Naya chapter, naya jaadugar! 🎩 Aaj se hum Trachtenberg ke niyam seekhenge. Pehla niyam hai 12 se guna ka: 12 ka pahaada yaad ho ya na ho, koi farak nahi. Bas har digit ka double karo aur uske padosi ko jodo. Har shabd (double, padosi, aage 0, carry) neeche bilkul shuru se samjhaaya hai, to aaraam se padhiye.
⏱ Before Timer: Pehle Purane Tareeke Se
Pehle school wala tareeka. 47 × 12 ko school mein do line mein karte hain, kyunki 12 = 10 + 2:
- 47 × 2 (column mein): ikai 2 × 7 = 14, neeche 4 likho, 1 carry. Dahai 2 × 4 = 8, carry ka 1 jodo: 8 + 1 = 9. Pehli line: 94.
- 47 × 10: peeche 0 lagao (Chapter 5): 470. Yeh doosri line hai.
- Dono line jodo: 470 + 94 = 564.
Teen kaam, do line, aur ek lamba jod. Aaj ki trick yeh sab ek hi line mein kar degi.
Ab phone ya ghadi mein stopwatch (samay naapne wali ghadi) chalu kijiye. Neeche wala sawaal isi school wale tareeke (column method) se kijiye, copy-pencil ke saath. Column method na aata ho to ungliyon pe gin kar bhi kar sakte hain. Jaise hi jawab mil jaaye, stopwatch rok dijiye aur apna time likh lijiye.
Mera time: ______ second
Jawab dekho
47 × 12 = 564
Abhi trick nahi dekhni! Post ke end mein After Timer mein yahi sawaal trick se karenge aur dono time ki tulna karenge. 😊
— 📌 charumam.com
✅ Pehle Yeh Aata Hai
Chahiye: guna ka matlab, carry wala jod (Chapter 2), aur ek-digit ka double. Sab neeche ke boxes mein dobara samjhaaya hai. Sabse zaroori box hai “Trachtenberg ke 5 shabd”: use do baar padhiye. Pehle naye shabd:
📖 Shabd jo is post mein aayenge
- Trachtenberg: ek engineer jinke banaye mann ke guna ke niyam hum is chapter aur agle chapter mein seekhenge (neeche box mein unki kahani).
- Niyam (rule): ek pakka tareeka jo har baar same tarah lagta hai.
- Double: number ko do baar lena: 7 ka double = 7 + 7 = 14.
- Padosi: jis digit pe kaam ho raha hai, uske thik right wala digit. 047 mein 4 ka padosi 7. Ikai ka koi padosi nahi (0 maano).
- Aage 0 lagana: number ke left mein ek 0 likhna: 47 → 047. Keemat wahi rehti hai, bas jawab ki ek extra jagah ke liye taiyaari.
- Right se left: ikai se kaam shuru karke left ki taraf badhna, jaise school ke column method mein.
- Carry (haath ka): jod 10 ya zyada aaye to sirf aakhri digit likho, baaki left wali jagah mein jodo. 16 → 6 likho, 1 carry.
- 12 = 10 + 2: 12 ko 10 aur 2 mein todte hain. Isi se trick bani hai (neeche “Yeh Kaam Kyun Karta Hai” mein).
📏 Bade numbers padhna aur round number mein jodna
Har digit ki ek jagah (place) hoti hai. Right se gino: pehli jagah Ikai (ones, 1 wale), uske left mein Dahai (tens, 10 wale), phir Saikda (hundreds, 100 wale), phir Hazaar (thousands, 1000 wale).
10 ka bundle: 10 ikai = 1 dahai (10). 10 dahai = 1 saikda (100). 10 saikde = 1 hazaar (1000). Har jagah apne right wali jagah se 10 guna badi hai.
| Number | Kaise padhein | Kis cheez se bana |
|---|---|---|
| 98 | attaanve | 9 dahai (90) + 8 ikai |
| 165 | ek sau painsath | 1 saikda (100) + 6 dahai (60) + 5 ikai |
| 999 | nau sau ninyaanve | 9 saikde (900) + 9 dahai (90) + 9 ikai |
| 1011 | ek hazaar gyaarah | 1 hazaar (1000) + 0 saikda + 1 dahai (10) + 1 ikai |
Round number mein chhota number jodna (place box niyam): chhote number ko right se likho, yaani ikai ke neeche ikai, dahai ke neeche dahai. Phir har dabbe mein jo hai, woh jawab mein utaar do. Ek digit wala number sirf ikai ka zero badalta hai, do digit wala ikai aur dahai dono.
- 100 + 65: 65 mein do digit, to 6 dahai mein aur 5 ikai mein: 1|6|5 = 165.
- 100 + 2: 2 sirf ikai mein, dahai ka 0 waise hi rehta hai: 1|0|2 = 102 (12 nahi, 120 nahi!).
- 200 + 6: 6 sirf ikai mein: 2|0|6 = 206.
- 1000 + 11: 11 mein do digit, to 1 dahai mein aur 1 ikai mein: 1|0|1|1 = 1011.
✖️ Guna kya hai? (Multiplication)
Guna (nishaan ×) matlab ek hi number ko baar-baar jodna. “6 × 4” ko bolte hain “6 guna 4”, matlab 4 ko 6 baar jodo:
6 × 4 = 4 + 4 + 4 + 4 + 4 + 4 = 24
Isi tarah 6 × 7 matlab 7 ko 6 baar jodo. Running total (har baar ka naya jod) dekho: 7, 14, 21, 28, 35, 42. To 6 × 7 = 42.
