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Chapter 1 “Fractions in Disguise” percentage ke baare mein hai. Percentage samajhne ke liye kuch chhoti skills chahiye: fraction ka “/”, decimal point, kuch symbols, aur thoda sa guna-bhaag. Agar inmein se kuch bhool gaye ho ya kabhi padha hi nahi, koi baat nahi! Yeh post ek “toolbox” hai. Har section 2–3 minute ka hai. Jo aata hai, skip karo; jo nahi aata, yahan seekho, phir chapter shuru karo.
📝 Shuru karne se pehle 5 chhote shabd (inka poora lesson neeche hai, par yeh pehle bhi kaam aayenge):
- ≈ = “lagbhag barabar”. Jab jawab bahut lamba ho (33.3333…), use chhota karke likhte hain: 33.33. Ise rounding kehte hain (rule: section 16).
- Indian commas: right se pehle 3 digit, phir 2-2 digit. 2,36,000 = do lakh chhattees hazaar (section 14).
- Even number: jo 2 se poora bat jaaye (0, 2, 4, 6, 8 pe khatam), jaise 18, 40, 96. Odd: 1, 3, 5, 7, 9 pe khatam.
- Positive number: 0 se bada number, jaise 5, 2.5, 100. (section 14)
- ≥ = “bada ya barabar”, ≤ = “chhota ya barabar” (section 11).
Chapter ke har post ke upar ek 🧰 “Atak Gaye?” box hai jo seedha yahan ke sahi section pe le aata hai.
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1️⃣ Fraction aur “/” ka matlab
3/4 padho “teen bata chaar” ya “4 mein se 3”. Ek roti ke 4 barabar tukde kiye aur unmein se 3 liye.
- Upar wala number (3) = numerator (ansh): kitne hisse liye.
- Neeche wala number (4) = denominator (har): poori cheez kitne barabar hisson mein bati.
- Beech ki “/” line ka matlab “÷” (bhaag) bhi hai: 3/4 = 3 ÷ 4.
Isliye 45/100 ka matlab: 100 barabar hisson mein se 45. Yahi percentage hai: 45% = 45/100.
Fraction ka guna kisi number se: 3/5 × 100 matlab “100 ka 3/5”. Pehle 100 ÷ 5 = 20 (ek hissa), phir 3 hisse = 3 × 20 = 60. Ya upar-upar guna karo: 3 × 100 = 300, phir neeche se bhaag: 300 ÷ 5 = 60. Dono same!
“Equivalent fractions” (barabar fractions): upar aur neeche dono ko ek hi number se guna karo to value nahi badalti. 3/5 = (3 × 20)/(5 × 20) = 60/100. Aadhi roti = 2/4 roti = 50/100 roti, sab same.
✍️ Khud try karo:
- 2/5 ka matlab kya hai?
- 100 ka 3/4 kitna?
- 7/20 ko denominator 100 wala banao.
Answers dekho
- 5 barabar hisson mein se 2.
- 100 ÷ 4 = 25, phir 3 × 25 = 75.
- 20 × 5 = 100, to upar bhi × 5: 35/100.
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2️⃣ Factor, HCF aur Simplify karna
Factor (gunankhand): jo number kisi number ko poora-poora baant de (baaki 0). 12 ke factors: 1, 2, 3, 4, 6, 12 (kyunki 12 ÷ 3 = 4, baaki 0). 5, 12 ka factor nahi (12 ÷ 5 mein baaki 2 bachta hai).
Number ko factors mein todna: 8 = 2 × 2 × 2. 20 = 2 × 2 × 5. 100 = 2 × 2 × 5 × 5. Yeh Post 5 mein kaam aayega.
Multiple (gunaj): kisi number ki table. 4 ke multiples: 4, 8, 12, 16, … (har ek 4 se poora bat jaata hai).
Common factor: jo dono numbers ka factor ho. 65 aur 100: 65 ke factors 1, 5, 13, 65; 100 ke factors 1, 2, 4, 5, 10, 20, 25, 50, 100. Dono mein common: 1 aur 5.
HCF (Highest Common Factor) = sabse bada common factor. HCF(65, 100) = 5.
Simplify karna (simplest form): upar aur neeche dono ko ek hi number se bhaag do, jab tak koi common factor (1 ke alawa) na bache. Sabse jaldi: HCF se ek baar mein bhaag do.
Common divisor dhoondhne ki tricks: dono even (yaani 2 se poore bat jaate hain: 0, 2, 4, 6, 8 pe khatam) → 2 se bhaag. Dono 0 ya 5 pe khatam → 5 se. Dono 0 pe khatam → 10 se.
