11 se Guna ka Trick aur uska Kyun
CharuMam ka Padosi-Jod Formula: 11 = 10 + 1. Number ko ek ghar khiskao, phir khud ko jodo. Beech mein padosi digits ka jod aa jaata hai 🏘️
🔥 Kahani: Chacha ka jaadu aur Golu ka 8157
Rohan ke chacha ki kirana dukaan hai. Ek din ek grahak ne 11 packet chai li, ₹34 ka ek. Chacha ne bina calculator ke turant bola: “₹374.” Rohan hairaan: “Chacha, itni jaldi kaise?”
Chacha hanse: “34 ke 3 aur 4 ko alag karo. Beech mein 3 + 4 = 7 rakh do. Ban gaya 374!”
Agle din school mein Rohan ne yeh jaadu Golu ko sikhaya. Golu ne turant try kiya: 11 × 87. “8 aur 7 ke beech 8 + 7 = 15. Toh jawaab… 8157!” Priya ne calculator pe check kiya: 957. 😮
Golu chidh gaya: “Trick hi galat hai!” Priya boli: “Trick galat nahi, tumhe uska kyun nahi pata. Kyun pata hota, toh 15 ka kya karna hai, woh bhi pata hota.” Aaj wahi kyun pakdenge: trick kahan se aati hai, carry kahan jaata hai, aur 3-digit numbers pe kaise chalti hai.
✅ Prerequisite Checker
Yeh post samajhne ke liye itna aana chahiye:
- Place value: 34 mein 3 ka matlab 30, aur 10 se guna karne pe har digit ek ghar baayein khisakta hai (MN-01: Place Value). Yeh sabse zaroori hai!
- Number ko tod ke jodna, jaise 340 + 34 (MN-09: Mental Addition)
- Guna = barabar groups ka jod: 11 × 34 matlab 11 baar 34 (MN-14: Guna = Baar-baar Jodna)
Bas! Agar tum 34 × 10 = 340 turant bol sakte ho, toh tum taiyaar ho. 🙂
📘 Concept: 11 ka asli raaz
11 packet = 10 packet + 1 packet. Toh 11 × 34 = 10 × 34 + 1 × 34 = 340 + 34.
Kyun tod sakte hain? 11 line mein 34-34 cheezein hain. Upar ki 10 line alag karo, neeche 1 line alag. Cheezein wahi hain, bas do dher mein dekh rahe hain. 10 ke saath guna aasan hai, aur 1 ke saath toh number khud hi hai.
3 dahai (30) ban gaya 3 sau (300), 4 ikai (4) ban gaya 4 dahai (40). Khaali ikai mein 0 aa gaya.
Kyun? 10 guna matlab har cheez 10 guna. 10 ikai = 1 dahai, 10 dahai = 1 sau. Isliye har digit ek ghar upar chadh jaata hai (MN-01 wala Ghar-Badlo).
3 4 0
+ 3 4
= 3 7 4
Ikai: 0 + 4 = 4. Dahai: 4 + 3 = 7. Sau: 3 (neeche kuch nahi) = 3. Jawaab 374.
Kyun beech mein 3 + 4? Kyunki khiske hue 34 ka 4 aur asli 34 ka 3, dono dahai ke ghar mein aakar mil gaye. Yahi “padosi digits ka jod” hai! Chacha ka jaadu bas yahi jod hai, shortcut mein.
2-digit number AB ke liye: A | A + B | B. Jaise 11 × 52 → 5 | 5 + 2 | 2 = 572.
Kyun kinare same rehte hain? Sau ke ghar mein sirf khiska hua 5 aata hai, ikai mein sirf asli 2. Unke saath jodne ko koi padosi hi nahi hai.
11 × 87 → 8 | 8 + 7 = 15 | 7. 15 ek ghar mein nahi aata. 5 dahai mein rakho, 1 ko baayein wale ghar mein jodo: 8 + 1 = 9. Jawaab 957.
Kyun baayein? 15 dahai = 150 = 1 sau + 5 dahai. Woh 1 sau, sau ke ghar ka hai. Check: 870 + 87 = 957 ✅. Golu ne 15 ko ek hi ghar mein thoons diya tha, isliye 8157 aaya.
11 × 253 → 2 | 2 + 5 | 5 + 3 | 3 = 2 | 7 | 8 | 3 = 2783.
