Guna = Baar-baar Jodna (Array se samjho)
CharuMam ka Line-Lagao Formula: cheezein barabar lines mein sajao. Kitni lines × har line mein kitne = kul. Guna bas barabar groups ka tez jod hai 🟡🟡🟡
🔥 Kahani: Priya ki mummy ka laddoo dabba
Diwali se pehle Priya ki mummy ne laddoo ka ek dabba bheja. Dabba khula toh andar laddoo line se sajaye hue the. Golu ne ungli rakh ke ginna shuru kiya: “1, 2, 3, 4… 17, 18… arre, kahan tak gina tha?” Phir se shuru. Teen baar ginti toot gayi.
Rohan hansa: “Ek-ek kyun gin raha hai? Dekh, har line mein 6 laddoo hain. Pehli line 6, doosri milake 12, teesri 18, chauthi 24. Bas!”
Priya ne aur chhota kar diya: “4 line, har line mein 6. Yaani 6 + 6 + 6 + 6, aur yahi toh hai 4 × 6 = 24.”
Golu ki aankhein badi ho gayin: “Matlab guna koi naya jaadu nahi? Bas jod hi hai, tez wala?” Bilkul! Aaj samjhenge ki guna kyun baar-baar jodna hai, aur cheezon ko lines (array) mein sajane se guna aankhon ke saamne kaise dikhne lagta hai.
✅ Prerequisite Checker
Yeh post samajhne ke liye itna aana chahiye:
- Jod aaram se aana: 6 + 6 = 12, 12 + 6 = 18 jaise (MN-09: Mental Addition)
- 10 ke ghar samajhna: 30 + 12 = 42, aur 10 ke guna mein peeche 0 kyun lagta hai (MN-01: Place Value)
- Thoda sa “ek kam karo, phir bharpaai karo” wala idea (MN-10: Compensation), sirf Example 2 ke liye
Bas! Agar tum 5, 10, 15, 20… bol ke gin sakte ho, toh tum taiyaar ho. 🙂
📘 Concept: Guna = barabar groups ka jod
4 × 6 ko padho: “4 baar 6“. Yaani 6 ko 4 baar jodo: 6 + 6 + 6 + 6 = 24.
Kyun yeh shorthand chahiye? Socho 25 baar 6 jodna ho. “6 + 6 + 6 + …” likhte-likhte copy bhar jaayegi. “25 × 6” teen akshar mein poori baat keh deta hai.
5 + 5 + 5 = 3 × 5 = 15 ✅. Par 5 + 7 + 4 ko “3 × 5” nahi bol sakte, kyunki teeno group alag size ke hain. Uska jawaab seedha jod hai: 16.
Kyun? “3 × 5” ka matlab hai “3 group, har ek mein 5″. Agar ek group mein 7 hai, toh yeh vaakya jhooth ho gaya. Guna sirf tab, jab har group same ho.
Laddoo dabbe jaisa: 4 line (rows), har line mein 6. Lines kitni = “kitni baar”, har line mein kitne = “har baar kitne”. Kul = 4 × 6.
Kyun array itna kaam ka hai? Bikhre hue 24 laddoo ginne mein galti hoti hai (Golu ki tarah). Line mein sajao toh ek nazar mein dikh jaata hai ki sab group barabar hain, aur sirf lines ginni padti hain.
Pehli line: 6. Doosri milake: 12. Teesri: 18. Chauthi: 24. Yahi toh 6 ka table hai: 6, 12, 18, 24!
Kyun? Har nayi line pe ek aur 6 jud jaata hai. Table koi ratne wali list nahi, woh array ki line-line ginti hai. 6 × 4 bhool gaye? 6, 12, 18, 24 bolo, jawaab mil jaayega.
7 × 6 yaad nahi? 7 lines ko 5 line + 2 line mein todo. 5 × 6 = 30, 2 × 6 = 12. Kul 30 + 12 = 42.
Kyun yeh chalta hai? Laddoo wahi hain, bas beech mein ek dhaaga daal ke do hisse kar diye. Dono hisson ke laddoo jodo toh poore wapas. Kuch gaya nahi, kuch aaya nahi. (Yahi soch aage Grid Method, MN-20, ki neev hai.)
Laddoo ki jagah 1 cm ke chhote khaane socho. 4 cm lamba, 3 cm chauda patta = 3 line, har line mein 4 khaane = 12 khaane. Isliye area = lambai × chaudai.
Kyun yeh jodna zaroori hai? Area ka formula koi alag rule nahi, yeh array ki ginti hi hai. Ek baar yeh dikh gaya toh formula bhoolna mushkil.
