair, etc.) added to CP

6.2 Core Formulas

Concept	Formula
Profit	Profit = SP − CP (when SP > CP)
Loss	Loss = CP − SP (when CP > SP)
Profit Percentage	Profit% = (Profit/CP) × 100
Loss Percentage	Loss% = (Loss/CP) × 100
Selling Price (given Profit%)	SP = CP × (100 + Profit%)/100
Selling Price (given Loss%)	SP = CP × (100 − Loss%)/100
Cost Price (given Profit%)	CP = SP × 100/(100 + Profit%)
Cost Price (given Loss%)	CP = SP × 100/(100 − Loss%)

Golden Rule: Profit and Loss percentages are ALWAYS calculated on the Cost Price, never on the Selling Price, unless the question explicitly states otherwise. This is the single most important rule in this chapter and the most common source of error.

6.3 Worked Examples

Example 8: A shopkeeper buys an article for Rs. 800 and sells it for Rs. 920. Find the profit percentage.

Solution: Profit = 920 − 800 = 120. Profit% = (120/800) × 100 = 15%.

Example 9: By selling an article for Rs. 570, a man loses 5%. Find the cost price.

Solution: CP = SP × 100/(100 − Loss%) = 570 × 100/95 = Rs. 600.

6.4 Combined Sale — Equal Selling Price Trick

A very frequently tested pattern: two items are sold at the same selling price, one at x% profit and the other at x% loss. In this case, there is ALWAYS a net loss overall, regardless of the actual selling price.

Shortcut Formula: When two articles are sold at the same SP, one at x% gain and other at x% loss, the overall result is always a loss, and the loss percentage is: Loss% = (x/10)²

Example 10: A man sells two horses at Rs. 1980 each. On one he gains 10% and on the other he loses 10%. Find his overall gain or loss percentage.

Solution: Using the shortcut: Loss% = (10/10)² = 1%. So there is a net loss of 1% (this is independent of the actual price, applicable purely because the percentages 10% and 10% are equal in magnitude).

6.5 False Weight / Dishonest Dealer Problems

A common TNPSC variant involves a shopkeeper claiming to sell at cost price but using a false (lighter) weight to cheat customers, thereby making a hidden profit.

Formula: If a dealer claims to sell at cost price but uses a weight that is less than the true weight, then:
Profit% = [True Weight − False Weight]/[False Weight] × 100

Example 11: A dishonest shopkeeper sells goods at cost price but uses a weight of 900 grams instead of 1 kg. Find his profit percentage.

Solution: Profit% = [(1000 − 900)/900] × 100 = (100/900) × 100 = 11.11%.

7. Discount

Discount problems extend Profit and Loss by introducing the concept of Marked Price — the price printed on the item before any reduction is applied.

Concept	Formula
Discount Amount	Discount = MP − SP
Discount Percentage	Discount% = (Discount/MP) × 100
Selling Price (given Discount%)	SP = MP × (100 − Discount%)/100
Marked Price (given SP and Discount%)	MP = SP × 100/(100 − Discount%)

Critical Distinction: Discount is always calculated on the Marked Price, whereas Profit/Loss is always calculated on the Cost Price. Mixing these two bases is the most common calculation error students make in combined Profit-Discount problems.

7.1 Successive Discounts

Just like successive percentage increases, successive discounts (e.g., "20% off, plus an additional 10% off") do NOT simply add up. Use the same successive percentage formula, treating both discounts as negative values.

Formula: For two successive discounts of x% and y%, the effective single discount is:
Effective Discount% = x + y − (xy/100)

Example 12: Find the single discount equivalent to two successive discounts of 20% and 10%.

Solution: Effective Discount = 20 + 10 − (20×10)/100 = 30 − 2 = 28%. (Note: NOT 30%, and always less than the simple sum.)

7.2 Worked Example Combining MP, CP, Discount, and Profit

Example 13: A trader marks his goods 40% above cost price and allows a discount of 10% on the marked price. Find his profit percentage.

Solution: Let CP = 100. MP = 100 + 40% of 100 = 140. SP = MP − 10% of MP = 140 × 0.90 = 126. Profit = 126 − 100 = 26. Profit% = 26%.

Shortcut using multiplying factors: SP = CP × (1 + markup%) × (1 − discount%). Here, SP = CP × 1.40 × 0.90 = CP × 1.26, directly giving 26% profit — this avoids writing out MP separately and is much faster in the exam.

8. Income and Expenditure Problems

These questions typically describe income and expenditure (or savings) using percentage relationships and ask you to find the missing value. The key relation to remember is:

Savings = Income − Expenditure

Example 14: A man's income is Rs. 40,000 per month and he spends 80% of it. Find his monthly savings.

