Percentage literally means "per hundred" — it is a ratio expressed as parts out of 100. When you say 35%, you mean 35 out of every 100 units. That is the entire foundation. Everything else — discounts, profit, elections, population change — is just this one idea applied to a context.
Here is the analogy that makes it click: imagine a cricket ground with exactly 100 seats. Whatever fraction of those seats is occupied gives you the percentage directly. If 60 seats are filled, that is 60%. If the ground now has 250 seats and 150 are filled, you scale: (150/250) × 100 = 60% — same answer, different total.
The three quantities in any percentage problem are:
The master formula is:
Rearranging gives you two more forms you'll use constantly:
Pick the form based on what is unknown. In most Agniveer Vayu questions, the Whole is unknown — you are given a percentage and its actual value, and asked for the total. The election-type question is the classic example of this pattern.
One thing beginners trip over: the base matters. A 20% increase followed by a 20% decrease does not bring you back to the original. The second 20% is applied to a larger number, so the net effect is a small loss. This asymmetry is why successive discount and percentage change questions are exam favourites.
If the result is positive, it is an increase. If negative, a decrease. The old value is always the base — not the new value, not the average.
Example: Price rises from 80 to 100.
(100 - 80) / 80 × 100 = 25% increase
Now reverse it: Price falls from 100 to 80.
(80 - 100) / 100 × 100 = -20% decrease
Same two numbers, different bases, different percentages. This is the most common confusion in the exam.
This is where Agniveer Vayu loves to test you. Two successive changes of a% and b% are NOT simply (a + b)%. The correct single equivalent change is:
Where a and b carry their signs (positive for increase, negative for decrease).
For two successive discounts of d1% and d2%:
Or using the formula: d1 + d2 - (d1 × d2)/100
Let's verify with 10% and 20% discounts:
10 + 20 - (10 × 20)/100 = 30 - 2 = 28%
And with 20% and 30%:
20 + 30 - (20 × 30)/100 = 50 - 6 = 44%
Both match the PYQs exactly. Memorise this formula — it is a direct mark on the paper.
You need these cold. Calculating 2/7 × 100 mid-exam costs 20 seconds. Knowing 2/7 ≈ 28.57% costs zero.
| Fraction | Percentage | |----------|------------| | 1/2 | 50% | | 1/3 | 33.33% | | 2/3 | 66.67% | | 1/4 | 25% | | 3/4 | 75% | | 1/5 | 20% | | 2/5 | 40% | | 3/5 | 60% | | 4/5 | 80% | | 1/6 | 16.67% | | 1/8 | 12.5% | | 3/8 | 37.5% | | 1/10 | 10% | | 1/12 | 8.33% |
Standard format: Candidate A gets X%, wins by Y votes. Find total votes.
The margin between two candidates is (X - (100 - X))% = (2X - 100)% of total votes.
Set that equal to the given margin and solve:
For 60% winner, margin percentage = 2(60) - 100 = 20%. So total = 1200 / 0.20 = 6000.
"After a 20% increase, the price is 600. What was the original price?"
Do NOT subtract 20% from 600. That is wrong.
Original × 1.20 = 600 Original = 600 / 1.20 = 500
The multiplier approach: increase by p% means multiply by (1 + p/100), decrease by p% means multiply by (1 - p/100). To reverse, divide by that multiplier.
Use a + b - ab/100 for two successive discounts or percentage changes. Works in both directions (use negative values for decreases if mixing increase/decrease).
Example: 10% and 20% discounts → 10 + 20 - (10×20)/100 = 30 - 2 = 28%. No multiplications of decimals needed.
Standard method (multiply 0.9 × 0.8 = 0.72, then 1 - 0.72 = 0.28): 5 steps. This formula: 3 steps. In the exam hall, that saves about 15 seconds per question, and these questions appear in pairs.
Whenever the winner's percentage W is given and you need total votes: margin% = 2W - 100. Divide the actual margin by this percentage (as a decimal).
Example: W = 60%, margin = 1200 votes → 2(60) - 100 = 20% → total = 1200 / 0.2 = 6000.
Standard method (write winner = 0.6T, loser = 0.4T, subtract, solve): 4 equations. This: 2 steps. Saves approximately 20 seconds.
"A number after X% increase becomes N. Find original." Divide N by (1 + X/100). Never subtract X% from N.