1 × 9 = 1 line mein 9 = 9. 0 × 9 = 0 line, har line mein 9 = koi laddoo hi nahi = 0. Aur 9 × 0 = 9 khaali dabbe = 0.
Kyun 9 + 0 = 9 par 9 × 0 = 0? Jod mein “9 laddoo aur 0 laddoo” milte hain, 9 bache. Guna mein “9 baar 0 laddoo” ka matlab hai har baar kuch nahi mila. 9 baar kuch nahi = kuch nahi.
🎯 CharuMam ka Line-Lagao Formula
“Barabar groups ko lines mein sajao. Kitni lines × har line mein kitne = kul. Array bada lage toh lines ko todo, hisse alag-alag gino, phir jodo.”
Laddoo dabba: 4 lines × 6 = 6 + 6 + 6 + 6 = 24 laddoo. Ek-ek ginne ki zaroorat hi nahi. ✅
🖼️ Picture se samjho: laddoo lines mein
Baayein taraf: 4 line, har line mein 6 laddoo. Har line pe chalti hui ginti dekho. Daayein taraf: 7 × 6 ko ek dhaage se 5 line + 2 line mein toda gaya hai.
✏️ Examples: aasan se mushkil tak
Example 1: 3 over mein kitni gendein?
Ek over = 6 gend. 3 over = 3 baar 6 = 6 + 6 + 6 = 18 gend. Guna mein: 3 × 6 = 18.
Kyun guna yahan theek hai? Har over mein barabar 6 gend hoti hain (no-ball, wide chhod do). Barabar groups = guna. Array socho: 3 line, har line mein 6 gol.
Example 2: Assembly mein 8 line, har line mein 9 bachche
Kul = 8 × 9. Do tareeke:
(a) Todo: 5 line + 3 line → 5 × 9 = 45, 3 × 9 = 27 → 45 + 27 = 72.
(b) Har line mein 1 bachcha aur khada karo toh 10 ho jaate: 8 × 10 = 80. Par har line mein 1 zyada gina, yaani 8 zyada → 80 − 8 = 72.
Kyun (b) chalta hai? Yeh MN-10 wali chaal hai: pehle aasan number (10) pe jao, phir jitna zyada liya utna wapas karo. 8 line thi, har line mein 1 zyada, toh kul 8 zyada.
Example 3: 12 × 7 (table 10 ke aage)
12 line ko 10 line + 2 line mein todo. 10 × 7 = 70, 2 × 7 = 14. Kul 70 + 14 = 84.
Check jod se: 70 + 7 + 7 = 84 ✅
Kyun 10 pe todna sabse aasan? 10 line × 7 = 7 ke 10 group = 70 (7 dahaai, MN-01). Bache 2 line ka guna chhota sa. Bade guna ko hamesha “10 wala hissa + chhota hissa” mein todo.
Example 4: 15 × 24 (bada array)
15 line, har line mein 24. Todo: 10 line + 5 line.
10 × 24 = 240. 5 line, 10 line ka aadha hai, toh 5 × 24 = 240 ka aadha = 120.
Kul: 240 + 120 = 360.
Kyun 5 line = 10 line ka aadha? 10 lines ko beech se do barabar hisson mein baanto, har hisse mein 5 lines. Toh 5 lines mein 240 ke aadhe laddoo. (Is chaal ka poora kyun MN-18 mein.) 24 ko 15 baar jodne se yeh kitna tez hai!
🧮 Khud karke dekho: CharuMam Array Machine
💡 Lines aur har line ke laddoo badlo: machine array banayegi, baar-baar jod aur table wali ginti dikhayegi. Phir “dhaaga” slider hilao: array do hisson mein tootega aur dono hisson ka jod wahi kul dega. 7 × 6 ko 5 pe todo, 12 × 7 ko 10 pe todo!
📕 Textbook yeh nahi batata
1. “4 × 6” ko do tarah padha jaata hai, aur dono theek hain. Is post mein humne padha “4 baar 6” (4 line, har line mein 6). Kai kitaabein aur teachers ise “4 ke 6 group” bhi padhte hain. Ghabrao mat: dono ka jawaab 24 hi aata hai. Laddoo ka dabba 90° ghuma do toh 6 line ho jaati hain, har line mein 4, aur laddoo wahi 24. Aisa kyun hota hai, yeh poora MN-15 mein.
2. Area ka formula array hi hai. Kamre ka farsh 4 m × 3 m hai toh 1 m × 1 m ki 12 tiles lagengi: 3 line, har line mein 4. “Lambai × chaudai” ratne ki cheez nahi, yeh tiles ki line-line ginti hai.