10 dahai = 1 saikda: 10 dande (har danda 10) milkar 100 bante hain, yaani 1 saikda. Isliye 24 dahai = 20 dahai + 4 dahai = 2 saikde + 4 dahai = 240. (Aur 10 saikde milkar 1 hazaar, yaani 1000, bante hain.)
Tukdon mein guna kyun kar sakte hain? 47 = 4 dande (40) + 7 cube (7). 47 ko 6 baar lena matlab neeche jaisi 6 line. Dande alag gin lo, cube alag gin lo, kul utna hi rahega:
6 × 47: 6 baar 40 = 240, 6 baar 7 = 42, kul 240 + 42 = 282
Guna mein carry 1 se bada ho sakta hai: jod mein carry aksar 1 hota hai, par guna mein bada bhi aa sakta hai. Column mein 6 × 7 = 42 aaya, yaani 4 dahai aur 2 ikai. To ikai mein 2 likho aur 4 dahai carry karo (aage le jao). Phir dahai mein 6 × 4 = 24, usme carry ke 4 jodo: 24 + 4 = 28. Jawab 282.
Guna mein jagah badlo, jawab wahi: 16 × 4 aur 4 × 16 ek hi cheez hain. Neeche dots dekho: 4 line, har line mein 16 dots. Line se gino to 4 baar 16; khade column se gino to 16 baar 4. Dots wahi hain, to jawab bhi wahi: 64. Isliye jo aasaan lage, us tarah jodo.
Copy pe column mein (28 + 15 jaisa):
- Ikai: 6 × 7 = 42. Neeche ikai mein 2 likho. 42 ke 4 dahai carry hain: chhota 4 dahai wale digit (4) ke upar likho.
- Dahai: 6 × 4 = 24, usme upar likha carry 4 jodo: 24 + 4 = 28. Neeche 28 likho (8 dahai mein, 2 saikde mein).
- Jawab: 282.
10 se guna: 3 × 10 matlab 10 ko 3 baar jodna: 10 + 10 + 10 = 30. Kisi number ko 10 se guna karo to woh har digit ek jagah left khisak jaata hai aur ikai mein 0 aa jaata hai: 3 → 30, 30 → 300.
👨🔬 Trachtenberg kaun the? Aur “niyam” ka matlab
Jakow Trachtenberg ek engineer the (machine aur pul jaisi cheezein banane wale). Doosre Vishwa Yuddh (World War 2) ke mushkil samay mein, jab woh jail mein band the, unhone apna dimaag tez rakhne ke liye mann mein guna karne ke chhote-chhote niyam banaye. Aaj in niyamon ko Trachtenberg system kehte hain.
Niyam (rule) matlab: ek pakka tareeka jo har baar same tarah lagta hai. Jaise “× 10 = peeche 0 lagao” ek niyam hai. Trachtenberg ke niyam mein pahaada (table) yaad nahi karna padta: bas har digit pe ek chhota kaam (double, jod, aadha) karna hota hai.
✌️ Double karna kya hai?
Double matlab number ko do baar lena, yaani number mein wahi number jodna. 7 ka double: 7 + 7 = 14. Yahi 2 × 7 bhi hai (GUNA box: 2 × 7 matlab 7 ko 2 baar jodo).
| Digit | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Double | 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 |
(Table mein upar digit hai, neeche uska double. Jaise 6 ke neeche 12, kyunki 6 + 6 = 12.) Is chapter mein hum sirf ek digit (0 se 9) ka double karenge, to sabse bada double 18 hai.
🧩 Trachtenberg ke 5 shabd (bahut dhyaan se)
1. Aage 0 lagao. Number ke left mein (shuru mein) ek 0 likh do: 47 → 047. Isse number ki keemat nahi badalti: 047 mein “0 saikda” hai, yaani koi saikda nahi, to 047 aur 47 ek hi hain (jaise ₹047 aur ₹47 ek hi). Yeh 0 isliye lagate hain kyunki jawab mein ek jagah zyada aa sakti hai, aur us jagah ke liye bhi ek digit chahiye.
2. Right se left chalo. Kaam ikai (sabse right wala digit) se shuru karo, phir dahai, phir saikda… bilkul school ke column method ki tarah. Kyun? Kyunki carry hamesha left wali jagah mein jaata hai, to pehle right wali jagah poori karni padti hai.
3. Padosi = thik right wala digit. Jis digit pe kaam kar rahe ho, uske thik right (daayein) mein jo digit baitha hai, woh uska padosi hai. 047 mein: 7 (ikai) ke right mein kuch nahi, to uska padosi nahi hai (0 maano). 4 ka padosi 7. Aage wale 0 ka padosi 4. Dhyaan: padosi hamesha right wala, left wala nahi.
4. Carry (haath ka). Jab kisi jagah ka jod 10 ya zyada aaye, to wahan sirf ikai wala digit (aakhri digit) likho, aur baaki hissa carry banakar left wali agli jagah ke jod mein mila do. Jaise jod 16 aaya: 6 likho, 1 carry. Jod 23 aaya: 3 likho, 2 carry (kyunki 23 = 2 dahai + 3). Is chapter mein carry 1 ya 2 tak aa sakta hai.
5. Aakhri kadam (aage wala 0). Sabse left wale 0 pe bhi niyam lagao. Wahan jo jod aaye, poora likh do (agar 11 aaye to 11), kyunki uske left mein aur koi jagah nahi bachi.
Aaj ka niyam ek line mein: har digit pe “double karo, padosi jodo”, right se left, carry ke saath.