Solution: Expenditure = 80% of 40000 = 32000. Savings = 40000 − 32000 = Rs. 8,000.

Example 15: A person's income increases by 20% and his expenditure increases by 10%. If his original income was Rs. 25,000 and original expenditure was Rs. 20,000, find the percentage increase in his savings.

Solution: Original savings = 25000 − 20000 = 5000. New income = 25000 × 1.20 = 30000. New expenditure = 20000 × 1.10 = 22000. New savings = 30000 − 22000 = 8000. Percentage increase in savings = [(8000−5000)/5000] × 100 = 60%.

Exam Tip: In income-expenditure problems, always calculate the new income and new expenditure separately first, then find savings by subtraction. Do NOT try to directly apply a percentage change formula to savings itself, since the percentage changes in income and expenditure are different and cannot be combined using the successive percentage formula (that formula only works when the SAME base undergoes successive changes).

9. Marks Problems (Percentage in Examinations)

This is a very TNPSC-specific and frequently tested category involving pass/fail marks, marks obtained versus maximum marks, and comparisons between students.

Concept	Formula
Percentage of marks obtained	(Marks Obtained/Maximum Marks) × 100
Passing marks	(Pass%/100) × Maximum Marks
Marks by which a candidate fails	Passing Marks − Marks Obtained

Example 16: A student secures 35% marks in an exam and fails by 20 marks. If passing marks are 40%, find the maximum marks.

Solution: The difference between pass% and obtained% represents the 20 marks: (40% − 35%) of Maximum Marks = 20. 5% of Maximum Marks = 20. Maximum Marks = 20 × (100/5) = 400.

Example 17: In an examination, a candidate needs 40% marks to pass. A student got 178 marks and failed by 22 marks. Find the maximum marks.

Solution: Passing marks = 178 + 22 = 200. Since 200 marks = 40% of Maximum Marks, Maximum Marks = 200 × (100/40) = 500.

Example 18: Ram scored 20% more marks than Shyam. If Shyam scored 250 marks, find Ram's marks, and find by what percent Shyam's marks are less than Ram's.

Solution: Ram's marks = 250 × 1.20 = 300. Shyam is less than Ram by [20/(100+20)] × 100 = 16.67%.

10. Shortcut Methods and Quick Tricks Summary

Situation	Quick Trick
Finding x% of a large round number	Convert x% into its simplest fraction equivalent and divide instead of multiplying by decimals
Two successive percentage changes	Net% = x + y + xy/100 (use negative sign for decrease)
Equal SP, opposite gain/loss %	Always a net loss of (common%/10)²
Price increase, expenditure constant	Reduce consumption by [x/(100+x)]×100 %
A is x% more than B → B compared to A	B is less than A by [x/(100+x)]×100 %
Successive Discounts	Effective% = x + y − xy/100
Chain of 3+ % changes	Multiply factors: (1±x/100)(1±y/100)(1±z/100)

11. Common Mistakes TNPSC Aspirants Make

• Adding successive percentage changes directly instead of using the correct compounding formula (e.g., assuming two 10% increases = 20%, when the answer is actually 21%).
• Calculating Profit/Loss percentage on Selling Price instead of Cost Price.
• Calculating Discount percentage on Cost Price instead of Marked Price.
• Confusing "A is x% more than B" with "B is x% less than A" — these use different bases and give different numeric answers.
• Forgetting to convert percentage to fraction/decimal correctly before multiplying (e.g., using 5 instead of 0.05 or 5/100).
• In marks problems, forgetting that "fails by n marks" means Passing Marks = Marks Obtained + n, not the other way round.
• Not reading carefully whether a discount is applied to Marked Price or the already-discounted price in multi-step discount problems.

12. TNPSC Previous Year Question Pattern Analysis

Based on the pattern of past TNPSC Group 1, Group 2, Group 2A, and Group 4 papers, questions from this chapter are usually asked in the following styles:

• Direct calculation: "Find x% of a given number" or "What percent is A of B" — appears almost every year in Group 4/VAO level papers as a scoring question.
• Percentage increase/decrease of population, production, price — commonly linked with real Tamil Nadu government data-style scenarios.
• Successive percentage change (two or three steps) — a Group 1/Group 2 favourite, often disguised as "price of an article is increased twice" type questions.
• Profit and Loss with false weight / dishonest dealer — appears very frequently at Group 2 and Group 2A level.
• Discount combined with profit percentage (marked price, cost price relation) — a classic two-in-one question type.
• Income-Expenditure-Savings triangle questions — regularly seen in Group 2 mains-level papers.
• Marks/Pass-Fail percentage problems — a VAO and Group 4 favourite due to relatable context.