3. “Baar-baar jodna” ki ek seema hai. 4 × 6 mein 6 ko 4 baar jodo, theek. Par ½ × 8 mein “aadhi baar jodo” ka kya matlab? Wahan guna ka matlab badal ke “8 ka aadha hissa lo” ho jaata hai = 4. Poore numbers ke liye baar-baar jodna perfect hai; fraction aur decimal mein guna ko “hissa lena / chhota-bada karna” ki tarah sochna padta hai (MN-26 aur MN-39 mein aayega).
🧠 Bachche yahan galti karte hain
❌ Galti 1: Alag-alag size ke groups ko guna bana dena: “3 thaile hain: 6, 8 aur 5 aam. Kul = 3 × 6 = 18.”
Yeh galat kyun hai? 3 × 6 ka matlab hai “teeno thailon mein 6-6”. Par ek mein 8 aur ek mein 5 hain. Groups barabar nahi toh guna nahi chalega. Sahi: 6 + 8 + 5 = 19.
❌ Galti 2: Guna ko jod samajh lena: “4 × 3 = 7”.
Kyun galat? 4 + 3 matlab “4 cheezein aur 3 cheezein” = 7. Par 4 × 3 matlab “4 line, har line mein 3” = 3 + 3 + 3 + 3 = 12. Array bana ke dekho: 4 line mein 3-3 gole, ginti 12 aati hai, 7 nahi.
❌ Galti 3: Array toda, par doosre hisse ka guna bhool gaye: “12 × 7 = 10 × 7 + 2 = 72”.
Kyun galat? 12 line ko 10 line + 2 line mein toda. 2 line mein sirf 2 laddoo nahi, 2 × 7 = 14 laddoo hain! Har hisse mein lines × har line ke laddoo. Sahi: 70 + 14 = 84.
🌍 Real Life mein kahan?
🥚 Ande ki tray: 5 line, har line mein 6 ande = 5 × 6 = 30 ande. Dukaan wala ek-ek nahi ginta, lines dekhta hai.
🏏 T20 mein bowler: Ek bowler zyada se zyada 4 over daal sakta hai = 4 × 6 = 24 gend (wide aur no-ball chhod ke).
🏫 Classroom: 7 line mein 5-5 bench = 35 bench. Har bench pe 2 bachche → 35 × 2 = 70 bachche.
📅 Calendar: 4 hafte × 7 din = 28 din. Calendar ka har mahina bhi ek array hai: line = hafta, har line mein 7 din!
🧱 Aangan ki tiles: 12 line mein 15-15 tiles = 10 × 15 + 2 × 15 = 150 + 30 = 180 tiles.
⚖️ Kab use karein, kab nahi
| ✅ Guna (Line-Lagao) use karo jab… | ⚠️ Dhyaan rakho / doosra tareeka socho jab… |
|---|---|
| Har group mein barabar cheezein hon: har dabbe mein 12 pencil | Groups alag-alag size ke hon (6, 8, 5): wahan seedha jod karo |
| Cheezein lines aur columns mein sajee hon: tiles, seats, tray, calendar | Fraction ya decimal ka guna ho (½ × 8): “baar-baar jodo” ki jagah “hissa lo” socho |
| Ek hi daam baar-baar lage: 3 kg × ₹40 = ₹120 | Number bade hon (37 × 48): baar-baar jodna bahut lamba, array todo ya Grid Method (MN-20) |
| ❌ Kabhi nahi: guna aur jod ko mila dena (4 × 3 ≠ 4 + 3), ya array todne ke baad kisi hisse ka guna bhool jaana. | |
📝 Self-Check Quiz
Q1. 6 + 6 + 6 + 6 + 6 ko guna mein kaise likhenge? (a) 6 × 6 (b) 5 × 6 (c) 6 + 5 (d) 5 × 5
✅ (b) 5 × 6: 6 paanch baar juda hai, toh “5 baar 6” = 30. Gin ke dekho: 6, 12, 18, 24, 30.
(a) galat: 6 × 6 = 36, yaani 6 chhe baar. Yahan sirf paanch 6 hain.
(c) galat: 6 + 5 = 11. Yeh “kitni baar” aur “har baar kitna” ko jod dena hai, guna nahi.
(d) galat: 5 × 5 = 25. Har group mein 6 hai, 5 nahi.
Q2. Ande ki tray mein 4 line, har line mein 7 ande. Kul kitne? (a) 11 (b) 28 (c) 24 (d) 47
✅ (b) 28: 4 × 7 = 7 + 7 + 7 + 7 = 28.