🔁 Chhota sa warm-up
- 8 ka double kitna?
- 0365 aur 365 mein kya farak hai?
- 0583 mein 5 ka padosi kaun hai?
Answers dekho
- 16 (8 + 8 = 16)
- Koi farak nahi. Aage ka 0 matlab “0 hazaar”, yaani kuch nahi.
- 8 (5 ke thik right mein 8 baitha hai)
Teeno aa gaye? To aaj ki trick ke saare auzaar taiyaar hain!
— 📌 charumam.com
🎬 Kahani: Kele Wale Bhaiya Aur Darjan
Sabzi mandi mein kele darjan ke hisaab se bikte hain. 1 darjan = 12 kele. Riya ki mummy ko school ke function ke liye 47 darjan kele chahiye the. Bhaiya ne poocha: “Kitne kele hue?” aur calculator dhoondhne lage.
Riya ne kaagaz pe 047 likha aur right se chalna shuru kiya: “7 ka double 14, 4 likho, 1 haath mein. 4 ka double 8, padosi 7 aur haath ka 1: 16, 6 likho, 1 haath mein. Aage ke 0 pe padosi 4 aur haath ka 1: 5.” Phir boli: “564 kele!”
Calculator ne bhi 564 dikhaya. Bhaiya hairaan! 😄 Riya ne kaha: “Yeh Trachtenberg ka niyam hai: double karo, padosi jodo.”
Aaj yahi seekhenge. ✌️
— 📌 charumam.com
🪄 Trick Step by Step: Aage 0, Phir Har Digit Ka Double + Padosi
💡 Is post ki ek badi baat: × 12 ka niyam: number ke aage 0 lagao. Phir right se left, har digit ka double karo aur usme uska padosi (thik right wala digit) jodo. Carry ho to agli jagah mein jodo. Aakhri (aage wale 0 ki) jagah pe jo aaye, poora likh do.
Step 1: Aage 0 lagao
Sawaal: 23 × 12. Number 23 ke aage 0: 023. (Keemat wahi: 0 saikda + 2 dahai + 3 ikai = 23.)
Step 2: Ikai: sirf double
Sabse right wala digit 3 hai. Uska double: 3 + 3 = 6. Iske right mein koi digit nahi, to padosi nahi. Neeche 6 likho.
Step 3: Dahai: double + padosi
Agla digit (left ki taraf) 2 hai. Uska double: 2 + 2 = 4. Uska padosi (thik right wala digit) 3 hai. Jodo: 4 + 3 = 7. Neeche 6 ke left mein 7 likho: abhi tak 76.
Step 4: Aage wala 0: 0 ka double + padosi
Ab aage wala 0. 0 ka double 0 hi hai. Uska padosi 2 hai. Jodo: 0 + 2 = 2. Sabse left mein 2 likho.
Step 5: Jawab padho, andaaze se check
Left se right: 2 | 7 | 6 = 276. Andaaza: 23 × 10 = 230, aur 12 thoda zyada hai 10 se, to jawab 230 se thoda bada hona chahiye. 276 ✅
23 × 12: number ke aage 0 lagao → 023. Ab right se left (ikai se shuru) har digit pe niyam:
| Kadam | Jagah | Digit | Kaam | Likho | Carry aage |
|---|---|---|---|---|---|
| 1 | Ikai | 3 | 3 ka double: 3 + 3 = 6; padosi nahi (ikai ke right mein koi digit nahi). Kul: 6 | 6 | 0 |
| 2 | Dahai | 2 | 2 ka double: 2 + 2 = 4; padosi 3 jodo. Jod: 4 + 3 = 7 | 7 | 0 |
| 3 | Saikda | 0 (aage wala) | aage wala 0: double 0 + 0 = 0; padosi 2 jodo. Jod: 0 + 2 = 2 | 2 | — |
Ab “Likho” wale digit neeche se upar, yaani left se right padho: 2 | 7 | 6 → 276.
Carry wala sawaal: 47 × 12 (dhyaan se)
Ab ek aisa sawaal jahan jod 10 se bada aata hai. Niyam wahi, bas carry ka dhyaan: jod 10 ya zyada ho to sirf aakhri digit likho, aur aage wala digit (1 ya 2) agle kadam ke jod mein mila do.
47 × 12: number ke aage 0 lagao → 047. Ab right se left (ikai se shuru) har digit pe niyam:
| Kadam | Jagah | Digit | Kaam | Likho | Carry aage |
|---|---|---|---|---|---|
| 1 | Ikai | 7 | 7 ka double: 7 + 7 = 14; padosi nahi (ikai ke right mein koi digit nahi). Kul: 14 | 4 | 1 |
| 2 | Dahai | 4 | 4 ka double: 4 + 4 = 8; padosi 7 jodo; haath ka carry 1. Jod: 8 + 7 + 1 = 16 | 6 | 1 |
| 3 | Saikda | 0 (aage wala) | aage wala 0: double 0 + 0 = 0; padosi 4 jodo; haath ka carry 1. Jod: 0 + 4 + 1 = 5 | 5 | — |
Ab “Likho” wale digit neeche se upar, yaani left se right padho: 5 | 6 | 4 → 564.
Dhyaan do: kadam 1 mein 14 aaya, to sirf 4 likha aur 1 “haath mein” rakha. Kadam 2 mein woh 1 jod diya (8 + 7 + 1). Bilkul Chapter 2 ke carry jaisa.
