A percent is just a fraction with a fixed denominator of 100 . The word "per-cent" literally means "per hundred" (Latin centum = hundred). So 37 % 37\% 37% 37 100 \dfrac{37}{100} 100 37
WHY do we bother with 100? Because comparing 17 40 \frac{17}{40} 40 17 9 20 \frac{9}{20} 20 9 42.5 % 42.5\% 42.5% 45 % 45\% 45% same measuring stick , so different quantities become directly comparable.
x % x\% x % x 100 \dfrac{x}{100} 100 x x × 0.01 x \times 0.01 x × 0.01
HOW to move between forms (fraction ↔ decimal ↔ percent):
fraction → ÷ and × 100 percent → ÷ 100 decimal \text{fraction} \xrightarrow{\ \div\ \text{and} \times 100\ } \text{percent} \xrightarrow{\ \div\,100\ } \text{decimal} fraction ÷ and × 100 percent ÷ 100 decimal
Fraction → percent: a b × 100 % \dfrac{a}{b} \times 100\% b a × 100% 3 8 × 100 = 37.5 % \dfrac{3}{8}\times100 = 37.5\% 8 3 × 100 = 37.5%
Percent → decimal: divide by 100. e.g. 37.5 % = 0.375 37.5\% = 0.375 37.5% = 0.375
Intuition Ask "A is what part of the whole B?"Percent is always relative to a base (the "whole", the 100%). First identify the base B, then express A as a fraction of it, then scale to 100.
Worked example What percent is 18 of 45?
Base B = 45 B=45 B = 45 A = 18 A=18 A = 18
p = 18 45 × 100 p = \dfrac{18}{45}\times100 p = 45 18 × 100 Why? Set p 100 = 18 45 \frac{p}{100}=\frac{18}{45} 100 p = 45 18 18 45 = 2 5 = 0.4 \dfrac{18}{45}=\dfrac{2}{5}=0.4 45 18 = 5 2 = 0.4 p = 0.4 × 100 = 40 % p = 0.4\times100 = 40\% p = 0.4 × 100 = 40%
Worked example Find 15% of 240
15 100 × 240 = 0.15 × 240 = 36 \dfrac{15}{100}\times 240 = 0.15\times240 = 36 100 15 × 240 = 0.15 × 240 = 36 Feynman trick: 10 % = 24 10\% = 24 10% = 24 5 % 5\% 5% = 12 =12 = 12 15 % = 24 + 12 = 36 15\% = 24+12 = 36 15% = 24 + 12 = 36 Why? Percentages add because they're linear in x x x
Worked example A shirt costs ₹800; you pay 12% GST. Total?
Tax = 12 % = 12\% = 12% 800 = 0.12 × 800 = ₹ 96 800 = 0.12\times800 = ₹96 800 = 0.12 × 800 = ₹96
Total = 800 + 96 = ₹ 896 = 800 + 96 = ₹896 = 800 + 96 = ₹896
Intuition Change is measured against the ORIGINALThe crucial idea: percentage change is always the change divided by the starting value , not the final value. That's what makes a rise from 100→150 a "50% increase" but a fall 150→100 only a "33.3% decrease" — different bases!
Worked example Price rises 250 → 300. % increase?
Change = 300 − 250 = 50 =300-250=50 = 300 − 250 = 50 = 250 =250 = 250
50 250 × 100 = 20 % \dfrac{50}{250}\times100 = 20\% 250 50 × 100 = 20% Why base 250? We ask how big the rise is compared to where we started .
Worked example Price falls 300 → 250. % decrease?
Change = 250 − 300 = − 50 =250-300=-50 = 250 − 300 = − 50 = 300 =300 = 300
− 50 300 × 100 = − 16. 6 ‾ % \dfrac{-50}{300}\times100 = -16.\overline{6}\% 300 − 50 × 100 = − 16. 6 % 16.7 % 16.7\% 16.7% Why not 20% again? The base changed to 300. Same jump, bigger base ⇒ smaller percent.
Worked example Successive changes: +10% then −10%V n = V o × 1.10 × 0.90 = 0.99 V o V_n = V_o \times 1.10 \times 0.90 = 0.99\,V_o V n = V o × 1.10 × 0.90 = 0.99 V o − 1 % \mathbf{-1\%} − 1% 0 % 0\% 0% Why? The second 10% is taken off a bigger number.
