Chapter 12 shuru! 🎉 Is chapter mein Trachtenberg ke 4 niyam: 6, 7, 8 aur 9 se guna, bina pahaade ke. Aaj pehla: × 6. Niyam mein do nayi baatein hain: padosi ka aadha aur taaq digit pe 5. Dono neeche shuru se samjhaayi hain: juft/taaq kya hai, aadha karke adhoora hissa kyun chhodte hain, aur 5 kahan se aata hai.
⏱ Before Timer: Pehle Purane Tareeke Se
Pehle school wala tareeka (column method). 357 × 6, right se:
- Ikai: 6 × 7 = 42. Neeche 2 likho, 4 carry.
- Dahai: 6 × 5 = 30, carry 4 jodo: 30 + 4 = 34. Neeche 4 likho, 3 carry.
- Saikda: 6 × 3 = 18, carry 3 jodo: 18 + 3 = 21. Neeche 21 likho.
- Jawab: 2142. Isme 6 ka pahaada (6 × 7, 6 × 5, 6 × 3) yaad hona chahiye, aur bade carry (4, 3) ka dhyaan.
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
357 × 6 = 2142
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: Chapter 11 ke Trachtenberg shabd (aage 0, padosi, right se left, carry), aur Chapter 5 ka aadha aur × 5. Sab neeche ke boxes mein dobara, aaj ke hisaab se. Pehle naye shabd:
📖 Shabd jo is post mein aayenge
- Juft (even) digit: 0, 2, 4, 6, 8: 2 barabar hisson mein bant jaate hain, kuch nahi bachta.
- Taaq (odd) digit: 1, 3, 5, 7, 9: baantne pe 1 bach jaata hai.
- Padosi ka aadha: padosi (thik right wala digit) ka aadha, aur agar ½ bache to use chhod do. 7 ka aadha → 3.
- Taaq pe 5: jis digit pe kaam ho raha hai, woh khud taaq ho to us kadam mein 5 aur jodo.
- 6 = 1 + 5: 6 ko 1 aur 5 mein todte hain: × 1 (digit khud) aur × 5 (× 10 ka aadha). Isi se niyam bana.
- Kadam ke teen hisse: × 6 ke har kadam pe: digit + (taaq ho to 5) + padosi ka aadha, aur carry ho to woh bhi.
📏 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.
⚖️ Juft aur taaq digit (even aur odd)
Chapter 5 mein dekha tha: jo number 2 barabar hisson mein bina kuch bache bant jaaye, woh juft (even) hai. Jo poora na bante, ek bach jaaye, woh taaq (odd) hai. Is chapter mein hum sirf ek digit (0 se 9) dekhenge:
| Juft digit (2 barabar hisse, kuch nahi bachta) | 0 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|
| Taaq digit (ek bach jaata hai) | 1 | 3 | 5 | 7 | 9 |
Yaad karne ka tareeka: 0, 2, 4, 6, 8 juft (do-do karke ginti: 2, 4, 6, 8). Baaki 1, 3, 5, 7, 9 taaq. Jaise 6 laddoo, 2 bachche: 3-3, kuch nahi bacha → juft. 7 laddoo: 3-3, aur 1 bach gaya → taaq. Aur 0? 0 laddoo baantne pe dono ko 0, kuch nahi bacha, to 0 juft hai.
✂️ “Padosi ka aadha”: aadha karo, adhoora hissa chhod do
Aadha matlab 2 barabar hisse karke ek hissa (Chapter 5). Juft digit ka aadha seedha: 8 ka aadha 4 (4 + 4 = 8). Par taaq digit ka aadha poora nahi aata: 7 ka aadha “3 aur ek aadha” (3½, padho “saadhe teen”).