Kyun do jod? 2530 + 253 likho: ikai 3, dahai 3 + 5, sau 5 + 2, hazaar 2. Teen digit hain toh do padosi jodiyan (2-5 aur 5-3). Hamesha daayein se shuru karo, taaki carry aaye toh seedha baayein jod sako.
🎯 CharuMam ka Padosi-Jod Formula
“11 × N = 10 × N + N. Kinare ke digit wahi, beech mein har padosi jodi ka jod. Daayein se chalo, 9 se zyada ho toh 1 baayein bhejo!”
Golu ka sawaal: 11 × 87 → 8 | 15 | 7 → 5 rakho, 1 carry → 9 | 5 | 7 = 957. ✅
🖼️ Picture se samjho: 11 × 34 aur 11 × 87
Upar 10 × N (ek ghar khiska hua), neeche N. Ek hi ghar mein aaye digits jud jaate hain. Beech wala ghar hi padosi-jod hai.
✏️ Examples: aasan se mushkil tak
Example 1: 11 × 52
Kinare: 5 aur 2. Beech: 5 + 2 = 7. Jawaab 572.
Check: 520 + 52 = 572 ✅
Kyun itna aasan? 5 + 2 = 7, jo 9 se chhota hai, toh koi carry nahi. Har ghar mein ek hi digit baitha.
Example 2: Dukaan pe 11 pen, ₹45 ka ek
11 × 45 → 4 | 4 + 5 | 5 = 4 | 9 | 5 = ₹495.
Check: 450 + 45 = 495 ✅
Kyun 9 pe ruk gaye? 9 ek ghar mein aa jaata hai, carry tab hi jab jod 10 ya zyada ho. 4 + 5 = 9 bilkul border pe hai, par andar hai.
Example 3: 11 × 68 (carry wala)
6 | 6 + 8 = 14 | 8. Daayein se: ikai 8. Dahai mein 14 ka 4, aur 1 carry baayein: 6 + 1 = 7. Jawaab 748.
Check: 680 + 68 = 748 ✅
Kyun 6 badal gaya? 14 dahai = 1 sau + 4 dahai. Woh extra 1 sau, sau ke ghar wale 6 mein judta hai. Kinara tabhi badalta hai jab beech se carry aaye.
Example 4: 11 × 789 (3-digit, do carry)
7 | 7 + 8 | 8 + 9 | 9 = 7 | 15 | 17 | 9. Ab daayein se:
Ikai: 9. Agla: 17 → 7 likho, 1 carry. Agla: 15 + 1 = 16 → 6 likho, 1 carry. Aakhri: 7 + 1 = 8.
Jawaab 8679. Check: 7890 + 789 = 8679 ✅
Kyun daayein se? Har carry baayein wale ghar mein jaata hai. Agar baayein se shuru karte, toh baad mein aane wala carry pehle likhe digit ko badal deta aur sab ulat-pulat ho jaata. Column jod bhi toh hum daayein se hi karte hain!
🧮 Khud karke dekho: CharuMam 11-Guna Machine
💡 Machine dikhayegi: kinare ke digit, har padosi jodi ka jod, daayein se carry, aur 10 × N + N se check. 99, 789 aur 4567 try karo!
📕 Textbook yeh nahi batata
1. 11 ki powers mein chhupa pattern. 11 × 11 = 121, 121 × 11 = 1331, 1331 × 11 = 14641. Padosi-jod lagao: 1331 → 1 | 1+3 | 3+3 | 3+1 | 1 = 1 | 4 | 6 | 4 | 1. Har agla number pichhle ke padosi jodne se banta hai! Yeh pattern tab tak saaf dikhta hai jab tak koi jod 9 se upar na jaaye. 14641 × 11 mein 6 + 4 = 10 aata hai, carry aa jaata hai, aur jawaab 161051 ban jaata hai.
2. Wahi idea 101 aur 1001 ke liye. 101 = 100 + 1, toh 101 × 34 = 3400 + 34 = 3434. Aur 1001 × 234 = 234000 + 234 = 234234. Number apne aap ko do ghar ya teen ghar khiska ke jodta hai. Isliye number “do baar likha” dikhta hai!
3. Trick ratne ki cheez nahi. Jo bachcha sirf “beech mein jod do” ratta hai, woh 87 pe atak jaata hai. Jo samajhta hai ki “10 × N + N” ho raha hai, woh carry, 3-digit, 4-digit, sab khud sambhal leta hai.