✍️ Khud try karo:
- 18/24 ko simplify karo.
- HCF(36, 45)?
- 35/100 simplest form?
Answers dekho
- Dono ÷ 6 → 3/4.
- 36: 1,2,3,4,6,9,12,18,36. 45: 1,3,5,9,15,45. Sabse bada common = 9.
- Dono ÷ 5 → 7/20.
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3️⃣ Kaatna (Cancel): bade guna ko chhota banana
Jab fraction ka guna kisi number se ho, to pehle kaat lo, phir guna karo. Kaatna = upar ke kisi number aur neeche ke kisi number ko ek hi number se bhaag dena.
WHY yeh sahi hai? Guna aur bhaag ka order badalne se jawab nahi badalta. 52 × 1500 ÷ 100 = 78,000 ÷ 100 = 780. Kaatne se bas chhote numbers mile.
✍️ Khud try karo:
- 64/100 × 4500 = ?
- 33/100 × 600 = ?
Answers dekho
- 4500 ÷ 100 = 45, to 64 × 45 = 2,880.
- 600 ÷ 100 = 6, to 33 × 6 = 198.
— 📌 charumam.com
4️⃣ Mixed numbers: 33 1/3 kya hai?
33 1/3 padho “taintees aur ek-tihaai”. Matlab 33 + 1/3. Jaise 2 1/2 roti = 2 poori roti + aadhi roti.
- 33 1/3 % = 33% + ek-tihaai pratishat ≈ 33.33% (≈ = lagbhag; 1/3 = 0.333… ko chhota karke likha, rounding).
- 16 2/3 = 16 + 2/3 ≈ 16.67.
- 11 1/9 = 11 + 1/9 ≈ 11.11.
Bada fraction → mixed: 100/3 = ? 100 ÷ 3 = 33, baaki 1. To 100/3 = 33 1/3. (Baaki upar, bhaajak neeche.)
Mixed → seedha fraction: 2 1/2 = (2 × 2 + 1)/2 = 5/2. 3 1/4 = (3 × 4 + 1)/4 = 13/4.
Check: 6 × 16 2/3 = 6 × 16 + 6 × 2/3 = 96 + 4 = 100 ✅
✍️ Khud try karo:
- 50/3 ko mixed number mein likho.
- 5 1/2 ko seedha fraction mein likho.
Answers dekho
- 50 ÷ 3 = 16, baaki 2 → 16 2/3.
- (5 × 2 + 1)/2 = 11/2.
— 📌 charumam.com
5️⃣ Place value aur decimal point
Har number ke digits alag-alag “gharon” mein rehte hain. Har ghar apne right wale ghar ka 10 guna hota hai.
| Hazaar | Sau | Das | Ek | . | Dasvaan (tenths) | Sauvaan (hundredths) | Hazaarvaan |
|---|---|---|---|---|---|---|---|
| 1000 | 100 | 10 | 1 | . | 1/10 | 1/100 | 1/1000 |
| 2 | 4 | 5 | 0 | . | 7 | 5 |
2450.75 = 2 hazaar + 4 sau + 5 das + 0 ek + 7 dasvaan (7/10) + 5 sauvaan (5/100).
Decimal point (.) “ek” ke ghar ke baad lagta hai. Uske baad wale hisse 1 se chhote hote hain. 0.5 = 5/10 = aadha. 0.25 = 25/100 = ek-chauthai. 0.07 = 7/100.
Har poore number mein decimal chhupa hota hai: 450 = 450.0 = 450.00. Point dikhta nahi, par “ek” ke ghar ke baad hota hai. Peeche zero lagane se value nahi badalti: 0.3 = 0.30 = 0.300.
✍️ Khud try karo:
- 0.6 ko fraction mein likho.
- 3.25 mein 2 kis ghar mein hai?
Answers dekho
- 6/10 = 3/5.
- Dasvaan (tenths): 2/10.
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6️⃣ × 10, ÷ 10, × 100, ÷ 100: point khiskao
| Kaam | Decimal point kidhar? | Example |
|---|---|---|
| ÷ 10 | 1 ghar baayein (left) | 450 → 45.0 = 45; 85 → 8.5; 7 → 0.7 |
| ÷ 100 | 2 ghar baayein (left) | 3700 → 37; 245 → 2.45; 1.5 → 0.015 |
| ÷ 1000 | 3 ghar baayein (left) | 3500 → 3.5; 4 → 0.004 |
| × 10 | 1 ghar daayein (right) | 2.5 → 25; 0.07 → 0.7 |
| × 100 | 2 ghar daayein (right) | 0.07 → 7; 0.375 → 37.5; 1.2 → 120 |
Yaad rakhne ki trick: bhaag se number chhota hota hai, to point left (chhoti taraf) jaata hai. Guna se number bada, to point right.