(a) galat: 4 + 7 = 11. Lines aur har line ke ande jod diye (Galti 2).
(c) galat: 24 = 4 × 6. Har line mein 7 hain, 6 nahi. Check: 24 + 4 = 28, har line ka 1 anda chhoot gaya.
(d) galat: 4 aur 7 ko bas paas-paas likh diya. 47 ande toh 4 line ke 7-7 se kahin zyada hain.
Q3. Inme se kaunsa guna mein NAHI likh sakte? (a) 9 + 9 + 9 (b) 5 + 5 + 5 + 5 (c) 4 + 6 + 8 (d) 12 + 12
✅ (c) 4 + 6 + 8: teeno group alag size ke hain, toh ek seedha guna nahi banta. Jawaab jod se: 18.
(a) galat (yeh guna ban sakta hai): 3 × 9 = 27.
(b) galat: 4 × 5 = 20.
(d) galat: 2 × 12 = 24.
Q4. 8 × 7 ko array tod ke kaise karenge? (a) 5 × 7 + 3 × 7 (b) 5 × 7 + 3 (c) 8 × 5 + 7 (d) 5 + 3 × 7
✅ (a) 5 × 7 + 3 × 7: 8 line = 5 line + 3 line. 35 + 21 = 56 = 8 × 7 ✅
(b) galat: 3 line ka guna bhool gaye (Galti 3): 35 + 3 = 38.
(c) galat: 8 × 5 = 40 (8 line mein 5-5), phir sirf 7 joda = 47. Har line mein 7 chahiye, yaani har line mein 2 aur: 8 × 2 = 16 jodna tha → 40 + 16 = 56.
(d) galat: pehle guna hota hai (MN-50), toh 5 + 21 = 26. Pehle 5 line ka bhi guna karna tha: 5 × 7.
Q5. 7 × 0 = ? (a) 7 (b) 0 (c) 70 (d) 1
✅ (b) 0: 7 line, har line mein 0 laddoo = koi laddoo nahi. 0 + 0 + 0 + 0 + 0 + 0 + 0 = 0.
(a) galat: 7 + 0 = 7 hota hai. Jod aur guna mila diya.
(c) galat: peeche 0 lagana 10 se guna ka kaam hai: 7 × 10 = 70.
(d) galat: 1 tab aata jab 7 ÷ 7 karte. 0 se guna kuch nahi deta.
💪 Practice Zone
Har sawaal mein pehle socho ki groups barabar hain ya nahi, phir array dimaag mein banao, aur bade ho toh todo.
- 7 + 7 + 7 + 7 ko guna mein likho aur jawaab batao.
- 5 × 8 ko table wali ginti (skip counting) se nikalo.
- Ek building mein 6 floor hain, har floor pe 4 flat. Kul kitne flat?
- 9 × 6 ko 5 line + 4 line mein tod ke nikalo.
- 13 × 6 = ? (10 line + 3 line)
- Classroom mein 7 line, har line mein 5 bench, har bench pe 2 bachche. Kul bachche?
- Kya 8 + 8 + 9 + 8 ko seedha ek guna mein likh sakte ho? Kul kitna?
- Aangan mein 12 line, har line mein 15 tiles. Kitni tiles?
Answer Key (click karo)
1) 4 × 7 = 28 · 2) 8, 16, 24, 32, 40 → 40 · 3) 6 × 4 = 24 flat · 4) 5 × 6 + 4 × 6 = 30 + 24 = 54 · 5) 10 × 6 + 3 × 6 = 60 + 18 = 78 · 6) 7 × 5 = 35 bench, 35 × 2 = 70 bachche · 7) Nahi, groups barabar nahi (ek mein 9). Par 4 × 8 + 1 = 32 + 1 = 33 · 8) 10 × 15 + 2 × 15 = 150 + 30 = 180 tiles
Answer Key — charumam.com exclusive
🚀 Challenge (HOTS)
Q1. Priya ki mummy 24 laddoo ko ek poore array mein sajana chahti hain (har line mein barabar laddoo, koi laddoo bahar nahi). Kitne alag-alag tarah ke array ban sakte hain? (1 line mein 24 bhi chalega.)
Solution
8 tarah ke (lines × har line mein): 1 × 24, 2 × 12, 3 × 8, 4 × 6, 6 × 4, 8 × 3, 12 × 2, 24 × 1.
Har baar check: guna 24 aana chahiye. 5 line kyun nahi? 5 × 4 = 20, 5 × 5 = 25, 24 beech mein atak gaya: ek line adhoori rahegi. Isi tarah 7, 9, 10, 11 line bhi nahi chalengi.