Common mistake "10% up then 10% down brings me back to start."Why it feels right: the two 10%s look like they cancel.
The steel-man: they would cancel if both were of the same base — but the down-10% acts on the raised value 1.1 V 1.1V 1.1 V 1.1 × 0.9 = 0.99 1.1\times0.9 = 0.99 1.1 × 0.9 = 0.99 1 % 1\% 1% Fix: always multiply the multipliers; never add/subtract raw percents across different bases.
Common mistake Using the wrong base in % change.Why it feels right: "increase and decrease look symmetric."
Fix: % change denominator is always the original (old) value . Going up 250→300 is 20%; the reverse trip 300→250 uses base 300, giving 16.7%.
Common mistake "% of a quantity gives a percent."Why it feels right: the question has a % sign so the answer 'should' too.
Fix: 15 % 15\% 15% quantity (₹36), not a percentage. Keep the units of Q Q Q
Mnemonic "OF times, IS over."
"OF " → multiply: "x% of Q" = x 100 × Q \frac{x}{100}\times Q 100 x × Q
"IS over OF ": "A is what % of B" = A B × 100 \frac{A}{B}\times100 B A × 100 is -number on top, of -number below).
Change: "New minus Old, over Old."
Recall Explain it to a 12-year-old (Feynman)Imagine every pizza is cut into 100 tiny slices no matter its size. A "percent" is just how many of those 100 slices you're talking about. If you have 40 slices out of 100, that's 40%. To find "20% of a pizza with 300 grams," you take 20 slices out of 100, i.e. 20 100 × 300 = 60 \frac{20}{100}\times300 = 60 100 20 × 300 = 60 extra 50 to where it started (250), so it grew by 20%. But if it shrinks back, you now compare 50 to the bigger 300, which is a smaller share — that's why going up and coming down aren't the same percent!
What does "percent" literally mean? Per hundred — a fraction with denominator 100.
Formula: A is what % of B? A B × 100 % \dfrac{A}{B}\times 100\% B A × 100% (is-number over of-number).
Formula: x% of a quantity Q? x 100 × Q \dfrac{x}{100}\times Q 100 x × Q (an amount, keeps Q's units).
Formula for percentage change? V n e w − V o l d V o l d × 100 \dfrac{V_{new}-V_{old}}{V_{old}}\times100 V o l d V n e w − V o l d × 100 , base = old value.
Multiplier for a 20% increase? Multiplier for a 20% decrease? Net effect of +10% then −10%? × 1.1 × 0.9 = 0.99 \times1.1\times0.9=0.99 × 1.1 × 0.9 = 0.99 ⇒ 1% net decrease.
Why isn't 300→250 a 20% decrease? Base is now 300, not 250;
50 300 × 100 = 16.7 % \frac{50}{300}\times100=16.7\% 300 50 × 100 = 16.7% .
15% of 240 using the 10%+5% trick? 24 + 12 = 36 24+12=36 24 + 12 = 36 .
Convert 3/8 to a percent. 3 8 × 100 = 37.5 % \frac38\times100 = 37.5\% 8 3 × 100 = 37.5% .
solve p over 100 = A over B
Percent = fraction over 100
Decimal via divide by 100
Tax and discount problems
Percent increase or decrease
Intuition Hinglish mein samjhoDekho, "percent" ka matlab hai "per hundred" — yaani har cheez ko imagine karo 100 barabar tukdon mein bata hua. 40 % 40\% 40% A B × 100 \frac{A}{B}\times100 B A × 100 x 100 × Q \frac{x}{100}\times Q 100 x × Q
Sabse important cheez hai percentage increase/decrease mein base . Change hamesha purani value se compare hota hai: new − old old × 100 \frac{\text{new}-\text{old}}{\text{old}}\times100 old new − old × 100
Ek classic trap: agar price pehle 10% badhe phir 10% ghate, log sochte hain wapas same ho gaya — galat! Multiplier lagao: 1.10 × 0.90 = 0.99 1.10\times0.90 = 0.99 1.10 × 0.90 = 0.99
Exam tip (80/20): teen formule ratto — is/of wala, x% of Q wala, aur change wala — aur hamesha ulta multiply karke check karo (jaise 40% of 45 = 18). Yeh chhoti si habit tumhare 80% percentage questions bacha legi.
