Fractions and percentagesFree

Har fraction ka ek base hota hai

Fraction batata hai ke kisi whole ko barabar hisson mein baant kar kitne hissay liye gaye hain. 3/4 mein denominator 4 whole ke barabar hissay aur numerator 3 liye gaye hissay batata hai. Denominator zero nahi ho sakta. Whole sirf ek cheez nahi: ek ribbon ki length, ek class ke students ya ek mahine ka budget bhi whole ho sakta hai.

Pehle base ka naam likhein. Agar ribbon 12 metre hai to us ka 3/4 hissa 9 metre hai. Doosri 20-metre ribbon ka 3/4 hissa 15 metre hai. Fraction ek jaisa hai, lekin quantity alag hai kyun ke base alag hai. Students ko count kar rahe hon to exact fraction se nikla count whole number hona chahiye; 3.5 students ko bina wajah round karna premise ko badal deta hai.

Equivalent fractions same proportion dikhate hain. Numerator aur denominator ko ek hi nonzero number se multiply ya divide karne se value nahi badalti: 6/8 = 3/4. Sirf numerator badalna value badalta hai. 7/4 = 1 + 3/4 ek whole se zyada hai; fraction zaroori nahi ke 1 se chhota ho.

Fraction, decimal aur percentage ek hi proportion hain

Percentage ka matlab har 100 mein se hai. 3/4 = 75/100 = 0.75 = 75%. Fraction ko decimal mein badalne ke liye numerator ko denominator se divide karein. Decimal ko percentage banane ke liye 100 se multiply karein; percentage ko decimal banane ke liye 100 se divide karein. 12.5% = 0.125 = 125/1000 = 1/8.

Comparison ke liye forms same kar lein: 2/5 = 0.40, 45% = 0.45, is liye 2/5 < 45%. Common denominator bhi kaam karta hai: 3/5 = 15/25 < 16/25. Lekin 40% of 200 aur 45% of 100 ki actual quantities 80 aur 45 hain. Proportion compare karna aur quantities compare karna alag steps hain.

150% = 1.5 = 3/2: output target ka 150% ho sakta hai, kyun ke output target se zyada ho sakti hai. Ek fixed class ke present students us class ke 150% nahi ho sakte. Percentage ki meaning context aur base se check hoti hai.

Hissay jorna, ghataana aur ek hissay ka hissa lena

Ek hi budget ka 1/4 transport aur 2/5 food par kharch ho to addition same whole par hai. Dono ko twentieths mein likhein:

1/4 + 2/5 = 5/20 + 8/20 = 13/20.

Denominators add nahi karte: fourths aur fifths barabar size ke hissay nahi hain. Common denominator unhein same unit mein laata hai. Bacha budget 1 − 13/20 = 20/20 − 13/20 = 7/20 hai. Subtraction bhi same-sized hissay compare karti hai. Yeh complement tab sahi hai jab stated categories overlap na karti hon aur baqi hissa isi whole ka ho.

Ab maan lein budget ka 3/5 books ke liye hai aur us book budget ka 2/3 textbooks par lagta hai. Doosra fraction original whole ki bajaye pehle selected hissay par hai:

Textbooks ka original-budget share = (2/3) × (3/5) = 6/15 = 2/5.

Multiplication mein numerators ko aur denominators ko multiply karein. Common factors cancel karke calculation chhoti ho sakti hai. Addition aur multiplication ka farq wording se samjhein: "whole ka yeh aur woh hissa" addition ho sakti hai; "is hissay ka itna hissa" multiplication hai.

Division multiplication ko reverse karti hai. Agar 3/4 metre ek piece ki length ho, 3/8 metre us piece ka kitna hissa hai? (3/8) ÷ (3/4) = (3/8) × (4/3) = 1/2. Reciprocal se multiply is liye karte hain ke (3/4) × (4/3) = 1; is se unknown share alag ho jata hai. Zero proportion se divide nahi kar sakte.

Part, whole aur proportion dhoondhna

Basic relation hai:

Part = proportion × whole.

Is mein proportion decimal ya fraction hai, percent symbol wala number seedha multiplier nahi. 240 items ka 35% 0.35 × 240 = 84 items hai, 35 × 240 nahi.

Isi relation ko reverse karein: 84 items kisi batch ka 35% hon to whole = 84/0.35 = 240. Check: 0.35 × 240 = 84. Agar 84 out of 240 items selected hon to proportion = 84/240 = 0.35 = 35%. Part aur whole dono ki units comparable hon; 500 gram ko 2 kilogram se divide karne se pehle 2 kilogram ko 2000 gram karein, phir share 500/2000 = 1/4 hai.

Whole nonzero hona chahiye. Zero students ki class mein attendance percentage 0/0 se defined nahi hoti. Agar proportion zero aur part zero hai, is information se whole recover nahi hota: 0% of 10 aur 0% of 100 dono zero hain.

Percentage change hamesha stated original base se hoti hai

Ek fee 400 se 460 rupees ho gayi. Amount increase 460 − 400 = 60 rupees hai. Percentage increase 60/400 × 100 = 15%: denominator original fee hai.

