Percentage Problems Explained — Formulas, Tricks and Interview Questions
Percentages show up everywhere — your salary hike, your bank's interest rate, election results, discount offers, your company's growth report. If numbers describe the world, percentages are how we compare and communicate those numbers in a way every human on the planet instantly understands. That's why every aptitude test — from campus placements to UPSC to big-tech hiring rounds — opens with percentage problems. They test whether you can think proportionally under pressure, which is exactly what real business decisions demand.
The frustrating thing is that most people know what a percentage IS but freeze the moment a question twists the idea slightly — 'what is the original price after a 20% increase?' or 'by what percent is A's salary more than B's?' These aren't hard problems. They just need one clear mental model and three or four formulas that you actually understand, not just memorise.
By the end of this article you'll be able to calculate any percentage value, find the original number before a percentage change, solve percentage increase and decrease problems, handle the classic 'successive percentage change' trap, and answer the trickiest interview percentage questions without a calculator. We'll build every formula from scratch using plain English before showing you the math — because once you see where a formula comes from, you never forget it.
The Foundation: What 'Percent' Actually Means and the One Core Formula
The word 'percent' breaks into two parts: 'per' (for every) + 'cent' (hundred). So 40% simply means 40 out of every 100. Nothing more.
Here's the single formula that every percentage calculation comes from:
Percentage = (Part / Whole) × 100
Flip it around and you get the two other forms you'll use constantly:
Part = (Percentage / 100) × Whole Whole = (Part / Percentage) × 100
Think of it as a triangle. Cover the thing you want to find — the other two pieces of the triangle tell you how to get it.
Example: In a class of 60 students, 45 passed. What percentage passed? Percentage = (45 / 60) × 100 = 75%
Example: 35% of what number is 84? Whole = (84 / 35) × 100 = 240
Example: What is 15% of 340? Part = (15 / 100) × 340 = 51
All three are the same formula, just rearranged. Burn this triangle into memory — every single percentage problem is one of these three questions in disguise.
═══════════════════════════════════════════════════ CORE PERCENTAGE FORMULA — ALL THREE FORMS ═══════════════════════════════════════════════════ FORM 1 — Find the Percentage Formula : Percentage = (Part / Whole) × 100 Problem : 18 out of 72 students are absent. What % are absent? Step 1 : Part = 18 (the piece we're looking at) Step 2 : Whole = 72 (the total group) Step 3 : Percentage = (18 / 72) × 100 = 0.25 × 100 = 25% Answer : 25% of students are absent. ─────────────────────────────────────────────────── FORM 2 — Find the Part (most common in exams) Formula : Part = (Percentage / 100) × Whole Problem : What is 35% of 480? Step 1 : Percentage = 35 Step 2 : Whole = 480 Step 3 : Part = (35 / 100) × 480 = 0.35 × 480 = 168 Answer : 35% of 480 = 168 ─────────────────────────────────────────────────── FORM 3 — Find the Whole (tricky in interviews) Formula : Whole = (Part / Percentage) × 100 Problem : 63 is 45% of what number? Step 1 : Part = 63 Step 2 : Percentage = 45 Step 3 : Whole = (63 / 45) × 100 = 1.4 × 100 = 140 Answer : 63 is 45% of 140 VERIFY : 45% of 140 = (45/100) × 140 = 63 ✓
Form 2 Answer : 35% of 480 = 168
Form 3 Answer : 63 is 45% of 140
Verification : 45% of 140 = 63 ✓
Percentage Increase and Decrease — The Direction Trap That Fools Everyone
This is where 80% of interview candidates slip up — not because the math is hard, but because they apply the percentage in the wrong direction.
The core formula is:
Percentage Change = ((New Value − Old Value) / Old Value) × 100
If the result is positive → it's an increase. Negative → a decrease.
Always divide by the OLD (original) value, never the new one. That's the trap.
Now let's flip it. Suppose you know the original value and the percentage change, and you want the new value:
After an increase of R%: New Value = Old Value × (1 + R/100) After a decrease of R%: New Value = Old Value × (1 − R/100)
The multiplier (1 + R/100) or (1 − R/100) is your best friend. It lets you jump straight to the answer in one step.
And if you know the new value but want the original:
Original = New Value / (1 + R/100) [after increase] Original = New Value / (1 − R/100) [after decrease]
This reverse-calculation is the most common 'hard question' in aptitude tests. Once you have the multiplier idea, it's just division.