Is chapter ka niyam: taaq digit ka aadha karo to adhoora (½) hissa chhod do, sirf poora hissa lo. 7 ka aadha → 3½ → 3. Is ½ ko “chhodna” galti nahi hai: woh ½ kahin gaya nahi, woh “taaq pe 5” wale niyam mein wapas aata hai (neeche “Yeh Kaam Kyun Karta Hai” mein dekhenge).
| Digit | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Aadha (½ chhod kar) | 0 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
(Upar digit, neeche uska aadha. Jaise 9 ke neeche 4: 9 ka aadha 4½, ½ chhoda, to 4. Aur 1 ka aadha ½ hai, ½ chhoda to 0.) Pakka tareeka: taaq digit ho to pehle 1 kam karo (juft ban jaayega), phir aadha: 7 → 6 → 3. 9 → 8 → 4.
🪙 × 5 = × 10 ka aadha (Chapter 5 yaad karo)
Chapter 5 mein seekha: kisi number × 5 = us number × 10, phir aadha, kyunki 5 = 10 ka aadha. Jaise 7 × 5: peeche 0 → 70, aadha → 35.
Noton se dekho: ₹70 matlab ₹10 ke 7 note. Inhe Aman aur Riya mein barabar baanto: dono ko 3-3 note (₹30), aur 1 note bach gaya. Use tod kar ₹5 Aman ko, ₹5 Riya ko. Har ek ko ₹30 + ₹5 = ₹35.
Dhyaan do: 7 ke aadhe se do cheezein mili: 3 note (dahai wala hissa, “7 ka aadha ½ chhod kar”) aur ek ₹5 (ikai mein), kyunki 7 taaq tha. Agar digit juft hota (jaise 6), to koi note nahi bachta: ₹60 ka aadha ₹30, ₹5 nahi. Yahi do baatein aaj ke niyam mein aayengi: “padosi ka aadha” aur “taaq pe 5”.
🧩 Trachtenberg ke 5 shabd (bahut dhyaan se)
1. Aage 0 lagao. Number ke left mein (shuru mein) ek 0 likh do: 357 → 0357. Isse number ki keemat nahi badalti: 0357 mein “0 hazaar” hai, yaani koi hazaar nahi, to 0357 aur 357 ek hi hain (jaise ₹0357 aur ₹357 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. 0357 mein: 7 (ikai) ke right mein kuch nahi, to uska padosi nahi hai (0 maano). 5 ka padosi 7. Aage wale 0 ka padosi 3. 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 (× 6, × 7) 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 “digit + padosi ka aadha, aur digit taaq ho to 5 aur”, right se left, carry ke saath.
🔁 Chhota sa warm-up
- 7 juft hai ya taaq?
- 9 ka aadha, ½ chhod kar?
- 8 × 5 = ?
Answers dekho
- Taaq (7 = 3 + 3 + 1, ek bachta hai)
- 4 (9 ka aadha 4½, ½ chhoda)
- 40 (80 ka aadha)
Teeno aa gaye? To × 6 ke saare auzaar taiyaar!
— 📌 charumam.com
🎬 Kahani: Chhe Samose Ki Plate
Canteen mein ek plate mein 6 samose aate hain. Sports day pe 357 plate ka order aaya. Canteen wali aunty ne poocha: “Kitne samose banaane honge?”
Aarav ko 6 ka pahaada pakka nahi tha. Par usne 0357 likha aur Trachtenberg ka niyam lagaya: “Har digit, plus padosi ka aadha, aur digit taaq ho to 5.” Ikai se chala: 12, phir 14, phir 11, phir 2.
“2142 samose!” Aunty ne calculator se check kiya: bilkul sahi! 🥟
Aarav: “Bina pahaade ke! Bas aadha karna aur 5 jodna.” Aaj yahi seekhenge. 6️⃣
— 📌 charumam.com
🪄 Trick Step by Step: Digit + Padosi Ka Aadha, Taaq Ho To + 5
💡 Is post ki ek badi baat: × 6 ka niyam: aage 0 lagao. Right se left, har digit pe: digit khud + padosi ka aadha (½ chhod kar). Agar digit khud taaq hai (1, 3, 5, 7, 9), to 5 aur jodo. Carry agle kadam mein. Ikai ka padosi nahi hota.