🧠 Bachche yahan galti karte hain
❌ Galti 1: Jod ko poora beech mein thoons dena: “11 × 87 = 8 | 15 | 7 = 8157.”
Yeh galat kyun hai? Ek ghar mein sirf ek digit baith sakta hai. 15 dahai = 1 sau + 5 dahai. Toh 5 dahai mein, 1 sau wale ghar mein: 8 + 1 = 9. Jawaab 957. Sense check bhi karo: 11 × 87 lagbhag 11 × 90 = 990 hona chahiye, 8157 toh bahut bada hai (MN-06 wala estimation)!
❌ Galti 2: Carry galat taraf bhejna: “11 × 68 = 6 | 14 | 8 → 6 | 4 | 9 = 649.”
Kyun galat? Carry hamesha bade ghar mein jaata hai, yaani baayein. 14 dahai mein jo extra 10 dahai hai, woh 1 sau hai, ikai nahi. Sahi: 6 + 1 = 7, jawaab 748. Check: 680 + 68 = 748.
❌ Galti 3: 3-digit number pe 2-digit wala trick seedha laga dena: “11 × 253 = 2 | 2 + 5 + 3 | 3 = 2103” ya “2 | 8 | 3 = 283.”
Kyun galat? Beech mein ek jod nahi, har padosi jodi ka alag jod chahiye: 2 + 5 = 7 aur 5 + 3 = 8. Jawaab 2 | 7 | 8 | 3 = 2783. Yaad rakho: 3 digit ka number × 11 = 4 (ya 5) digit ka jawaab, kyunki 10 × N ek ghar aage chala gaya.
🌍 Real Life mein kahan?
🏏 Cricket team: Ek team mein 11 khiladi. Har khiladi ki jersey ₹56 ki. Kul: 5 | 11 | 6 → 616, yaani ₹616.
🚌 School picnic: 11 bachche, har ek ka ticket ₹75. 7 | 12 | 5 → carry → ₹825.
📱 Recharge: ₹19 ka daily pack 11 din chalaya: 1 | 10 | 9 → ₹209.
🛒 Dukaan: 11 kg aata, ₹43 kilo: 4 | 7 | 3 → ₹473. Chacha ki tarah, bina calculator!
📅 Mahine: Kisi cheez ki kist ₹250 mahina, 11 mahine: 2 | 7 | 5 | 0 → ₹2750.
⚖️ Kab use karein, kab nahi
| ✅ Padosi-Jod use karo jab… | ⚠️ Koi aur raasta socho jab… |
|---|---|
| Kisi bhi number ko 11 se guna karna ho: 11 × 52 = 572 | 12, 13 jaise numbers ho: yeh trick sirf 11 ka hai. Wahan bhi todo: 12 × 34 = 340 + 68 = 408 |
| 2-digit ya 3-digit number ho, mann mein jaldi chahiye | Bahut bada number ho (6-7 digit): seedha 10 × N + N column mein likh ke jodo, carry ka hisaab saaf rahega |
| Dukaan, ticket, team jaise “11 cheezein” wale sawaal | Number decimal mein ho (11 × 4.5): pehle 11 × 45 = 495 nikaalo, phir decimal sahi jagah lagao (49.5), dhyaan se (MN-03) |
| 110 ya 1100 se guna: pehle 11 se, phir 0 lagao (110 × 23 = 253 ke saath 0 = 2530) | Answer ka sirf andaaza chahiye: 11 × 87 lagbhag 11 × 90 = 990, itna kaafi |
| ❌ Kabhi nahi: 15 ya 17 jaise jod ko ek ghar mein likhna, ya carry ko daayein bhejna. | |
📝 Self-Check Quiz
Q1. 11 × 43 = ? (a) 473 (b) 437 (c) 54 (d) 443
✅ (a) 473: 4 | 4 + 3 | 3 = 473. Check: 430 + 43 = 473.
(b) galat: 437 mein padosi-jod 7 ko beech ki jagah aakhir mein likh diya.
(c) galat: 54 = 11 + 43. Guna ki jagah jod kar diya.
(d) galat: 443 mein beech mein jod nahi kiya, bas 4 dobara likh diya.
Q2. 11 × 76 = ? (a) 7136 (b) 836 (c) 736 (d) 737
✅ (b) 836: 7 | 13 | 6 → 3 rakho, 1 carry baayein → 8 | 3 | 6. Check: 760 + 76 = 836.
(a) galat: 13 ko ek hi ghar mein thoons diya.