Ek-digit × 12 (12 ka pahaada bina ratte)
Ek digit wale number ke aage 0 lagao (7 → 07), aur wahi do kadam: ikai ka double, phir aage wale 0 pe padosi (+ carry).
| Sawaal | Ikai (digit khud) | Aage wala 0 | Jawab |
|---|---|---|---|
| 2 × 12 | 2 ka double: 2 + 2 = 4; padosi nahi (ikai ke right mein koi digit nahi). Kul: 4 → likho 4 | aage wala 0: double 0 + 0 = 0; padosi 2 jodo. Jod: 0 + 2 = 2 → likho 2 | 24 |
| 3 × 12 | 3 ka double: 3 + 3 = 6; padosi nahi (ikai ke right mein koi digit nahi). Kul: 6 → likho 6 | aage wala 0: double 0 + 0 = 0; padosi 3 jodo. Jod: 0 + 3 = 3 → likho 3 | 36 |
| 4 × 12 | 4 ka double: 4 + 4 = 8; padosi nahi (ikai ke right mein koi digit nahi). Kul: 8 → likho 8 | aage wala 0: double 0 + 0 = 0; padosi 4 jodo. Jod: 0 + 4 = 4 → likho 4 | 48 |
| 5 × 12 | 5 ka double: 5 + 5 = 10; padosi nahi (ikai ke right mein koi digit nahi). Kul: 10 → likho 0, carry 1 | aage wala 0: double 0 + 0 = 0; padosi 5 jodo; haath ka carry 1. Jod: 0 + 5 + 1 = 6 → likho 6 | 60 |
| 6 × 12 | 6 ka double: 6 + 6 = 12; padosi nahi (ikai ke right mein koi digit nahi). Kul: 12 → likho 2, carry 1 | aage wala 0: double 0 + 0 = 0; padosi 6 jodo; haath ka carry 1. Jod: 0 + 6 + 1 = 7 → likho 7 | 72 |
| 7 × 12 | 7 ka double: 7 + 7 = 14; padosi nahi (ikai ke right mein koi digit nahi). Kul: 14 → likho 4, carry 1 | aage wala 0: double 0 + 0 = 0; padosi 7 jodo; haath ka carry 1. Jod: 0 + 7 + 1 = 8 → likho 8 | 84 |
| 8 × 12 | 8 ka double: 8 + 8 = 16; padosi nahi (ikai ke right mein koi digit nahi). Kul: 16 → likho 6, carry 1 | aage wala 0: double 0 + 0 = 0; padosi 8 jodo; haath ka carry 1. Jod: 0 + 8 + 1 = 9 → likho 9 | 96 |
| 9 × 12 | 9 ka double: 9 + 9 = 18; padosi nahi (ikai ke right mein koi digit nahi). Kul: 18 → likho 8, carry 1 | aage wala 0: double 0 + 0 = 0; padosi 9 jodo; haath ka carry 1. Jod: 0 + 9 + 1 = 10 → likho 10 | 108 |
🆘 Atak gaye? Aise socho
- Shuru kahan se? Hamesha ikai (sabse right) se. Pehle aage 0 lagana mat bhoolo: 47 → 047.
- Padosi kaun? Jis digit pe ho, uske right wala. 047 mein 4 ka padosi 7 (0 nahi).
- Jod 10 se bada? Sirf aakhri digit likho, baaki carry: 16 → 6 likho, 1 aage. 22 → 2 likho, 2 aage.
- Aakhri kadam: aage wale 0 ka double 0 hai, to wahan bas padosi + carry. Jo aaye poora likho.
- Check: number × 10 + number × 2. 47 × 10 = 470, 47 × 2 = 94, 470 + 94 = 564 ✅
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Chaabi: 12 = 10 + 2. To 47 × 12 matlab “47 ko 10 baar lo, aur 2 baar aur”: 47 × 10 + 47 × 2.
- × 10 wala hissa: 47 × 10 = 470. Dekho kya hua: har digit ek jagah left khisak gaya. 7 pehle ikai mein tha, ab dahai mein aa gaya. 4 dahai mein tha, ab saikde mein. Yaani har khaane mein ab woh digit aa gaya jo pehle uske thik right mein tha: yahi padosi hai!
- × 2 wala hissa: har digit apni hi jagah pe do baar (double): ikai mein 7 + 7 = 14, dahai mein 4 + 4 = 8.
- Dono jodo, khaane-khaane mein: har khaane mein “double + padosi”. Ikai: 14 + 0 = 14. Dahai: 8 + 7 = 15. Saikda: 0 + 4 = 4.
- Carry theek karo: ikai ka 14 → 4 likho, 1 dahai mein. Dahai 15 + 1 = 16 → 6 likho, 1 saikde mein. Saikda 4 + 1 = 5. Jawab 564.
Neeche chitra mein yahi khaane dikh rahe hain. Isliye niyam “double + padosi” koi jaadu nahi, bas × 10 aur × 2 ko ek saath karna hai.
Aage 0 kyun?
Upar chitra mein sabse left wala khaana (Saikda) dekho. 47 mein saikda tha hi nahi, par × 10 karne pe 4 saikde mein khisak aaya. Is naye khaane ke liye hi hum shuru mein 0 lagate hain: 047. Agar 0 na lagaayein, to yeh aakhri kadam bhool jaate hain aur jawab 64 jaisa adhoora aata hai.
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Tareeke Se Check)
Example 1: 31 × 12 = ?
Aage 0: 031. Ikai 1: double 1 + 1 = 2 → 2 likho. Dahai 3: double 3 + 3 = 6, padosi 1; 6 + 1 = 7 → 7 likho. Aage wala 0: padosi 3; 0 + 3 = 3 → 3 likho (aakhri kadam). Jawab: 3 | 7 | 2 → 372.