13. TNPSC Exam Strategy for This Topic

• Memorise the fraction-percentage conversion table thoroughly — this alone can cut your solving time in half for direct calculation questions.
• Always identify the base (CP for profit/loss, MP for discount, B for "A is % more/less than B" comparisons) before applying any formula.
• For successive percentage/discount questions, train yourself to use the x + y + xy/100 formula instead of adding directly — practice at least 15-20 such questions until it becomes automatic.
• In the exam hall, if time permits, verify your answer using the reverse method (e.g., after finding CP, check whether applying the given profit% actually gives back the SP mentioned in the question).
• Practice mental percentage calculations daily (10%, 20%, 25%, 5%, 12.5% of numbers) to build calculation speed, since TNPSC papers are strictly time-bound.
• Attempt marks-based and income-expenditure questions after profit-loss and discount, since they combine multiple percentage concepts and are best solved once the basics are fully automatic.

14. Model Multiple Choice Questions (MCQs) with Solutions

#	Question	Options	Answer with Solution
1	What is 15% of 640?	(a) 86 (b) 96 (c) 106 (d) 90	(b) 96 — 15% of 640 = (15/100)×640 = 96
2	If the price of an item is increased by 20% and then decreased by 20%, the net effect is:	(a) No change (b) 4% increase (c) 4% decrease (d) 40% decrease	(c) 4% decrease — Net = 20 − 20 + (20×−20)/100 = −4%
3	A man buys a table for Rs. 1,200 and sells it for Rs. 1,320. His profit percentage is:	(a) 8% (b) 10% (c) 12% (d) 15%	(b) 10% — Profit = 120, Profit% = (120/1200)×100 = 10%
4	The marked price of an article is Rs. 500. A discount of 15% is given. Find the selling price.	(a) Rs. 425 (b) Rs. 435 (c) Rs. 450 (d) Rs. 480	(a) Rs. 425 — SP = 500×0.85 = 425
5	A's salary is 20% more than B's. B's salary is what percent less than A's?	(a) 20% (b) 16.67% (c) 25% (d) 15%	(b) 16.67% — Using [x/(100+x)]×100 = [20/120]×100 = 16.67%
6	By selling a fan for Rs. 475, a shopkeeper loses 5%. What is the cost price?	(a) Rs. 490 (b) Rs. 495 (c) Rs. 500 (d) Rs. 510	(c) Rs. 500 — CP = 475×100/95 = 500
7	Two successive discounts of 10% and 20% are equal to a single discount of:	(a) 30% (b) 28% (c) 26% (d) 32%	(b) 28% — 10+20−(10×20)/100 = 30−2 = 28%
8	A candidate needs 33% marks to pass. He gets 125 marks and fails by 40 marks. Find the maximum marks.	(a) 450 (b) 480 (c) 500 (d) 520	(c) 500 — Passing marks = 165; 33% of Max = 165; Max = 500

15. Important Observations for Quick Revision

• Percentage is always relative to a base — always identify the base before calculating.
• Profit/Loss% → always on CP. Discount% → always on MP.
• Successive % changes are never simply additive; always use x + y + xy/100.
• "More than" and "less than" comparisons are NOT symmetric — the base shifts depending on direction.
• Equal SP with equal and opposite profit/loss % always results in a net LOSS of (common%/10)².
• In price-consumption-expenditure constant problems, use [x/(100±x)]×100 depending on direction of price change.

16. Final Revision Points

1. Percentage = fraction with denominator 100.
2. x% of N = (x/100) × N.
3. Percentage Change = [(New − Old)/Old] × 100.
4. Successive % Change = x + y + xy/100 (use correct signs).
5. Profit% = (Profit/CP)×100; Loss% = (Loss/CP)×100.
6. Discount% = (Discount/MP)×100.
7. SP = CP×(100±%)/100 for profit/loss; SP = MP×(100−Discount%)/100 for discount.
8. Dishonest dealer profit% = [(True Wt − False Wt)/False Wt]×100.
9. Savings = Income − Expenditure (calculate new income and expenditure separately, then subtract).
10. Marks problems: Passing Marks = Marks Obtained + Marks failed by.

17. Conclusion

Percentage and its applications — Profit & Loss, Discount, Income-Expenditure, and Marks problems — together form one of the most scoring and consistently tested chapters in TNPSC Aptitude examinations for Group 1, Group 2, Group 2A, Group 4, and VAO posts. The concepts are not individually difficult, but the successive-change and base-comparison traps are specifically designed to test careful reading and formula application rather than raw calculation skill. Aspirants who master the fraction-percentage table, internalise the correct bases for each formula (CP for profit/loss, MP for discount), and practise the successive percentage formula until it becomes automatic will be able to solve nearly every question in this chapter within 30-45 seconds — a decisive advantage in the time-constrained TNPSC examination environment. Regular practice with previous year questions in this exact pattern is strongly recommended before attempting the final exam.