═══════════════════════════════════════════════════ PERCENTAGE INCREASE & DECREASE — WORKED EXAMPLES ═══════════════════════════════════════════════════ PROBLEM 1 — Find the % Change A phone costs ₹12,000. After a price hike it now costs ₹13,800. By what percentage did the price increase? Formula : % Change = ((New − Old) / Old) × 100 Step 1 : New = 13800, Old = 12000 Step 2 : Change = 13800 − 12000 = 1800 Step 3 : % Change = (1800 / 12000) × 100 = 0.15 × 100 = 15% Answer : The price increased by 15%. ─────────────────────────────────────────────────── PROBLEM 2 — Find the New Value after a Change A salary of ₹45,000 is increased by 12%. What is the new salary? Multiplier Method (fastest): New Salary = 45000 × (1 + 12/100) = 45000 × 1.12 = ₹50,400 Verify: 12% of 45000 = 5400 → 45000 + 5400 = 50400 ✓ ─────────────────────────────────────────────────── PROBLEM 3 — Find the ORIGINAL value (the interview trap) After a 20% price reduction, a laptop costs ₹36,000. What was the original price? Common WRONG approach (don't do this!): Wrong: Original = 36000 + 20% of 36000 = 36000 + 7200 = 43200 ✗ Why wrong? The 20% was applied to the ORIGINAL price, not ₹36,000. Correct approach: New Price = Original × (1 − 20/100) 36000 = Original × 0.80 Original = 36000 / 0.80 = ₹45,000 Verify: 20% of 45000 = 9000 → 45000 − 9000 = 36000 ✓ Answer: The original price was ₹45,000. ─────────────────────────────────────────────────── PROBLEM 4 — % Decrease A stock fell from ₹250 to ₹190. What is the % decrease? % Change = ((190 − 250) / 250) × 100 = (−60 / 250) × 100 = −24% Answer : The stock decreased by 24%.
Problem 2 Answer : New salary = ₹50,400
Problem 3 Answer : Original price = ₹45,000
Problem 4 Answer : Stock decreased by 24%
Successive Percentage Changes — Why 10% + 10% is NOT 20%
Here's a classic interview trick: 'A value is increased by 10% and then increased again by 10%. What is the total percentage increase?' Most people say 20%. The real answer is 21%. Let's see why.
When you apply two percentage changes one after another, the second change is applied to the ALREADY-CHANGED value, not the original. So the percentages don't simply add up.
The formula for two successive percentage changes of A% and B%:
Net % Change = A + B + (A × B) / 100
The extra term (A × B / 100) is the 'compounding effect' — the percentage of a percentage.
For three or more changes, just chain the multipliers: Final Value = Original × (1 + A/100) × (1 + B/100) × (1 + C/100)
Where negative values represent decreases.
This is the secret behind why compound interest beats simple interest, why back-to-back discounts work differently than a single combined discount, and why a 50% loss followed by a 50% gain still leaves you below your starting point. Once you see the compounding pattern, these problems become straightforward.
═══════════════════════════════════════════════════ SUCCESSIVE PERCENTAGE CHANGES — THE COMPOUNDING TRICK ═══════════════════════════════════════════════════ PROBLEM 1 — Classic Interview Question A population of 20,000 increases by 10% in year 1 and increases by 10% in year 2. What is the total % increase? What is the final population? WRONG (naive) answer: 10 + 10 = 20% → 24,000 CORRECT — Multiplier Method: After Year 1 : 20000 × 1.10 = 22,000 After Year 2 : 22000 × 1.10 = 24,200 Net % Change using formula: = A + B + (A × B) / 100 = 10 + 10 + (10 × 10) / 100 = 20 + 1 = 21% Verify: 21% of 20000 = 4200 → 20000 + 4200 = 24,200 ✓ ─────────────────────────────────────────────────── PROBLEM 2 — Increase then Decrease (very common in exams) A shirt's price is increased by 30% and then decreased by 20%. What is the net change? Formula : Net % = A + B + (A × B)/100 Here A = +30, B = −20 Net % = 30 + (−20) + (30 × (−20)) / 100 = 30 − 20 + (−600) / 100 = 10 − 6 = +4% Answer: Net increase of 4%. Verify with ₹100 base: After 30% increase : 100 × 1.30 = ₹130 After 20% decrease : 130 × 0.80 = ₹104 Change : (104 − 100) = ₹4 = 4% increase ✓ ─────────────────────────────────────────────────── PROBLEM 3 — The Painful Trap (50% loss then 50% gain) An investor's portfolio loses 50%, then gains 50%. Are they back to even? Net % = (−50) + 50 + ((−50) × 50) / 100 = 0 + (−2500 / 100) = −25% Verify with ₹10,000: After 50% loss : 10000 × 0.50 = ₹5,000 After 50% gain : 5000 × 1.50 = ₹7,500 Net change : −₹2,500 = −25% loss Answer: No! They are DOWN 25%, not back to even. ─────────────────────────────────────────────────── PROBLEM 4 — Three Successive Changes A property value rises 20%, then rises 15%, then falls 10%. Find the net % change from an original value of ₹5,00,000. Final Value = 500000 × 1.20 × 1.15 × 0.90 = 500000 × 1.242 = ₹6,21,000 Net % Change = ((621000 − 500000) / 500000) × 100 = (121000 / 500000) × 100 = 24.2% Answer: Net increase of 24.2%. Final value = ₹6,21,000.