✔️ Normal tareeke se check: 31 × 10 = 310, 31 × 2 = 62, 310 + 62 = 372 ✅
Example 2: 56 × 12 = ?
Aage 0: 056. Ikai 6: double 6 + 6 = 12 → 2 likho, carry 1. Dahai 5: double 5 + 5 = 10, padosi 6, carry 1; 10 + 6 + 1 = 17 → 7 likho, carry 1. Aage wala 0: padosi 5, carry 1; 0 + 5 + 1 = 6 → 6 likho (aakhri kadam). Jawab: 6 | 7 | 2 → 672.
✔️ Normal tareeke se check: 56 × 10 = 560, 56 × 2 = 112, 560 + 112 = 672 ✅
Example 3: 85 × 12 = ? (carry 2)
Aage 0: 085. Ikai 5: double 5 + 5 = 10 → 0 likho, carry 1. Dahai 8: double 8 + 8 = 16, padosi 5, carry 1; 16 + 5 + 1 = 22 → 2 likho, carry 2. Aage wala 0: padosi 8, carry 2; 0 + 8 + 2 = 10 → 10 likho (aakhri kadam). Jawab: 10 | 2 | 0 → 1020.
Dhyaan: dahai mein jod 22 aaya, to 2 likha aur 2 carry kiya (22 = 2 dahai + 2). Aakhri kadam pe 10 aaya, to poora 10 likha.
✔️ Normal tareeke se check: 85 × 10 = 850, 85 × 2 = 170, 850 + 170 = 1020 ✅
Example 4: 9 × 12 = ?
Aage 0: 09. Ikai 9: double 9 + 9 = 18 → 8 likho, carry 1. Aage wala 0: padosi 9, carry 1; 0 + 9 + 1 = 10 → 10 likho (aakhri kadam). Jawab: 10 | 8 → 108.
✔️ Normal tareeke se check: 12 ko 9 baar jodo: 12, 24, 36, 48, 60, 72, 84, 96, 108 ✅
— 📌 charumam.com
🕉️ Vedic Maths Connection
Trachtenberg system Vedic Maths se alag hai (yeh 20vi sadi mein Europe mein bana), par dono ek hi baat pe tike hain: place value. Vedic Maths ka ek sutra hai “Urdhva-Tiryagbhyam”, matlab “seedha aur tirchha”. 47 × 12 pe woh aise chalta hai: ikai mein 7 × 2 = 14 (seedha), dahai mein 4 × 2 + 7 × 1 = 8 + 7 = 15 (tirchha), saikde mein 4 × 1 = 4. (Yahan pehle dono guna karo, phir jodo.) Dekho: “4 × 2 + 7 × 1” bilkul aaj ka “4 ka double + padosi 7” hai! Carry ke baad wahi 564. Yaani 12 ke saath dono tareeke ek hi kaam karte hain.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| 2-digit × 12 (47 × 12) | ✅ Haan | Teen chhote kadam, ek line. |
| Ek-digit × 12 (7 × 12) | ✅ Haan | Ikai ka double, phir padosi: 84. |
| Darjan ka hisaab (23 darjan) | ✅ Bahut kaam ka | 1 darjan = 12, to 23 × 12 = 276. |
| 3 ya 4-digit × 12 (3142 × 12) | ✅ Agle post mein | Wahi niyam, bas zyada kadam aur carry 2 tak (Post 11.2). |
| × 11 (47 × 11) | ➡️ Chapter 4 / Post 11.3 | Double nahi, sirf digit + padosi. |
| × 13, × 15 jaise | ❌ Is niyam se nahi | 12 ka niyam sirf 12 ke liye. 13 ke liye school wala tareeka (× 10 + × 3). |
— 📌 charumam.com
❌→✅ Common Mistakes
Galti 1: Padosi galat taraf se liya
Sawaal: 47 × 12
❌ Galat: 7 ke saath uske left wala 4 joda, 4 ke saath aage ka 0 → jawab 98.
✅ Sahi: padosi hamesha right wala. Ikai ka padosi koi nahi, 4 ka padosi 7: jawab 564.
Galti 2: Carry bhool gaye
Sawaal: 47 × 12
❌ Galat: 14 ka 4 likha par 1 aage nahi jodha; 8 + 7 = 15 ka 5 likha, phir 4 → 454.
✅ Sahi: 8 + 7 + 1 = 16 → 6, phir 4 + 1 = 5: 564.
Galti 3: Aage ka 0 (aakhri kadam) bhool gaye
Sawaal: 47 × 12
❌ Galat: sirf do kadam kiye → 64.
✅ Sahi: aage 0 lagao (047) aur uska kadam bhi karo: 0 + 4 + 1 = 5 → 564. Andaaza bhi batata hai: 47 × 12 to 470 se bada hona chahiye.
Galti 4: Double karna bhool gaye (× 11 kar diya)
Sawaal: 47 × 12
❌ Galat: har digit + padosi (bina double) → 517. Yeh to 47 × 11 hai!
✅ Sahi: 12 ke liye double + padosi: 564.
🧠 Rule of thumb: Aage 0 → right se left → har digit ka double + padosi (right wala) → carry aage → aakhri kadam poora likho.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 sawaal, sab × 12. Target: 5 minute. Copy pe aage 0 lagakar karo, phir dheere-dheere mann mein.
- 6 × 12 = ?
- 8 × 12 = ?
- 11 × 12 = ?
- 13 × 12 = ?
- 21 × 12 = ?
- 24 × 12 = ?
- 32 × 12 = ?
