Percentage is just a fraction with denominator 100. When you write 35%, you mean 35/100 or 0.35. That is the whole foundation — everything else is an application of this one idea.
Here is the analogy that makes it stick: imagine a 100-unit measuring rod. "Percent" means "per that rod." If something covers 35 units on that rod, it is 35%. If the rod itself shrinks or stretches (i.e., the base changes), the same physical amount will show a different percentage reading. That is why "percentage of what" is the most important phrase in every percentage problem.
Three core ideas you need before everything else:
1. Percent ↔ Fraction ↔ Decimal conversions. Slow conversion is the silent killer. If you are still computing 12.5% as 12.5/100, you are losing 10–15 seconds per question. 12.5% = 1/8 — know that cold.
The critical fraction-to-percent table to have automatic recall for:
| Fraction | Percentage | | --- | --- | | 1/2 | 50% | | 1/3 | 33.33% | | 1/4 | 25% | | 1/5 | 20% | | 1/6 | 16.67% | | 1/7 | 14.28% | | 1/8 | 12.5% | | 1/9 | 11.11% | | 3/4 | 75% | | 2/3 | 66.67% |
2. Percentage change formula.
Percentage change = (Change / Original) × 100
The trap: most mistakes happen when the denominator is wrong — people use the new value instead of the original (base). Burn this in: percentage change is always over the original.
3. Reverse percentage. If after a 20% increase the value is 120, the original is 120 / 1.20 = 100. You divide by the multiplier, not subtract 20% from 120 (that gives 96, which is wrong). This reverse-percent slip costs candidates marks constantly.
Stop writing percentage formulas. Start writing multipliers. Every percentage operation has a one-number multiplier:
Why does this matter? Because chains of percentage operations become chains of multiplications, which you can collapse at once.
If a quantity undergoes two successive percentage changes of a% and b%, the net percentage change is:
Net % change = a + b + (ab/100)
Where a is positive for increase and negative for decrease.
Example: Price increases by 10%, then decreases by 5%.
Net = 10 + (−5) + (10 × −5)/100 = 10 − 5 − 0.5 = 4.5% increase.
This formula is also the most efficient way to handle population growth / compound changes over two periods. For three or more periods, use the multiplier method: 1.10 × 0.95 × ...
When two variables are multiplied together (length × breadth, price × consumption), and both change:
Net % change = a + b + (ab/100)
Here a and b are the individual changes. If length increases 20% and breadth decreases 10%:
Net = 20 + (−10) + (20 × −10)/100 = 20 − 10 − 2 = 8% increase.
This is identical to successive change — because area = l × b is mathematically the same structure.
Given a final value and the percentage change applied, find the original:
Original = Final value / Multiplier
If after a 26% profit, selling price = Rs. 441:
Cost price = 441 / 1.26 = 350
The common trap: students subtract 26% of 441 and get 326.34. Wrong base. You must divide.
Structure: "X% on A, Y% on B, Z% on C. Saves Rs. N. Find income."
Step 1: Sum the spent percentages. Step 2: Savings % = 100 − spent %. Step 3: (Savings %) of Income = Rs. N → solve for Income.
Income = N × (100 / Savings%)
Chain problems: A is 20% more than B; B is 25% less than C. What is A as a % of C?
Always fix one value at 100 and chain through.
A is 90% of C. Direct calculation, no formula needed.
For expenditure = price × consumption to remain constant when price changes by r%:
Reduction in consumption = r / (100 + r) × 100
Price up 25%: reduction = 25/125 × 100 = 20%
Memorise this fraction structure. Standard method (algebraic setup) takes 40–50 seconds. With this formula: under 10 seconds.
The chain is: Cost Price → Marked Price → Selling Price (after discount).
SP = CP × (1 + profit%/100)SP = MP × (1 − discount%/100)So: MP = SP / (1 − discount%/100)
When CP, profit%, and discount% are all given — always find SP first, then work backwards to MP. Do not try to set up simultaneous equations.
When price increases by r%, the consumption reduction to keep spending constant is always r/(100+r) × 100%.
Why: Expenditure = P × C = constant. New P = P(1 + r/100). New C = C / (1 + r/100). Reduction fraction = (r/100)/(1 + r/100) = r/(100+r).
Micro-example: Price up 25% → reduction = 25/125 = 1/5 = 20%. Zero algebra.
Speed: Standard algebraic method: ~45 seconds. This pattern: ~8 seconds.
Two percentage changes a% and b% applied successively give a net change of a + b + ab/100 (negatives for decreases).
Micro-example: +20% then −10% → 20 − 10 + (20×−10)/100 = 10 − 2 = 8% increase. Same answer as multiplier method (1.20 × 0.90 = 1.08), but no decimal multiplication needed.
Use this when both changes are clean numbers. Standard method (compute intermediate value, then compute final): ~50 seconds. Formula: ~12 seconds.
When a value has already been increased by r%, the original = Final / (1 + r/100).
Micro-example: Selling price Rs. 441 at 26% profit. Students wrongly compute 441 − 26% of 441 = 441 − 114.66 = 326.34. Correct: 441 / 1.26 = 350. A full Rs. 23.66 error — likely means the wrong answer choice.
Rule: The "subtract the percentage" instinct is always wrong for reverse problems. Divide by the multiplier. This applies in profit, discount, population, and salary chain problems uniformly.
Speed: Wrong method gives wrong answer — so this is a correctness gain, not just a time gain.