Zero wala shortcut: agar number zero pe khatam ho, to ÷ 10 = ek zero hatao (450 → 45), ÷ 100 = do zero hatao (20,000 → 200).
Khaali ghar mein 0 bharo: 1.5 ÷ 100 → point 2 ghar left: 0.015 (beech mein 0 bharna pada).
✍️ Khud try karo:
- 2,450 ÷ 10 = ?
- 0.045 × 100 = ?
- 1440 ÷ 100 = ?
Answers dekho
- 245.
- 4.5.
- 14.4.
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7️⃣ Decimal jodna aur ghatana
Rule ek hi hai: decimal points ko ek line mein khada karo, khaali jagah mein 0 bharo, phir normal jodo/ghatao.
| Jodna: 45 + 22.5 | Ghatana: 4.5 − 0.45 |
|---|---|
45.0 + 22.5 ------ 67.5 | 4.50 − 0.45 ------ 4.05 |
Dekho: dono points ek hi seedhi line (column) mein hain. Isi tarah upar-neeche likh ke jodo/ghatao.
Paise se socho: ₹45 + ₹22.50 = ₹67.50. ₹4.50 − ₹0.45 = ₹4.05.
✍️ Khud try karo:
- 450 − 22.5 = ?
- 12.5 + 2.5 = ?
Answers dekho
- 450.0 − 22.5 = 427.5.
- 15.0 = 15.
— 📌 charumam.com
8️⃣ Decimal ka guna
Tareeka 1: Todke (decimal × poora number). 1.5 × 40 = (1 × 40) + (0.5 × 40) = 40 + 20 = 60. (0.5 = aadha, to 40 ka aadha = 20.)
(0.18 × 2,000 = 360 kaise? Neeche Tareeka 2 se: point hatao → 18 × 2,000 = 36,000. 0.18 mein point ke baad 2 digit the, to 36,000 mein point 2 ghar left → 360.00 = 360.)
Tareeka 2: Point hatao, guna karo, point wapas lagao.
- Points bhool ke normal guna karo.
- Dono numbers mein point ke baad kitne digits the, gin ke jodo.
- Jawab mein right se utne ghar ginke point lagao.
✍️ Khud try karo:
- 0.9 × 0.9 = ?
- 1.5 × 0.5 = ?
- 12.5 × 22 = ?
Answers dekho
- 9 × 9 = 81, 2 digits → 0.81.
- 15 × 5 = 75, 2 digits → 0.75.
- 125 × 22 = 2750, 1 digit → 275.0 = 275.
— 📌 charumam.com
9️⃣ Decimal se bhaag (÷ 1.25, ÷ 1.18)
Rule: bhaag mein upar-neeche (dono numbers) ko ek hi number se guna karne se jawab nahi badalta. To bhaajak (jisse bhaag de rahe ho) ka decimal hata do.
Percentage wala aasaan raasta (1% method): “÷ 1.18” ka matlab hai “118% diya hai, 100% chahiye”. 118% = 2,360 → 1% = 2,360 ÷ 118 = 20 → 100% = 20 × 100 = 2,000. Chapter mein hum yahi tareeka zyada use karenge!
✍️ Khud try karo:
- 27,000 ÷ 0.9 = ?
- 1,500 ÷ 1.2 = ?
Answers dekho
- Dono × 10 → 2,70,000 ÷ 9 = 30,000.
- Dono × 10 → 15,000 ÷ 12 = 1,250.
— 📌 charumam.com
🔟 Lambi bhaag: 3/8 ko decimal kaise banayein?