- 34 × 12 = ?
- 41 × 12 = ?
- 45 × 12 = ?
- 52 × 12 = ?
- 63 × 12 = ?
- 67 × 12 = ?
- 78 × 12 = ?
- 99 × 12 = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- 72
- 96
- 132
- 156
- 252
- 288
- 384
- 408
- 492
- 540
- 624
- 756
- 804
- 936
- 1188
Kaise kiya (kuch):
Aage 0: 078. Ikai 8: double 8 + 8 = 16 → 6 likho, carry 1. Dahai 7: double 7 + 7 = 14, padosi 8, carry 1; 14 + 8 + 1 = 23 → 3 likho, carry 2. Aage wala 0: padosi 7, carry 2; 0 + 7 + 2 = 9 → 9 likho (aakhri kadam). Jawab: 9 | 3 | 6 → 936.
Aage 0: 099. Ikai 9: double 9 + 9 = 18 → 8 likho, carry 1. Dahai 9: double 9 + 9 = 18, padosi 9, carry 1; 18 + 9 + 1 = 28 → 8 likho, carry 2. Aage wala 0: padosi 9, carry 2; 0 + 9 + 2 = 11 → 11 likho (aakhri kadam). Jawab: 11 | 8 | 8 → 1188.
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⏰ One-Minute Challenge
Sirf 1 minute (60 second). Timer lagao, aur jitne ho sakein utne sawaal mann mein (bina copy ke) karo. Sirf jawab likhna hai.
- 3 × 12 = ?
- 5 × 12 = ?
- 10 × 12 = ?
- 12 × 12 = ?
- 14 × 12 = ?
- 20 × 12 = ?
- 22 × 12 = ?
- 25 × 12 = ?
- 33 × 12 = ?
- 50 × 12 = ?
| Rating | Kitne sahi, kitne time mein |
|---|---|
| 🌱 Beginner | 60 second mein 5 se 7 sahi. Badhiya shuruaat! |
| 🚀 Pro | 60 second mein 8 ya 9 sahi |
| 🏆 Champion | 10 mein 10 sahi, 60 second ke andar |
Answers dekho
- 36
- 60
- 120
- 144
- 168
- 240
- 264
- 300
- 396
- 600
Dhyaan:
Aage 0: 025. Ikai 5: double 5 + 5 = 10 → 0 likho, carry 1. Dahai 2: double 2 + 2 = 4, padosi 5, carry 1; 4 + 5 + 1 = 10 → 0 likho, carry 1. Aage wala 0: padosi 2, carry 1; 0 + 2 + 1 = 3 → 3 likho (aakhri kadam). Jawab: 3 | 0 | 0 → 300.
10 × 12, 20 × 12, 50 × 12 jaise sawaal mein peeche 0 hai: 1 × 12 = 12, phir peeche 0 → 120. Isi tarah 2 × 12 = 24 → 240, 5 × 12 = 60 → 600.
5 se kam sahi? Koi baat nahi! Trick wala section ek baar aur padho aur kal phir try karo. Har din thoda better, bas yahi chahiye. 💪
— 📌 charumam.com
📝 10 MCQs (Har Option Ka Kaaran)
MCQ matlab ek sawaal ke chaar jawab (options) diye hote hain, unme se ek sahi chunna hai.
Q1. 47 × 12 = ?
Jawab aur kaaran dekho
Sahi jawab: (a) 564
- (a) sahi ✅: 047: 7 + 7 = 14 (4 likho, 1 carry); 8 + 7 + 1 = 16 (6 likho, 1 carry); 0 + 4 + 1 = 5 → 564.
- (b) galat ❌: 517: double karna bhool gaye, sirf digit + padosi kiya; yeh 47 × 11 hai.
- (c) galat ❌: 454: carry aage nahi jodha (14 ka 1 aur 16 ka 1 chhod diye).
- (d) galat ❌: 64: aage wale 0 ka kadam bhool gaye.
Q2. × 12 ke niyam mein “padosi” kaun hai?
Jawab aur kaaran dekho
Sahi jawab: (d) Thik right wala digit
- (a) galat ❌: Left wala nahi; left wala lene se 47 × 12 ka jawab 98 aata hai, jo galat hai.
- (b) galat ❌: Sabse pehla digit sirf aakhri kadam (aage wale 0) ka padosi hota hai, har digit ka nahi.
- (c) galat ❌: Sirf ikai ka padosi nahi hota (0 maante hain); baaki digits ka padosi asli digit hai.
- (d) sahi ✅: Haan: jis digit pe kaam ho raha hai, uske thik right wala. (× 10 se wahi is khaane mein aata hai.)
Q3. 23 × 12 = ?
Jawab aur kaaran dekho
Sahi jawab: (c) 276
- (a) galat ❌: 46: sirf double kiye (23 × 2), padosi nahi jode.
- (b) galat ❌: 266: dahai mein padosi 3 ki jagah wahi digit 2 jod diya (4 + 2 = 6), phir 0 + 2 = 2.
- (c) sahi ✅: 023: 3 + 3 = 6; 2 + 2 + 3 = 7; 0 + 2 = 2 → 276.
- (d) galat ❌: 253: double nahi kiya; yeh 23 × 11 hai.
Q4. 56 × 12 = ?
Jawab aur kaaran dekho
Sahi jawab: (b) 672
- (a) galat ❌: 72: aage wale 0 ka kadam bhool gaye.
- (b) sahi ✅: 056: 6 + 6 = 12 (2 likho, 1 carry); 10 + 6 + 1 = 17 (7 likho, 1 carry); 0 + 5 + 1 = 6 → 672.