Problem 2: Net change = +4% increase
Problem 3: Net change = -25% (still a loss!)
Problem 4: Net change = +24.2%, Final value = ₹6,21,000
Percentage Comparison Problems — 'More Than' vs 'Less Than' vs 'Of'
One of the most misread question types in aptitude tests involves comparative percentages. The difference between 'A is what percent MORE than B' and 'A is what percent OF B' is not just grammar — the answers are completely different numbers.
Here are the four question types you'll see:
Type 1 — 'A is what % of B?' Answer = (A / B) × 100
Type 2 — 'A is what % more than B?' Answer = ((A − B) / B) × 100 (You divide by B because B is the base/reference)
Type 3 — 'A is what % less than B?' Answer = ((B − A) / B) × 100 (Still divide by B — it's always the reference value)
Type 4 — 'By what % should A be increased to equal B?' Answer = ((B − A) / A) × 100 (Now A is the base because you're changing A)
The golden rule: the denominator is always the BASE — the thing you're comparing FROM or changing FROM. Read the question carefully to identify what the 'reference' is.
This catches candidates constantly because when you're not sure, your instinct picks the wrong denominator and your answer is off.
═══════════════════════════════════════════════════ PERCENTAGE COMPARISON — ALL FOUR QUESTION TYPES ═══════════════════════════════════════════════════ Setup: Ravi earns ₹40,000/month. Priya earns ₹50,000/month. ─────────────────────────────────────────────────── TYPE 1 — 'Ravi's salary is what % of Priya's?' Formula : (Ravi / Priya) × 100 Answer : (40000 / 50000) × 100 = 80% Reading : Ravi earns 80% of what Priya earns. ─────────────────────────────────────────────────── TYPE 2 — 'Priya's salary is what % MORE than Ravi's?' Base : Ravi (we are comparing TO Ravi's salary) Formula : ((Priya − Ravi) / Ravi) × 100 Answer : ((50000 − 40000) / 40000) × 100 = (10000 / 40000) × 100 = 25% Reading : Priya earns 25% MORE than Ravi. Note : NOT 20%! The base is Ravi's salary (40k), not Priya's. ─────────────────────────────────────────────────── TYPE 3 — 'Ravi's salary is what % LESS than Priya's?' Base : Priya (we are comparing TO Priya's salary) Formula : ((Priya − Ravi) / Priya) × 100 Answer : ((50000 − 40000) / 50000) × 100 = (10000 / 50000) × 100 = 20% Reading : Ravi earns 20% LESS than Priya. ★ KEY INSIGHT: 'MORE than' and 'LESS than' give DIFFERENT answers for the same two numbers (25% vs 20%) because the BASE changes. ─────────────────────────────────────────────────── TYPE 4 — 'By what % must Ravi's salary increase to match Priya's?' Base : Ravi (he is the one changing) Formula : ((Target − Current) / Current) × 100 Answer : ((50000 − 40000) / 40000) × 100 = 25% Reading : Ravi needs a 25% raise to match Priya. Verify : 40000 × 1.25 = 50000 ✓ ─────────────────────────────────────────────────── SUMMARY TABLE: Question Type | Formula | Answer ─────────────────────────|──────────────────────|─────── What % OF Priya? | (40k/50k) × 100 | 80% What % MORE than Ravi? | (10k/40k) × 100 | 25% What % LESS than Priya? | (10k/50k) × 100 | 20% % increase needed? | (10k/40k) × 100 | 25%
Type 2: Priya earns 25% more than Ravi
Type 3: Ravi earns 20% less than Priya
Type 4: Ravi needs a 25% raise to match Priya
| Problem Type | Formula to Use | Base (Denominator) | When You See This... |
|---|---|---|---|
| Find % of a number | (Percentage / 100) × Whole | Always 100 | 'What is 35% of 480?' |
| Find what % one number is of another | (Part / Whole) × 100 | The 'whole' or reference value | '18 out of 72 is what %?' |
| % Increase / Decrease | ((New − Old) / Old) × 100 | Always the OLD (original) value | 'Price changed from X to Y, find % change' |
| Find original before % change | Original = New / (1 ± R/100) | N/A — you're solving for it | 'After 20% off it costs X, find original' |
| Successive % changes | Net% = A + B + (AB/100) OR chain multipliers | Starting original value | 'Applied 2 or more changes in sequence' |
| A is % more than B | ((A − B) / B) × 100 | B — the reference you're comparing TO | 'How much more than B is A?' |
| A is % less than B | ((B − A) / B) × 100 | B — still the reference | 'How much less than B is A?' |
🎯 Key Takeaways
- Every percentage problem is one of three things: finding the Part, finding the Whole, or finding the Percentage itself — all three come from the one formula: Part = (Percentage / 100) × Whole.