Original W mein p% increase ho to new value W + (p/100)W = (1 + p/100)W hai. Decrease mein new value (1 − p/100)W hai. 400 mein 15% increase ka multiplier 1.15 hai; 15% decrease ka multiplier 0.85 hai. Ordinary nonnegative quantities mein 100% decrease zero karta hai; us se zyada decrease negative value dega aur context ki justification chahega.

Percentage points ka matlab do percentage readings ka seedha difference hai. Attendance 40% se 50% ho to 10 percentage points barhi, lekin relative increase (50 − 40)/40 × 100 = 25% hai. Yahan original percentage reading 40 base hai. Dono answers alag sawalon ke hain.

Changed value se original value kaise nikalein

Sale ke baad price 720 rupees hai aur original price par 10% reduction hui thi. New price original ka 90% hai:

0.90W = 720, is liye W = 720/0.90 = 800 rupees.

Check: 800 ka 10% 80 hai aur 800 − 80 = 720. 720 mein 10% add karna 792 deta hai; woh original ka 10% nahi, changed value ka 10% hai.

Increase ko bhi multiplier se reverse karein. Output 25% increase ke baad 300 units ho to original 300/1.25 = 240 units hai. Check: 240 + 60 = 300.

100% decrease ka multiplier zero hai. Final zero aur decrease 100% ho to original uniquely recover nahi hota: koi bhi nonnegative original zero ban sakta hai. Final nonzero ho to yeh stated conditions ek doosre se contradict karti hain.

Check yourself 10% reduction ke baad original recover karne ke liye final ko 1.10 se multiply kyun nahi karte?Try first, then view the answer

Answer

Reduction original base ki thi. Final original ka 0.90 hai, is liye final/0.90 original deta hai; 1.10 se multiplication final base par ek naya increase lagati hai.

Successive steps mein base badal sakta hai

Budget ka 1/4 pehle kharch hua, phir bache budget ka 1/3 kharch hua. Pehle remaining 3/4 hai; doosra kharch original ka (1/3) × (3/4) = 1/4 hai. Total kharch 1/4 + 1/4 = 1/2, remaining 1/2 hai. 1/4 + 1/3 add karna ghalat hai kyun ke doosra fraction remaining base ka hai.

100 rupees mein 10% increase ke baad 110, phir 10% decrease ke baad 99 rupees hote hain. Overall multiplier 1.10 × 0.90 = 0.99; net decrease 1% hai. Equal percentages cancel nahi huin kyun ke doosre step ka base 110 tha. Do successive 10% increases ka multiplier 1.10 × 1.10 = 1.21, yani 21% overall increase hai.

Agar question dono increases ko same original value par calculate karne ko kehta hai, tab unke change amounts add ho sakte hain. Faisla wording ke base se karein, ek memorised shortcut se nahi.

Alag groups ka combined share

Team A mein 40 people hain aur 25% selected hain: 10 people. Team B mein 60 people hain aur 50% selected hain: 30 people. Groups disjoint hain, yani koi person dono mein count nahi hota. Total selected 40 aur total people 100, is liye combined share 40% hai.

Combined share = (pA × A + pB × B)/(A + B).

Yahan pA, pB fraction/decimal proportions aur A, B group sizes hain. (25% + 50%)/2 = 37.5% groups ko equal weight deta hai, jab ke sizes equal nahi. Equal group sizes par simple average valid hai; equal selection proportions par bhi combined proportion wahi rahegi, chahe sizes alag hon.

Absolute sizes na hon lekin A:B = 2:3 ho to A=2k, B=3k likh sakte hain. Upar wali proportions ke saath combined share (0.25×2 + 0.50×3)/5 = 40% hai; k cancel hota hai. Lekin actual selected count 2k hai aur k ke baghair woh count nahi nikalta. Size ratio share ke liye kaafi ho sakta hai, total people ke liye nahi.

Agar group sizes aur unka ratio dono missing hon, distinct selection proportions se overall share bhi fixed nahi hoti. A aur B ke sizes 40,40 hon to selected 10+20=30 out of 80, yani 37.5%. Sizes 40,80 hon to selected 10+40=50 out of 120, yani 41 2/3%. Dono given percentages satisfy karte hain. Do valid results non-uniqueness sabit karte hain.

Past Paper se connection

Supplied attendance question mein boys ka 1/3 aur girls ka 1/2 present hai. Boys B aur girls G hon to present count B/3 + G/2 hai. Sizes nahi di gayin: B=60,G=60 par 50 present, aur B=90,G=60 par 60 present hain. Is liye total uniquely determined nahi; equal groups assume karna missing information invent karega.

Supplied reverse-increase question mein original x ka 20% add hone se 102 banta hai. 1.20x=102, is liye x=85; 85+17=102 forward check hai. Yeh same original-base reasoning hai jo decrease, kisi aur percentage aur multiple steps mein bhi kaam karti hai.

References

Fraction foundations aur operations ke liye OpenStax Prealgebra 2e §4.1, §4.2 aur §4.5 dekhein. Percent conversion aur original-base applications §6.1 aur §6.2 mein hain. Is lesson ke group aur successive-step calculations isi part/whole relation se derive kiye gaye hain.