- (c) galat ❌: 616: double nahi kiya; yeh 56 × 11 hai.
- (d) galat ❌: 562: carry nahi jode.
Q5. 85 × 12 mein dahai ka jod 22 aata hai. Ab kya karein?
Jawab aur kaaran dekho
Sahi jawab: (a) 2 likho, 2 carry
- (a) sahi ✅: 22 = 2 dahai + 2, to 2 likho aur 2 aage. Jawab 1020.
- (b) galat ❌: 22 poora sirf aakhri kadam pe likhte hain, beech mein nahi.
- (c) galat ❌: 22 mein 2 dahai hain, 1 nahi; 1 carry se aakhri kadam 0 + 8 + 1 = 9 aata aur jawab 920 (galat).
- (d) galat ❌: Carry chhodne se jawab 820 aata, jo galat hai (85 × 12 = 1020).
Q6. Aage 0 kyun lagate hain?
Jawab aur kaaran dekho
Sahi jawab: (d) Jawab mein ek jagah zyada aati hai
- (a) galat ❌: Aage 0 se keemat nahi badalti: 047 = 47.
- (b) galat ❌: Peeche 0 lagane se 10 guna hota hai, aage 0 se nahi.
- (c) galat ❌: Iske bina aakhri kadam chhoot jaata hai (47 × 12 → 64), to yeh zaroori hai.
- (d) sahi ✅: × 10 se har digit ek jagah left khisakta hai, to jawab mein ek naya khaana banta hai; uske liye 0.
Q7. 9 × 12 = ?
Jawab aur kaaran dekho
Sahi jawab: (c) 108
- (a) galat ❌: 18: sirf 9 ka double kiya, aage wala kadam bhool gaye.
- (b) galat ❌: 99: double nahi kiya; yeh 9 × 11 hai.
- (c) sahi ✅: 09: 9 + 9 = 18 (8 likho, 1 carry); 0 + 9 + 1 = 10 → poora 10 likho → 108.
- (d) galat ❌: 98: aakhri kadam mein carry 1 nahi joda (0 + 9 = 9).
Q8. Kaunsa sawaal is niyam (double + padosi) se hoga?
Jawab aur kaaran dekho
Sahi jawab: (b) 34 × 12
- (a) galat ❌: 15 ke liye bhi nahi: 340 + 170 = 510 (yahan 170 = 34 × 5).
- (b) sahi ✅: 12 ka niyam: 034 → 8; 6 + 4 = 10 (0 likho, 1 carry); 0 + 3 + 1 = 4 → 408.
- (c) galat ❌: 13 ke liye yeh niyam nahi; school tareeka: 340 + 102 = 442 (yahan 102 = 34 × 3).
- (d) galat ❌: 21 = 12 nahi! 34 × 21 = 714 (school tareeka: 680 + 34, jahan 680 = 34 × 20).
Q9. 99 × 12 = ?
Jawab aur kaaran dekho
Sahi jawab: (a) 1188
- (a) sahi ✅: 099: 18 (8, carry 1); 18 + 9 + 1 = 28 (8, carry 2); 0 + 9 + 2 = 11 → 1188.
- (b) galat ❌: 1089: double nahi kiya; yeh 99 × 11 hai.
- (c) galat ❌: 978: carry nahi jode.
- (d) galat ❌: 88: aage wale 0 ka kadam bhool gaye.
Q10. □ × 12 = 276. □ = ? (□ = woh number jo dhoondhna hai, yaani khaali dabba)
Jawab aur kaaran dekho
Sahi jawab: (d) 23
- (a) galat ❌: 24 × 12 = 288 (024: 8; 4 + 4 = 8; 0 + 2 = 2), 276 nahi.
- (b) galat ❌: 22 × 12 = 264 (022: 4; 4 + 2 = 6; 0 + 2 = 2), 276 nahi.
- (c) galat ❌: 276 to jawab hai; □ woh number hai jise 12 se guna kiya.
- (d) sahi ✅: 23 × 12: 023 → 6, 7, 2 → 276 ✅. Andaaza: □ 20 aur 30 ke beech hoga, kyunki 20 × 12 = 240 aur 30 × 12 = 360.
— 📌 charumam.com
🧠 Challenge: HOTS Aur Olympiad Style
HOTS matlab Higher Order Thinking Skills, yaani aise sawaal jinme sirf trick nahi, thoda dimaag bhi lagana padta hai. Olympiad ek maths competition (pratiyogita) hota hai.
Q1. Ek darjan = 12. School ki 34 class ko ek-ek darjan pencil chahiye. Kul kitni pencil?
Solution dekho
34 × 12. 034: ikai 4 + 4 = 8; dahai 3 + 3 = 6, padosi 4: 6 + 4 = 10 → 0 likho, 1 carry; aage 0: 0 + 3 + 1 = 4. Jawab 408 pencil.
Check: 34 × 10 = 340, 34 × 2 = 68, 340 + 68 = 408 ✅
Q2. 12 × 12 = 144 aur 21 × 21 = 441. Ulta number, ulta jawab! Kya 13 × 13 aur 31 × 31 mein bhi aisa hai?
Solution dekho
13 × 13 = 169 (school: 130 + 39, jahan 39 = 13 × 3). 31 × 31 = 961 (310 + 310 + 310 + 31 = 961). Haan! 169 ka ulta 961 ✅.
Aaj ki trick se 12 × 12 bhi: 012 → 2 + 2 = 4; 1 + 1 + 2 = 4; 0 + 1 = 1 → 144.