Step 1: Aage 0 lagao
Sawaal: 24 × 6. Aage 0: 024.
Step 2: Ikai: digit + (taaq ho to 5)
Ikai 4 hai. Iska padosi nahi (right mein kuch nahi). 4 juft hai, to 5 nahi. Bas 4. Neeche 4 likho.
Step 3: Dahai: digit + padosi ka aadha
Agla digit 2. Juft hai, to 5 nahi. Padosi 4 ka aadha 2. Jodo: 2 + 2 = 4. Neeche 4 likho (pehle wale 4 ke left mein).
Step 4: Aage wala 0: sirf padosi ka aadha
0 khud 0 hai (juft, to 5 nahi). Padosi 2 ka aadha 1. Jodo: 0 + 1 = 1.
Step 5: Jawab padho aur check
Left se right: 1 | 4 | 4 = 144. Check: 24 × 6 = 24 × 5 + 24 = 120 + 24 = 144 ✅ (24 × 5: 240 ka aadha, yaani 120.)
24 × 6: number ke aage 0 lagao → 024. Ab right se left (ikai se shuru) har digit pe niyam:
| Kadam | Jagah | Digit | Kaam | Likho | Carry aage |
|---|---|---|---|---|---|
| 1 | Ikai | 4 | digit 4; 4 juft hai, to 5 nahi; padosi nahi (ikai ke right mein koi digit nahi). Kul: 4 | 4 | 0 |
| 2 | Dahai | 2 | digit 2; 2 juft hai, to 5 nahi; padosi 4 ka aadha 2. Jod: 2 + 2 = 4 | 4 | 0 |
| 3 | Saikda | 0 (aage wala) | aage wala 0: digit 0; padosi 2 ka aadha 1. Jod: 0 + 1 = 1 | 1 | — |
Ab “Likho” wale digit neeche se upar, yaani left se right padho: 1 | 4 | 4 → 144.
Taaq digit wala sawaal: 357 × 6 (dhyaan se)
Yahan teeno digit taaq hain (3, 5, 7), to har kadam pe 5 judega. Aur padosi taaq ho to uske aadhe ka ½ chhodna hai.
357 × 6: number ke aage 0 lagao → 0357. Ab right se left (ikai se shuru) har digit pe niyam:
| Kadam | Jagah | Digit | Kaam | Likho | Carry aage |
|---|---|---|---|---|---|
| 1 | Ikai | 7 | digit 7; 7 taaq hai, to 5 jodo; padosi nahi (ikai ke right mein koi digit nahi). Jod: 7 + 5 = 12 | 2 | 1 |
| 2 | Dahai | 5 | digit 5; 5 taaq hai, to 5 jodo; padosi 7 ka aadha 3 (aadha hissa chhod diya); haath ka carry 1. Jod: 5 + 5 + 3 + 1 = 14 | 4 | 1 |
| 3 | Saikda | 3 | digit 3; 3 taaq hai, to 5 jodo; padosi 5 ka aadha 2 (aadha hissa chhod diya); haath ka carry 1. Jod: 3 + 5 + 2 + 1 = 11 | 1 | 1 |
| 4 | Hazaar | 0 (aage wala) | aage wala 0: digit 0; padosi 3 ka aadha 1 (aadha hissa chhod diya); haath ka carry 1. Jod: 0 + 1 + 1 = 2 | 2 | — |
Ab “Likho” wale digit neeche se upar, yaani left se right padho: 2 | 1 | 4 | 2 → 2142.
Do baatein alag rakho: “5 jodo” digit khud ko dekh kar (jis pe kaam ho raha hai). “Aadha” padosi ka. Jaise dahai wale 5 pe: 5 khud taaq → +5; padosi 7 → aadha 3.