- When reversing a percentage change, always DIVIDE by the multiplier (e.g. ÷ 0.80 to undo a 20% drop) — never add the percentage back to the final value, because the percentage was never applied to that final value.
- Two successive percentage changes never simply add up — use Net% = A + B + (AB/100) to account for the compounding. A 50% loss followed by a 50% gain still leaves you 25% below where you started.
- In comparison questions ('A is what % more than B'), the denominator is always the BASE — the thing you are comparing FROM. 'More than B' means B is in the denominator. 'More than A' means A is in the denominator. Get this wrong and your answer will be completely different from the correct one, even though the arithmetic looks fine.
⚠ Common Mistakes to Avoid
- ✕Mistake 1: Adding successive percentages directly — If something increases by 10% then 10% again, beginners write 10 + 10 = 20%. The real answer is 21% because the second 10% is taken on the already-increased value. Fix it: always use Net% = A + B + (AB/100) for two changes, or chain multipliers like Original × 1.10 × 1.10 = Original × 1.21.
- ✕Mistake 2: Using the WRONG base when reversing a percentage change — If a price drops 20% to reach ₹36,000, beginners add 20% back to ₹36,000 to get ₹43,200. That's wrong because the 20% was taken from the ORIGINAL price, not ₹36,000. Fix it: divide the final value by the multiplier — ₹36,000 ÷ 0.80 = ₹45,000 — always divide, never add back.
- ✕Mistake 3: Swapping the denominator in 'more than' vs 'less than' questions — If Priya earns ₹50k and Ravi earns ₹40k, many candidates say 'Ravi earns 25% less than Priya' (using ₹40k as the base). The correct answer is 20% less, because the base when comparing Ravi to Priya is Priya's salary (₹50k). Fix it: the denominator is always the value that follows the words 'than' or 'of' in the question — identify that word and its associated number first.
Interview Questions on This Topic
- QA shopkeeper marks a product 40% above its cost price and then offers a 20% discount. What is his net profit or loss percentage? (Hint: start with a cost price of ₹100 to make the math clean.)
- QRohan's income is 25% more than Sneha's. By what percentage is Sneha's income less than Rohan's? Most candidates say 25% — why is that wrong and what is the correct answer?
- QA number is first increased by 20% and then the result is decreased by 20%. Is the final number the same as the original? If not, what is the net percentage change? Explain why this happens.
Frequently Asked Questions
What is the formula for percentage increase and decrease?
The formula is: Percentage Change = ((New Value − Old Value) / Old Value) × 100. A positive result means an increase, a negative result means a decrease. The key rule is always to divide by the OLD (original) value, never the new one. To find the new value directly, multiply the original by (1 + R/100) for an increase or (1 − R/100) for a decrease.
Why does a 50% increase followed by a 50% decrease not bring you back to the original value?
Because the second percentage is applied to a different number. If you start at 100, a 50% increase gives 150. A 50% decrease on 150 gives 75 — not 100. The increase worked on 100 but the decrease worked on 150, so the amounts in rupees were different even though the percentages were the same. This is the compounding effect: successive percentages multiply, they don't add.
What is the difference between 'A is 25% more than B' and 'B is 25% less than A'?
They are NOT the same thing. 'A is 25% more than B' means the difference is 25% of B (B is the base). 'B is 25% less than A' would mean the difference is 25% of A (A is the base). Since A and B are different numbers, dividing the same difference by different bases gives different percentages. If A = 125 and B = 100: A is 25% more than B, but B is only 20% less than A.
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