3/8 = 3 ÷ 8. Par 3 mein 8 ek baar bhi nahi jaata! To 3 ko 3.000 likho (peeche zero lagane se value nahi badalti) aur dasvaan, sauvaan ghar mein bhaag chalao.
| Kadam | Kya kiya | Jawab mein likha |
|---|---|---|
| 1 | 3 ÷ 8 = 0 (poora nahi jaata). Ab point aata hai. | 0. |
| 2 | Ek 0 neeche laao: 30 ÷ 8 = 3, kyunki 8 × 3 = 24. Baaki 30 − 24 = 6 | 0.3 |
| 3 | Agla 0 neeche: 60 ÷ 8 = 7, kyunki 8 × 7 = 56. Baaki 60 − 56 = 4 | 0.37 |
| 4 | Agla 0 neeche: 40 ÷ 8 = 5, kyunki 8 × 5 = 40. Baaki 0, ruk jao! | 0.375 |
Poore numbers ki lambi bhaag: 50,000 ÷ 125 (left se ek-ek digit lo):
| Kadam | Kya kiya | Jawab mein likha |
|---|---|---|
| 1 | 5 mein 125 nahi jaata, 50 mein bhi nahi. To pehle teen digit lo: 500. | – |
| 2 | 500 ÷ 125 = 4, kyunki 125 × 4 = 500. Baaki 500 − 500 = 0 | 4 |
| 3 | Agla digit 0 neeche: 0 ÷ 125 = 0 | 40 |
| 4 | Aakhri digit 0 neeche: 0 ÷ 125 = 0. Digits khatam! | 400 |
Check: 125 × 400 = 50,000 ✅. Ek aur: 4600 ÷ 70 → dono se ek zero hatao → 460 ÷ 7. 46 ÷ 7 = 6 (7 × 6 = 42), baaki 4 → 40 ÷ 7 = 5 (35), baaki 5 → ab point lagao, 50 ÷ 7 = 7 (49), baaki 1 → … = 65.7…
Kabhi baaki 0 nahi hota: 1 ÷ 3 → 10 ÷ 3 = 3 baaki 1 → phir 10 ÷ 3 = 3 baaki 1 … hamesha. To 1/3 = 0.333… (“…” matlab 3 aage chalta hi rahega).
✍️ Khud try karo:
- 1/4 ko lambi bhaag se decimal banao.
- 2/3 = ?
Answers dekho
- 1.00 ÷ 4: 10 ÷ 4 = 2 baaki 2, 20 ÷ 4 = 5 baaki 0 → 0.25.
- 0.666… (6 baar-baar).
— 📌 charumam.com
1️⃣1️⃣ Symbols ki chhoti dictionary
| Symbol | Padho | Example |
|---|---|---|
| > | “se bada hai” | 75 > 72 (muh bade number ki taraf khulta hai) |
| < | “se chhota hai” | 25 < 50 |
| ≥ / ≤ | “bada ya barabar” / “chhota ya barabar” | agla digit ≥ 5 matlab 5, 6, 7, 8 ya 9; marks ≤ 100 |
| = / ≠ | “barabar” / “barabar nahi” | 0.6 = 60%, par 0.6 ≠ 60 |
| ≈ | “lagbhag barabar” | 2/3 ≈ 0.67 (round kiya hua) |
| ± | “jodo ya ghatao” | 100 ± 15 = 115 (profit) ya 85 (loss) |
| @ | “ki dar se” | CGST @ 9% = 9% ki dar se CGST |
| 3 : 2 | “3 is to 2” (ratio) | har 3 ke saath 2; pehla doosre ka 1.5 guna |
| … | “aage aise hi chalta rahe” | 0.333… ; 7, 17, 27, … 97 |
| → | “isliye / phir” | 20% = 36 → 1% = 1.8 |
| ( ) | “pehle bracket ke andar ka kaam” | (60 + 70) ÷ 2 = 130 ÷ 2 = 65 |
| − number ke aage | “minus / ghaate wala” | −₹40 = ₹40 ka ulta (dena pada) |
Order of work: pehle bracket, phir × aur ÷ (left se right), phir + aur −. 2 + 3 × 4 = 2 + 12 = 14.
Average (ausat): sab jodo, ginti se bhaag do. 60 aur 70 ka average = (60 + 70) ÷ 2 = 65.
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1️⃣2️⃣ Powers: 1.1³ ka matlab
Chhota upar likha number batata hai kitni baar khud se guna karna hai.
- 5² (“5 ka square”) = 5 × 5 = 25
- 1.1² = 1.1 × 1.1 = 1.21
- 1.1³ = 1.1 × 1.1 × 1.1 = 1.21 × 1.1 = 1.331
- 0.9³ = 0.9 × 0.9 × 0.9 = 0.81 × 0.9 = 0.729
Saal-dar-saal badhne wale sawaalon (Post 11, 12) mein “3 saal” ka matlab hota hai “3 baar guna”, isliye wahan power dikhti hai. Par hum har jagah saal-dar-saal table bhi banayenge, taaki power ke bina bhi samajh aaye.