Ek-digit × 6 (6 ka pahaada bina ratte)
| Sawaal | Ikai (digit khud) | Aage wala 0 | Jawab |
|---|---|---|---|
| 2 × 6 | digit 2; 2 juft hai, to 5 nahi; padosi nahi (ikai ke right mein koi digit nahi). Kul: 2 → likho 2 | aage wala 0: digit 0; padosi 2 ka aadha 1. Jod: 0 + 1 = 1 → likho 1 | 12 |
| 3 × 6 | digit 3; 3 taaq hai, to 5 jodo; padosi nahi (ikai ke right mein koi digit nahi). Jod: 3 + 5 = 8 → likho 8 | aage wala 0: digit 0; padosi 3 ka aadha 1 (aadha hissa chhod diya). Jod: 0 + 1 = 1 → likho 1 | 18 |
| 4 × 6 | digit 4; 4 juft hai, to 5 nahi; padosi nahi (ikai ke right mein koi digit nahi). Kul: 4 → likho 4 | aage wala 0: digit 0; padosi 4 ka aadha 2. Jod: 0 + 2 = 2 → likho 2 | 24 |
| 5 × 6 | digit 5; 5 taaq hai, to 5 jodo; padosi nahi (ikai ke right mein koi digit nahi). Jod: 5 + 5 = 10 → likho 0, carry 1 | aage wala 0: digit 0; padosi 5 ka aadha 2 (aadha hissa chhod diya); haath ka carry 1. Jod: 0 + 2 + 1 = 3 → likho 3 | 30 |
| 6 × 6 | digit 6; 6 juft hai, to 5 nahi; padosi nahi (ikai ke right mein koi digit nahi). Kul: 6 → likho 6 | aage wala 0: digit 0; padosi 6 ka aadha 3. Jod: 0 + 3 = 3 → likho 3 | 36 |
| 7 × 6 | digit 7; 7 taaq hai, to 5 jodo; padosi nahi (ikai ke right mein koi digit nahi). Jod: 7 + 5 = 12 → likho 2, carry 1 | aage wala 0: digit 0; padosi 7 ka aadha 3 (aadha hissa chhod diya); haath ka carry 1. Jod: 0 + 3 + 1 = 4 → likho 4 | 42 |
| 8 × 6 | digit 8; 8 juft hai, to 5 nahi; padosi nahi (ikai ke right mein koi digit nahi). Kul: 8 → likho 8 | aage wala 0: digit 0; padosi 8 ka aadha 4. Jod: 0 + 4 = 4 → likho 4 | 48 |
| 9 × 6 | digit 9; 9 taaq hai, to 5 jodo; padosi nahi (ikai ke right mein koi digit nahi). Jod: 9 + 5 = 14 → likho 4, carry 1 | aage wala 0: digit 0; padosi 9 ka aadha 4 (aadha hissa chhod diya); haath ka carry 1. Jod: 0 + 4 + 1 = 5 → likho 5 | 54 |
🆘 Atak gaye? Aise socho
- 5 kab jodna hai? Jab woh digit khud taaq ho jis pe kaam ho raha hai. Padosi taaq ho to nahi.
- Padosi ka aadha: taaq padosi pe pehle 1 kam karo, phir aadha: 7 → 6 → 3.
- Ikai pe: padosi nahi, to bas digit (+5 agar taaq).
- Aage wala 0: 0 juft hai, to sirf padosi ka aadha + carry. 0 aaye to likhna nahi.
- Check: number × 5 + number (× 5 matlab × 10 ka aadha). 24 × 6 = 120 + 24 = 144.
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Chaabi: 6 = 1 + 5. To 357 × 6 = 357 × 1 + 357 × 5. Aur × 5 matlab × 10 ka aadha (Chapter 5).
- × 1 wala hissa: har digit apne khaane mein waisa hi. Yeh niyam ka “digit khud” hai.