Example: After 25% increase → 500. Original = 500 / 1.25 = 400.
Trap method (500 × 0.75 = 375): wrong answer in 5 seconds. Correct method: 1 division step. Zero extra time, zero error — just the right setup.
Pre-load fractions 1/2 through 1/12 (see table in Deep Dive). When a question gives you a percentage like 12.5% or 37.5%, immediately convert to fraction (1/8 or 3/8) and work with clean integers instead of decimals.
Example: 37.5% of 240 → (3/8) × 240 = 90. Done in one step vs. 0.375 × 240 long multiplication. Saves 10-15 seconds and eliminates decimal errors.
If a value increases by X% then decreases by X%, the net change is always a decrease of X²/100 %.
Example: +20% then -20% → net = -(20²/100)% = -4%. The value does NOT return to original.
Standard method (compute each step separately): 4 multiplications. This formula: 1 squaring. Saves about 20 seconds and eliminates the common "it cancels out" trap.
When a percentage question appears, identify the type in the first 5 seconds:
Type 1 — Find the percentage: Use (Part/Whole) × 100. Confirm which is the "whole" (it is always the original or reference value).
Type 2 — Find the part: Use (Percentage/100) × Whole. Convert ugly percentages to fractions first.
Type 3 — Find the whole (election, original price): Use Part = (Percentage/100) × Whole, rearrange to Whole = Part × 100 / Percentage. For elections, use the (2W - 100) shortcut.
Type 4 — Successive changes: Count how many percentage changes are applied one after another. If two, use a + b - ab/100. If three, apply the formula twice in sequence.
Type 5 — Reverse percentage: Identify the final value and the percentage applied. Divide by the multiplier (1 ± p/100).
One rule above all: in every percentage problem, write down the base first. Half the errors in this topic happen because students apply a percentage to the wrong number.
Why this question: The most direct test of successive discount. Appears almost every year. Students who add 10 + 20 = 30% lose a mark in under 5 seconds.
Solving path: Apply the formula a + b - ab/100 = 10 + 20 - (10 × 20)/100 = 30 - 2 = 28%. Alternatively: original price = 100, after 10% discount = 90, after 20% on that = 90 × 0.8 = 72. Single discount = (100 - 72)/100 = 28%. Both routes give 28%. The formula is faster by 2-3 steps.
Why this question: Tests the election problem structure. The trap is dividing 1200 by 60 (getting 20 instead of 6000). You must recognise that the margin is a percentage of the total, not the winner's share.
Solving path: Winner = 60%, loser = 40%, gap = 20% of total. Set 0.20 × T = 1200, so T = 6000. Using the shortcut: (2 × 60 - 100) = 20%, total = 1200 / 0.20 = 6000. Answer: 6000 (option B).
Why this question: Same type as the first PYQ but with larger discount values. Tests whether you know the formula or are just adding percentages. Placed in a different year to show this pattern is not a fluke — it is a recurring examiner favourite.
Solving path: a + b - ab/100 = 20 + 30 - (20 × 30)/100 = 50 - 6 = 44%. Or: start with 100, after 20% discount = 80, after 30% on 80 = 80 × 0.7 = 56. Equivalent single discount = 100 - 56 = 44%. Answer: 44% (option C).
Adding successive discounts directly: 10% + 20% = 30% is wrong. The second discount applies to the already-reduced price, so the effective discount is always less than the simple sum. Use a + b - ab/100.
Using the wrong base for percentage change: Always use the original (old) value as the base. "A increased from 80 to 100" is a 25% increase, not 20%. Divide by 80, not 100.
Reversing by subtracting the percentage: If a price after a 20% increase is 600, the original is NOT 600 × 0.80 = 480. It is 600 / 1.20 = 500. Subtracting the percentage from the new value is one of the most common errors at this level.
Confusing margin with winner's votes: In election problems, the "margin" is the difference between winner and loser votes, not the winner's total. Margin = (Winner% - Loser%) × Total, not Winner% × Total.
Assuming equal and opposite changes cancel: A 20% rise followed by a 20% fall is not zero net change — it is a 4% net loss. The formula X²/100 gives the loss percentage. Never assume they cancel.
Forgetting to convert percentage to decimal before multiplying: Writing 0.20 × T = 1200 is correct. Writing 20 × T = 1200 gives T = 60, which is nonsensical in context. Check your decimal placement every time.