Q3. Ek number ko 12 se guna kiya to jawab ka ikai wala digit 4 aaya. Number ki ikai kya ho sakti hai? (Ek-digit soch se)
Solution dekho
Jawab ki ikai = number ki ikai ka double ka aakhri digit (ikai ka koi padosi nahi). Kaunse digit ka double 4 pe khatam hota hai? 2 (2 + 2 = 4) aur 7 (7 + 7 = 14). To ikai 2 ya 7.
Check: 32 × 12 = 384, 47 × 12 = 564: dono 4 pe khatam ✅
Q4. Riya: “45 × 12 = 540.” Aarav: “Nahi, 495.” Kaun sahi?
Solution dekho
045: ikai 5 + 5 = 10 → 0 likho, 1 carry; dahai 4 + 4 = 8, padosi 5, carry 1: 8 + 5 + 1 = 14 → 4 likho, 1 carry; aage 0: 0 + 4 + 1 = 5. Jawab 540: Riya sahi. Aarav ne × 11 kar diya (45 × 11 = 495).
Check: 450 + 90 = 540 ✅
— 📌 charumam.com
🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | Darjan ke sawaal, cost × 12 (12 mahine ka kiraya), guna jaldi check karna. |
| Maths Olympiad | Ulte sawaal (□ × 12) aur pattern (12 × 12 = 144, 21 × 21 = 441). |
| NTSE / MAT | Mental maths aur number series. |
| SSC / Bank (aage) | Simplification mein × 12 ka tez hisaab. |
Shabdon ka matlab: Olympiad = school ke bachchon ki maths pratiyogita (competition). NTSE = Class 10 ke bachchon ka ek national scholarship (vazifa) exam. MAT = Mental Ability Test, yaani sochne-samajhne ki pariksha (NTSE ka ek hissa). SSC / Bank exam = badon ke sarkari naukri ke exam. Simplification = lambe hisaab ko chhota karke jawab nikaalna. Class 6–8 = school ki chhathi se aathvi kaksha.
— 📌 charumam.com
👩🏫 CharuMam Ka Tip
Mantra: “aage 0, double + padosi, carry aage.” Pehle 3 din copy pe table bana kar karo (jaise upar), phir mann mein. Roz 5 sawaal. Agle post mein lambe number: 3142 × 12, aur carry 2 wale! ✌️
— 📌 charumam.com
⏱ After Timer: Ab Trick Se!
Wahi sawaal jo shuru mein kiya tha. Stopwatch phir chalu kijiye, aur is baar aaj ki trick se kijiye:
Trick se kaise karein (dekho)
Aage 0: 047. Right se left: ikai 7 ka double 14 → 4 likho, 1 carry. Dahai 4 ka double 8, padosi 7 aur carry 1: 8 + 7 + 1 = 16 → 6 likho, 1 carry. Aage wala 0: padosi 4 aur carry 1: 0 + 4 + 1 = 5 → 5. Jawab: 5 | 6 | 4 = 564 ✅
| Before time | After time | Kitna tez? |
|---|---|---|
| ____ second | ____ second | Before − After = ____ second bache! |
Pehli baar mein time zyada bhi lag sakta hai, ghabraiye nahi! Nayi trick ko 3 se 4 din ki practice chahiye. Phir yeh purane tareeke se kaafi tez ho jaati hai. 🚀
— 📌 charumam.com
🤖 Aur Practice: AI Tutor Se Poochho
Neeche wali line copy kijiye aur CharuMam AI Tutor (tutor.charumam.com) mein paste kar dijiye. Tutor aapko naye sawaal dega aur galti hone pe samjhayega:
— 📌 charumam.com
❓ FAQs (Bachchon Ke Asli Sawaal)
1. Kya 12 ka pahaada yaad karna zaroori hai?
Nahi. Is niyam se sirf ek-digit ka double (0 se 9) aana chahiye. Double ki table upar box mein hai.
2. Right se left kyun, left se kyun nahi?
Carry hamesha left wali jagah mein jaata hai. Right se chaloge to carry pehle hi pata hoga aur seedha jod doge.
3. Padosi galti se left wala le liya to?
Jawab galat aayega (47 × 12 → 98). Yaad rakho: × 10 se har digit left khisakta hai, to har khaane mein right wala padosi aata hai.
4. Check kaise karein?
Number × 10 + number × 2. 47 × 12: 470 + 94 = 564. Ya andaaza: jawab number × 10 se thoda bada.
5. Agle post mein kya hai?
Post 11.2: 3-digit aur 4-digit × 12, carry 2 tak, aur jawab mein 0 aaye to kya.
— 📌 charumam.com
🟢 Recap: Aaj Kya Seekha
- 12 = 10 + 2: isliye har khaane mein padosi (× 10 se) + double (× 2 se).
- Pehle aage 0: 47 → 047.
- Right se left: ikai ka sirf double (padosi nahi).
- Baaki har digit: double + padosi (right wala) + carry.
- Jod 10 ya zyada: aakhri digit likho, baaki (1 ya 2) carry.
- Aakhri kadam (aage ka 0): padosi + carry, poora likho.
- Check: × 10 + × 2 (470 + 94 = 564).
— charumam.com 🎯
👉 Aage Kya?
Ab lambe number ki baari: Post 11.2 mein 3142 × 12 = 37704 aur 899 × 12 = 10788, jahan carry 2 bhi aayega. Wahi niyam, bas thode zyada kadam!
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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- 🌐 Website: charumam.com
- 🤖 AI Tutor (doubt poochho, 24×7): tutor.charumam.com
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