- × 5 wala hissa, pehla kadam (× 10): 357 → 3570. Har digit ek khaana left khisak gaya, to har khaane mein ab uska padosi aa gaya (Chapter 11 jaisa).
- × 5 wala hissa, doosra kadam (aadha): ab 3570 ka aadha, khaane-khaane. Har khaane mein aaye padosi ka aadha lo: yahi “padosi ka aadha”. Agar padosi taaq tha, to uska ½ bacha.
- Bacha ½ kahan gaya? “Taaq pe 5”! “× 5 = × 10 ka aadha” box ke noton ki tarah: taaq digit ke aadhe mein ek ₹10 ka note toot kar ₹5 + ₹5 ban jaata hai, aur woh ₹5 ek khaana right mein aata hai. Woh right wala khaana kaunsa? Wahi jahan woh digit asal mein baitha tha (× 10 ne use left bheja tha). Isliye jo digit khud taaq hai, uske apne khaane mein 5 judta hai.
Teeno hisse jodo: digit + padosi ka aadha + (taaq ho to 5). Neeche chitra mein 357 × 6 ke saare khaane hain.
Sabse chhota example: 7 × 6
7 × 6 = 7 + 7 × 5 = 7 + 35. 35 kaise bana? ₹70 ka aadha: 3 note (dahai mein 3) aur ₹5 (ikai mein, kyunki 7 taaq). Ikai: digit khud 7, aur taaq ka 5: 7 + 5 = 12 → 2 likho, 1 carry. Dahai (aage wala 0): padosi 7 ka aadha 3 + carry 1 = 4. Jawab 42 ✅. Bilkul “× 5 = × 10 ka aadha” box ka chitra!
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Tareeke Se Check)
Example 1: 48 × 6 = ?
Aage 0: 048. Ikai 8: 8 juft → 8 likho. Dahai 4: 4 juft, padosi 8 ka aadha 4; 4 + 4 = 8 → 8 likho. Aage wala 0: padosi 4 ka aadha 2; 0 + 2 = 2 → 2 likho (aakhri kadam). Jawab: 2 | 8 | 8 → 288.
✔️ Normal tareeke se check: 48 × 5: 480 ka aadha 240. Phir 240 + 48 = 288 ✅
Example 2: 9 × 6 = ?
Aage 0: 09. Ikai 9: 9 taaq → +5; 9 + 5 = 14 → 4 likho, carry 1. Aage wala 0: padosi 9 ka aadha 4, carry 1; 0 + 4 + 1 = 5 → 5 likho (aakhri kadam). Jawab: 5 | 4 → 54.
✔️ Normal tareeke se check: 6 ko 9 baar jodo: 6, 12, 18, 24, 30, 36, 42, 48, 54 ✅
Example 3: 503 × 6 = ? (beech mein 0)
Aage 0: 0503. Ikai 3: 3 taaq → +5; 3 + 5 = 8 → 8 likho. Dahai 0: 0 juft, padosi 3 ka aadha 1; 0 + 1 = 1 → 1 likho. Saikda 5: 5 taaq → +5, padosi 0 ka aadha 0; 5 + 5 + 0 = 10 → 0 likho, carry 1. Aage wala 0: padosi 5 ka aadha 2, carry 1; 0 + 2 + 1 = 3 → 3 likho (aakhri kadam). Jawab: 3 | 0 | 1 | 8 → 3018.
✔️ Normal tareeke se check: Column: 6 × 3 = 18 (8, carry 1); 6 × 0 = 0, 0 + 1 = 1; 6 × 5 = 30 → 3018 ✅
Example 4: 1234 × 6 = ?
Aage 0: 01234. Ikai 4: 4 juft → 4 likho. Dahai 3: 3 taaq → +5, padosi 4 ka aadha 2; 3 + 5 + 2 = 10 → 0 likho, carry 1. Saikda 2: 2 juft, padosi 3 ka aadha 1, carry 1; 2 + 1 + 1 = 4 → 4 likho. Hazaar 1: 1 taaq → +5, padosi 2 ka aadha 1; 1 + 5 + 1 = 7 → 7 likho. Aage wala 0: padosi 1 ka aadha 0; 0 + 0 = 0 → aage ka 0, nahi likhte. Jawab: 0 | 7 | 4 | 0 | 4 → 7404.