Agar ghumaye hue dabbe ko same maano (4 × 6 aur 6 × 4 ek hi dabba, bas 90° ghuma), toh 4 alag shape: 1 × 24, 2 × 12, 3 × 8, 4 × 6.
Kyun yeh kaam ka hai? Jo numbers 24 ka poora array banate hain (1, 2, 3, 4, 6, 8, 12, 24) unhe 24 ke factors kehte hain. Array se factors aankh se dikhte hain (MN-25 mein aage).
Q2. Golu ko pata hai 7 × 8 = 56. Bina table dekhe 8 × 8 aur 7 × 9 nikalo, aur batao kyun.
Solution
8 × 8 = 64 aur 7 × 9 = 63.
7 × 8 = 7 line, har line mein 8. Ek line aur jodo → 8 line: 8 laddoo aur aaye → 56 + 8 = 64.
Har line mein ek laddoo aur jodo (yaani ek column) → 7 line mein 9-9: har line mein 1 zyada, kul 7 zyada → 56 + 7 = 63.
Kyun dono alag? Nayi line mein “har line jitne” (8) aate hain, par naye column mein “jitni lines” (7) aate hain. Array saamne ho toh yeh farak saaf dikhta hai, aur ek yaad kiya guna das aur gunon ki chaabi ban jaata hai.
❓ FAQ
Agar guna bas jod hai, toh tables ratne kyun padte hain?
Kyunki table baar-baar jodne ka bachaaya hua jawaab hai. 7 × 8 har baar 8 + 8 + 8… karke nikaaloge toh time lagega. Ek baar samajh lo ki table array ki line-line ginti hai, phir yaad karo. Bhool jao toh array todo (5 × 8 + 2 × 8 = 40 + 16 = 56) aur wapas pa lo. Ratna aur samajhna dushman nahi, dost hain. 🙂
Koi bhi number × 0 = 0 kyun hota hai?
Kyunki 0 baar kuch bhi jodo, ya kitni bhi baar 0 jodo, kuch nahi milta. 9 × 0 = 9 khaali dabbe. 0 × 9 = 0 line, yaani dabba hi nahi. Dono mein laddoo zero. Jod mein 0 kuch nahi badalta (9 + 0 = 9), par guna mein 0 sab kuch mita deta hai.
Array mein lines pehle bolein ya har line ke laddoo?
Is post mein humne “lines × har line mein” bola. Kahin ulta bhi bolte hain. Jawaab nahi badalta, kyunki dabba ghumane se laddoo nahi badalte. Bas ek sawaal mein ek hi tareeka rakho, taaki confuse na ho. Iska poora kyun agle post (MN-15) mein.
Bade guna ko 10 pe hi kyun todte hain?
Kyunki 10 se guna sabse aasan hai: 10 × 7 = 70 (bas 7 dahaai, MN-01). Baaki bacha hissa chhota hota hai, jaise 12 × 7 mein sirf 2 × 7. Par zaroori nahi: 8 × 9 ko 5 + 3 mein bhi tod sakte ho. Jahan tumhe aasan lage, wahan dhaaga daalo. Bas har hisse ka guna karna mat bhoolna.
🔁 Recap
- Guna = barabar groups ka jod: 4 × 6 = 6 + 6 + 6 + 6 = 24
- Sharat: har group same size ka ho. 6 + 8 + 5 ko guna nahi bana sakte
- Array = barabar groups ko lines mein sajana. Lines × har line mein = kul
- Table = array ki line-line ginti (6, 12, 18, 24…)
- Line-Lagao Formula: bada array todo (7 × 6 = 5 × 6 + 2 × 6), har hisse ka guna karo, phir jodo
- Area = lambai × chaudai, kyunki tiles bhi ek array hain
- × 1 = wahi number; × 0 = 0 (khaali dabbe)
💖 CharuMam Motivation
Golu ne 24 laddoo teen baar gine aur teeno baar ginti toot gayi. Laddoo kam nahi the, beta, bas bikhra hua soch raha tha. Jaise hi laddoo lines mein dikhe, wahi kaam 5 second ka ho gaya. 🙂
Zindagi mein bhi bada kaam bikhra hua lage toh dar lagta hai: “itna sara syllabus!” Usko lines mein sajao. Roz ek chhota, barabar hissa: roz 20 minute, roz ek topic. 30 din × 20 minute = 600 minute, yaani 10 ghante ki padhaai, bina thake. Barabar chhote kadam baar-baar jodo, yahi zindagi ka guna hai! 🌱
— CharuMam
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