✍️ Khud try karo:
- 2³ = ?
- 1.2² = ?
Answers dekho
- 2 × 2 × 2 = 8.
- 1.2 × 1.2 = 1.44.
— 📌 charumam.com
1️⃣3️⃣ Letters (x, p, r) formula mein kya karte hain?
Formula mein letter ek khaali dabba hai jismein koi bhi number daal sakte ho. “x% = x/100” matlab: jo bhi number % ke saath ho, use 100 ke upar likh do. x = 45 daalo: 45% = 45/100.
- Do cheezein saath likhi hon to guna: “pr” = p × r. “1.5x” = 1.5 × x.
- Bracket ka matlab pehle andar ka kaam: p(1 + r) = p × (1 + r).
- Hum chapter mein formula ke saath hamesha asli number wala hal bhi dikhayenge. Letters Class 8 ke algebra chapters mein detail se aayenge, yahan sirf “shortcut likhne ka tareeka” hain.
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1️⃣4️⃣ Zaroori shabd: lakh, crore, prime, positive
Indian number system (commas): right se pehle 3 digit, phir 2-2 digit.
| Number | Naam |
|---|---|
| 1,000 | ek hazaar |
| 1,00,000 | ek lakh (100 hazaar) |
| 10,00,000 | das lakh |
| 1,00,00,000 | ek crore (100 lakh) |
1.5 crore = 1 crore + aadha crore = 1,50,00,000. 50 lakh = 50,00,000. 24,500 = chaubees hazaar paanch sau.
Prime number: jiske sirf do factors hon: 1 aur khud. 2, 3, 5, 7, 11, 13 … prime hain. 1 prime nahi (uska ek hi factor hai). 9 prime nahi (9 = 3 × 3).
Even / odd: 2 se poora bate = even (2, 4, 6…), na bate = odd (1, 3, 5…).
Positive / negative number: 0 se bada number positive hai (3, 12.5, 1000). 0 se chhota number negative hai, uske aage “−” lagta hai (−40 = 40 ka ulta, jaise ₹40 dena pada). 0 na positive, na negative. Chapter mein daam, marks, log, sab positive numbers hain.
— 📌 charumam.com
1️⃣5️⃣ Units badalna
| Badi unit | Chhoti unit | Badi → chhoti |
|---|---|---|
| 1 kg | 1000 g | × 1000 (0.4 kg = 400 g, 8.5 kg = 8500 g) |
| 1 m | 100 cm | × 100 (2 m = 200 cm) |
| 1 km | 1000 m | × 1000 (4 km = 4000 m) |
| 1 litre | 1000 ml | × 1000 |
| 1 ghanta | 60 minute | × 60 (216 minute = 3 ghante 36 minute, kyunki 3 × 60 = 180, 216 − 180 = 36) |
| 1 din | 24 ghante = 1440 minute | 24 × 60 |
| 1 GB | 1000 MB (lagbhag) | mobile data ki unit |
Chhoti → badi: ulta bhaag karo: 450 g = 450 ÷ 1000 = 0.45 kg. Percentage lagane se pehle dono cheezon ki unit same karo.
— 📌 charumam.com
1️⃣6️⃣ Rounding (≈): ek hi rule
Jab jawab lamba ho (91.666…), use chhota karke likhte hain aur “≈” lagate hain.
- Tay karo kahan tak rakhna hai (jaise point ke baad 1 digit).
- Uske agle digit ko dekho.
- Agla digit 5 ya zyada → pichhla digit 1 badhao. 4 ya kam → waise hi chhod do.
Zaroori: beech ke kadamon mein round mat karo; sirf aakhri jawab ek baar round karo. Log, bachche, cheezein poore numbers mein hote hain, isliye unka aakhri jawab poore number tak round hota hai.
— 📌 charumam.com
🟢 Toolbox Recap
- “/” = bhaag; upar numerator, neeche denominator.
- Simplify = upar-neeche ek hi number se bhaag (HCF se sabse jaldi).
- ÷ 100 = point 2 ghar left; × 100 = point 2 ghar right.
- Decimal guna: points hatao, guna karo, digits gin ke point lagao.
- Decimal se bhaag: dono ko 10/100 se guna karke point hatao, ya 1% method.
- > bada, < chhota, ≈ lagbhag, ± jodo/ghatao, 1.1³ = 1.1 × 1.1 × 1.1.
- 1 lakh = 1,00,000; 1 crore = 1,00,00,000; prime = sirf 1 aur khud se bate.
— charumam.com 🎯
👉 Aage Kya?
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