1.1.18
Ek percent bas fraction with a fixed denominator of 100 hota hai. "Per-cent" word ka literally matlab hai "per hundred" (Latin centum = hundred). Toh 37% 100 37
HUM 100 kyun lete hain? Kyunki 40 17 20 9 42.5% 45% ek hi measuring stick deti hain, isliye alag-alag quantities directly comparable ban jaati hain.
x % 100 x x × 0.01
HOW to move between forms (fraction ↔ decimal ↔ percent):
fraction ÷ and × 100 percent ÷ 100 decimal
Fraction → percent: b a × 100% 8 3 × 100 = 37.5%
Percent → decimal: 100 se divide karo. e.g. 37.5% = 0.375
Intuition Poocho "A, poore B ka kitna hissa hai?"Percent hamesha ek base ke relative hoti hai (woh "whole", woh 100%). Pehle base B identify karo, phir A ko uska fraction express karo, phir 100 par scale karo.
Worked example 18, 45 ka kitna percent hai?
Base B = 45 A = 18
p = 45 18 × 100 Kyun? 100 p = 45 18 45 18 = 5 2 = 0.4 p = 0.4 × 100 = 40%
Worked example 240 ka 15% nikalein
100 15 × 240 = 0.15 × 240 = 36 Feynman trick: 10% = 24 5% = 12 15% = 24 + 12 = 36 Kyun? Percentages add hoti hain kyunki woh x
Worked example Ek shirt ₹800 ki hai; tum 12% GST bharte ho. Total?
Tax = 12% 800 = 0.12 × 800 = ₹96
Total = 800 + 96 = ₹896
Intuition Change ORIGINAL ke against measure hota haiCrucial idea: percentage change hamesha change divided by starting value hoti hai, final value se nahi. Yahi wajah hai ki 100→150 ka rise "50% increase" hai lekin 150→100 ka fall sirf "33.3% decrease" hai — alag-alag bases!
Worked example Price 250 → 300 badhti hai. % increase?
Change = 300 − 250 = 50 = 250
250 50 × 100 = 20% Base 250 kyun? Hum poochhte hain ki rise kitni badi hai jahan se humne start kiya tha uske compare mein .
Worked example Price 300 → 250 girti hai. % decrease?
Change = 250 − 300 = − 50 = 300
300 − 50 × 100 = − 16. 6 % 16.7% 20% phir kyun nahi? Base 300 ho gaya. Same jump, bada base ⇒ choti percent.
Worked example Successive changes: +10% phir −10%V n = V o × 1.10 × 0.90 = 0.99 V o − 1% 0% Kyun? Doosra 10% ek bade number se liya jaata hai.
Common mistake "10% upar phir 10% neeche — main wapas start par aa jaaunga."Kyun sahi lagta hai: dono 10% cancel hote lagte hain.
Steel-man: woh cancel hote agar dono same base ke hote — lekin down-10% raised value 1.1 V 1.1 × 0.9 = 0.99 1% Fix: hamesha multipliers ko multiply karo; kabhi bhi alag-alag bases ke raw percents ko add/subtract mat karo.
Common mistake % change mein galat base use karna.Kyun sahi lagta hai: "increase aur decrease symmetric lagte hain."
Fix: % change ka denominator hamesha original (old) value hota hai. 250→300 jaana 20% hai; reverse trip 300→250 base 300 use karta hai, toh 16.7% aata hai.
Common mistake "% of a quantity ek percent deta hai."Kyun sahi lagta hai: sawaal mein % sign hai toh answer mein bhi 'hona chahiye'.
Fix: ₹240 ka 15% quantity hai (₹36), percentage nahi. Q
Mnemonic "OF times, IS over."
"OF " → multiply: "x% of Q" = 100 x × Q
"IS over OF ": "A is what % of B" = B A × 100 is -number upar, of -number neeche).
Change: "New minus Old, over Old."
Recall Isse ek 12-saal ke bachche ko samjhao (Feynman)Socho ki har pizza, chahe kisi bhi size ka ho, 100 tiny slices mein kata gaya hai. Ek "percent" bas itna hai ki tum un 100 slices mein se kitni slices ki baat kar rahe ho. Agar tumhare paas 100 mein se 40 slices hain, toh woh 40% hai. "300 grams wale pizza ka 20%" nikalne ke liye, tum 100 mein se 20 slices lete ho, yaani 100 20 × 300 = 60 extra 50 ko wahan se compare karte ho jahan se shuru hua tha (250), toh yeh 20% badhi. Lekin agar woh wapas aati hai, toh ab tum 50 ko bade 300 se compare karte ho, jo ek chota hissa hai — yahi wajah hai ki upar jaana aur neeche aana same percent nahi hote!
"Percent" ka literally kya matlab hai? Per hundred — denominator 100 wala fraction.
Formula: A, B ka kitna % hai? B A × 100%
Formula: quantity Q ka x%? 100 x × Q
Percentage change ka formula? V o l d V n e w − V o l d × 100
20% increase ka multiplier? 1.20
20% decrease ka multiplier? 0.80
+10% phir −10% ka net effect? × 1.1 × 0.9 = 0.99
300→250, 20% decrease kyun nahi hai? Base ab 300 hai, 250 nahi; 300 50 × 100 = 16.7%
10%+5% trick se 240 ka 15%? 24 + 12 = 36
3/8 ko percent mein convert karo. 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