(c) galat: 13 ka 3 rakha, par 1 carry bhool gaye.
(d) galat: carry daayein ikai mein bhej diya (6 + 1 = 7). Carry hamesha baayein jaata hai.
Q3. 11 wala trick kaam kyun karta hai? (a) Kyunki 11 = 10 + 1, toh 11 × N = 10 × N + N (b) Kyunki 11 prime number hai (c) Kyunki 11 odd hai (d) Kyunki 11 ke dono digit same hain, isliye jaadu hai
✅ (a): 10 × N ek ghar khiska hua N hai, aur usme N jodne se padosi digits ek ghar mein milte hain.
(b) galat: 13 bhi prime hai, par 13 pe yeh trick nahi chalta.
(c) galat: 9, 13, 15 bhi odd hain, unpe yeh trick nahi chalta.
(d) galat: koi jaadu nahi! Dono digit 1 hain isliye 10 × N aur 1 × N bante hain, yahi asli wajah hai jo (a) mein likhi hai.
Q4. 11 × 234 = ? (a) 2574 (b) 274 (c) 2340 (d) 25740
✅ (a) 2574: 2 | 2 + 3 | 3 + 4 | 4 = 2 | 5 | 7 | 4. Check: 2340 + 234 = 2574.
(b) galat: sirf ek padosi jodi (3 + 4) li, 2 + 3 wali chhod di. Aur jawaab 234 se bhi kam kaise ho sakta hai?
(c) galat: 2340 sirf 10 × 234 hai, 1 × 234 jodna bhool gaye.
(d) galat: aakhir mein faltu 0 laga diya. Yeh 110 × 234 hota.
Q5. Priya ne 11 × 95 kiya. Beech mein 9 + 5 = 14 aaya. Sahi jawaab? (a) 1045 (b) 9145 (c) 945 (d) 946
✅ (a) 1045: 9 | 14 | 5 → 4 rakho, 1 carry → 9 + 1 = 10 → 10 | 4 | 5 = 1045. Check: 950 + 95 = 1045.
(b) galat: 14 ko ek ghar mein likh diya.
(c) galat: carry bhool gaye. 945 toh 950 se bhi kam hai, jabki 10 × 95 hi 950 hai!
(d) galat: carry daayein bhej diya (5 + 1 = 6).
💪 Practice Zone
Har sawaal mein: kinare likho, padosi jodo, daayein se carry dekho. Phir 10 × N + N se check karo.
- 11 × 23 = ?
- 11 × 61 = ?
- 11 × 58 = ?
- 11 × 99 = ?
- 11 × 47 = ?
- 11 × 132 = ?
- 11 × 345 = ?
- Class ke 11 bachchon ne school trip ke liye ₹85-₹85 diye. Kul kitne paise jama hue?
Answer Key (click karo)
1) 2 | 5 | 3 = 253 · 2) 6 | 7 | 1 = 671 · 3) 5 | 13 | 8 → 638 · 4) 9 | 18 | 9 → 9 | 8 + carry → 10 | 8 | 9 = 1089 · 5) 4 | 11 | 7 → 517 · 6) 1 | 4 | 5 | 2 = 1452 · 7) 3 | 7 | 9 | 5 = 3795 · 8) 8 | 13 | 5 → ₹935 (check: 850 + 85 = 935)
Answer Key — charumam.com exclusive
🚀 Challenge (HOTS)
Q1. Golu ne dekha: 11 × 37 = 407. Beech mein 0! Aise saare 2-digit numbers dhoondo jinhe 11 se guna karne pe 3-digit jawaab aaye aur uska beech wala digit 0 ho. Kyun yahi numbers?
Solution
Beech mein 0 tab aata hai jab padosi-jod theek 10 ho: 10 ka 0 beech mein, 1 carry baayein. Toh dono digits ka jod 10 hona chahiye: 19, 28, 37, 46, 55, 64, 73, 82 (aur 91).
Jawaab: 209, 308, 407, 506, 605, 704, 803, 902. Har jawaab ka pehla digit, number ke pehle digit se 1 zyada hai (carry ki wajah se).
91 × 11 = 9 | 10 | 1 → 10 | 0 | 1 = 1001, yeh 4-digit ban gaya, isliye list mein nahi aata.
Kyun? Jod 10 se kam ho toh beech mein woh jod hi aata hai (0 nahi, jab tak dono digit 0 na hon). Jod 10 se zyada ho (jaise 11) toh beech mein 1, 2… aata hai. Sirf theek 10 pe beech khaali (0) hota hai.