When a percentage is a "clean" fraction (12.5% = 1/8, 16.67% = 1/6, 33.33% = 1/3), substitute the fraction immediately instead of computing decimals.
Micro-example: What is 37.5% of 480? Recognise 37.5% = 3/8. So answer = 480 × 3/8 = 180. No long multiplication.
The 10 fractions in the table above cover roughly 60% of the percentage values seen in SBI Clerk questions. For each one you know cold, you save 15–20 seconds of decimal arithmetic per usage.
If all numbers in a ratio are increased (or decreased) by the same percentage, the ratio does not change.
Micro-example: Ratio 3:4, both increased by 25%. New values = 3.75 and 5. Ratio = 3.75:5 = 3:4. Answer is the original ratio.
Use elimination: whenever you see "ratio … increased/decreased by same %" in the options, the answer is the original ratio. You do not need to compute anything. Standard calculation method: ~30 seconds. Recognition + elimination: ~5 seconds.
Read the question and identify the structure first — before touching numbers.
Is a final value given and the original asked? → Reverse percentage. Divide final by multiplier. Never subtract.
Are two quantities multiplied (area, revenue, expenditure)? → Use a + b + ab/100. Assign negatives to decreases.
Is it a chain comparison (A vs B vs C)? → Fix the anchor (usually C or the last reference) at 100, chain forward.
Is it income-savings? → Find savings%, then Income = Savings amount × (100/Savings%).
Is it a ratio with a uniform % change? → Ratio is unchanged. Pick that option immediately.
Is it marked price with profit and discount? → Find SP first via CP and profit%. Then find MP = SP / (1 − discount rate).
Is it price-consumption with fixed expenditure? → Reduction = r/(100+r) × 100.
One more check before confirming: verify the base. When computing percentage change, is your denominator the original value? If no — fix it.
Why this question: The classic income-savings reverse problem. Tests whether you find savings% correctly and apply reverse percentage cleanly.
Solving path: Total expenditure = 30 + 25 + 15 = 70%. So savings = 30% of income. You are told 30% = Rs. 1,200. Income = 1,200 × (100/30) = 1,200 × 10/3 = Rs. 4,000. Note: 100/30 simplifies to 10/3 — no calculator needed.
Why this question: The two-variable (area) successive change template. Appears regularly in both standalone and DI contexts.
Solving path: a = +20, b = −10. Net = 20 + (−10) + (20 × −10)/100 = 10 − 2 = 8% increase. Done in under 10 seconds with the formula.
Why this question: The price-consumption inverse. High-frequency template. Tests whether you know the r/(100+r) pattern.
Solving path: Price up 25%. Reduction = 25/(100+25) × 100 = 25/125 × 100 = 20%. If you set up P×C = 1.25P × new C instead, you get the same answer but in 40 extra seconds.
Why this question: Chain salary comparison. Tests whether you set the right anchor and chain correctly.
Solving path: Anchor C = 100. B = 100 × 0.75 = 75. A = 75 × 1.20 = 90. A/C = 90/100 = 90%. The trap is reversing the chain — students sometimes compute A first, then go to C incorrectly.
Why this question: Marked price with profit and discount — combines two reverse-percentage steps. Extremely common in SBI Clerk.
Solving path: SP = 350 × 1.26 = 441. (Quick check: 350 × 1.26 = 350 + 350×0.26 = 350 + 91 = 441.) MP = 441 / 0.9 = 490. (441 / 0.9 = 4410 / 9 = 490.) Both steps use divide-by-multiplier, not subtract.
Why this question: Successive change over two periods — the population variant. Tests the a + b + ab/100 formula with a decrease in the second period.
Solving path: a = +10, b = −5. Net = 10 + (−5) + (10×−5)/100 = 5 − 0.5 = 4.5% increase. Clean application of the formula.
Why this question: Ratio invariance under equal percentage change. Tests whether you recognise the pattern and skip computation.
Solving path: Both numbers in ratio 3:4 are increased by the same 25%. The multiplier is identical for both — it cancels in the ratio. Answer = 3:4. Use elimination: spot the original ratio in the options and confirm.
Why this question: Markup + discount profit calculation. One of the most repeated templates across SBI, IBPS, and RRB Clerk.
Solving path: CP = 100. MP = 140. SP = 140 × 0.85 = 119. Profit% = (119−100)/100 × 100 = 19%. The shortcut: SP multiplier = 1.40 × 0.85 = 1.19 directly → profit = 19%. One multiplication, no intermediate value needed.
Using the wrong base for percentage change. Percentage change = change / original, not change / new value. Dividing by the new value is a trap set by almost every option list.
Subtracting percentage from the final value in reverse problems. If a price after 20% increase is 120, the original is 120/1.20 = 100, not 120 − 24 = 96. Always divide by the multiplier.
Sign error in the successive change formula. When one change is a decrease, plug in a negative value for b. Writing a + b + ab/100 with b = +5 when the question says "decreased by 5%" gives a wrong answer by exactly 10/100 × 2b.
Treating "A is 20% more than B" the same as "B is 20% less than A." They are not the same. If A = 1.20B, then B = A/1.20 ≈ 0.833A — B is approximately 16.67% less than A, not 20%. This confusion appears directly in chain-comparison questions.
Forgetting to check which direction the chain comparison goes. In "A vs C" problems, students often compute "C as % of A" instead of "A as % of C." Read the question's exact phrasing before writing the final ratio.
In Marked Price problems, finding MP before SP. Always find SP from CP and profit% first. Trying to set up MP directly from CP and discount% without knowing SP leads to an extra unknown and a longer equation.