Q2. Rohan kehta hai: “11 × 11 = 121, toh 111 × 11 = 1221 aur 1111 × 11 = 12221.” Kya pattern sahi hai? 11 × 11 × 11 bhi nikalo padosi-jod se.
Solution
Haan, sahi hai. 111 × 11 = 1 | 1+1 | 1+1 | 1 = 1221 (check: 1110 + 111 = 1221). 1111 × 11 = 1 | 2 | 2 | 2 | 1 = 12221 (check: 11110 + 1111 = 12221).
11 × 11 × 11 = 121 × 11 = 1 | 1+2 | 2+1 | 1 = 1331 (check: 1210 + 121 = 1331).
Kyun? Har padosi jodi 1 + 1 = 2 hai, aur kinare 1 hi rehte hain. Koi jod 9 se upar nahi gaya, toh koi carry nahi, isliye pattern itna saaf hai. Lekin 11111111111 (gyarah 1) tak bhi yeh 1, 2, 2, … 2, 1 hi dega, kyunki 1 + 1 kabhi 10 nahi hota!
❓ FAQ
Beech mein dono digits ka jod hi kyun aata hai, guna kyun nahi?
Kyunki 11 × N asal mein 10 × N + N hai, aur yeh ek jod hai. 10 × N mein N ek ghar khisak jaata hai, toh N ka pehla digit agle digit ke upar aa baithta hai. Ek ghar mein do digit aaye, toh unhe jodte hain. Isliye padosi ka jod, guna nahi.
Carry hamesha baayein hi kyun jaata hai?
Kyunki baayein wala ghar 10 guna bada hai. Dahai ke ghar mein 15 aaya matlab 15 dahai = 150 = 1 sau + 5 dahai. Woh 1 sau, sau ke ghar mein hi jaayega, ikai mein nahi. Yeh wahi niyam hai jo har column jod mein chalta hai (MN-01 aur MN-13 mein ghar ki baat dekhi thi).
Kya 12 ya 21 ke liye bhi aisa trick hai?
Wahi soch chalti hai: number ko todo. 12 = 10 + 2, toh 12 × 34 = 340 + 68 = 408. 21 = 20 + 1, toh 21 × 34 = 680 + 34 = 714. Bas ab padosi ko seedha nahi, double karke jodna padta hai, isliye 11 jitna chhota shortcut nahi banta. Asli chaabi “todo aur jodo” hai, 11 sirf sabse aasan case hai.
Jawaab sahi hai, yeh kaise pakka karein?
Do tareeke. Pehla: 10 × N + N seedha jod ke dekho (11 × 87 = 870 + 87 = 957). Doosra: andaaza lagao. 11 × N hamesha 10 × N se thoda zyada hota hai. 11 × 87 ka jawaab 870 se thoda upar hona chahiye, toh 957 sense banata hai aur 8157 nahi (MN-06: Estimation).
🔁 Recap
- 11 = 10 + 1, isliye 11 × N = 10 × N + N (11 × 34 = 340 + 34)
- 10 × N mein har digit ek ghar baayein khisakta hai
- Shortcut: kinare wahi, beech mein padosi-jod: 11 × 52 = 5 | 7 | 2 = 572
- Jod 10 ya zyada ho toh carry baayein: 11 × 87 = 957, 8157 nahi
- 3-digit mein har padosi jodi ka alag jod: 11 × 253 = 2 | 7 | 8 | 3 = 2783
- Hamesha daayein se shuru karo, taaki carry sahi ghar mein jaaye
- Check: 10 × N + N, ya andaaza (jawaab 10 × N se thoda zyada)
- Same idea: 101 × 34 = 3434, 1001 × 234 = 234234
💖 CharuMam Motivation
Golu ne trick seekhi, par uska kyun nahi seekha. Isliye pehli hi carry pe atak gaya. Priya ke paas trick kam thi, par samajh zyada. Isliye woh har naye number pe bhi sahi rahi. 🙂
Zindagi mein bhi shortcuts bahut milenge, beta. Shortcut lena galat nahi, par yeh zaroor poocho: “Yeh kaam kyun karta hai?” Jo kyun jaanta hai, woh tab bhi raasta dhoondh leta hai jab shortcut kaam karna band kar de. Trick yaad rakho, par samajh ko apna dost banao. 🌱
— CharuMam
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