Dhyaan: aakhri kadam pe padosi 1 ka aadha 0 aaya, to aage koi digit nahi likha.
✔️ Normal tareeke se check: 1234 × 5: 12340 ka aadha 6170. Phir 6170 + 1234 = 7404 ✅
— 📌 charumam.com
🕉️ Vedic Maths Connection
Trachtenberg system Vedic Maths se alag hai, par dono place value pe chalte hain. Vedic Maths mein bhi × 5 ke liye “× 10 karo, phir aadha” wali soch milti hai (sutra = chhota niyam; sutra “Ardha” yaani aadha ki soch se jodte hain). Trachtenberg ne isi ko khaane-khaane kar diya: “padosi ka aadha” aur “taaq pe 5”. Aur 6 = 5 + 1 wala tod, Vedic ke “paas wale aasaan number se socho” jaisa hai.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| Koi bhi number × 6 (357 × 6) | ✅ Haan | Pahaade ki zaroorat nahi. |
| Ek-digit × 6 (7 × 6) | ✅ Haan | 7 + 5 = 12, phir 3 + 1 = 4 → 42. |
| Juft digit wale number (2468 × 6) | ✅ Aasaan | 5 kahin nahi judta, bas digit + aadha padosi. |
| × 60, × 600 | ✅ Haan | × 6, phir peeche 0 ya 00. |
| × 7 | ➡️ Post 12.2 | Digit ki jagah double. |
| × 5 | ✅ Bonus | Wahi niyam, bas “digit khud” mat jodo (HOTS dekho). |
— 📌 charumam.com
❌→✅ Common Mistakes
Galti 1: 5 padosi ko dekh kar joda
Sawaal: 47 × 6
❌ Galat: ikai 7 pe 5 nahi joda, aur dahai 4 pe joda (kyunki uska padosi 7 taaq hai) → 7; 4 + 3 + 5 = 12; 0 + 2 + 1 = 3 → 327.
✅ Sahi: 5 sirf tab jab digit khud taaq. Ikai 7 taaq: 7 + 5 = 12 (2, carry 1). Dahai 4 juft: 4 + 3 + 1 = 8. Aage wala 0: yahan 7 nahi aata (woh galti thi); 0 pe sirf padosi 4 ka aadha 2 aata hai → 2. Jawab 282. Check: 47 × 5 = 235, 235 + 47 = 282 ✅
Galti 2: Taaq digit pe 5 bhool gaye
Sawaal: 357 × 6
❌ Galat: kahin 5 nahi joda → 1587.
✅ Sahi: 3, 5, 7 teeno taaq → har kadam +5 → 2142.
Galti 3: Padosi ka aadha nahi, poora padosi
Sawaal: 357 × 6
❌ Galat: padosi poora joda (× 11 jaisa) → 4482.
✅ Sahi: padosi ka aadha: 7 → 3, 5 → 2, 3 → 1. Jawab 2142.
Galti 4: Taaq padosi ka ½ round karke upar le gaye
Sawaal: 7 × 6
❌ Galat: 7 ka aadha “4” maan liya → aage 0: 4 + 1 = 5 → 52.
✅ Sahi: ½ chhodo: 7 ka aadha 3. Jawab 42. (Woh ½ pehle hi “taaq pe 5” mein gin liya.)
🧠 Rule of thumb: × 6: digit + padosi ka aadha (½ chhodo) + 5 agar digit khud taaq. Right se left, carry aage.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 sawaal, sab × 6. Target: 6 minute. Pehle har digit ke neeche “J” (juft) ya “T” (taaq) likh lo, phir niyam.
- 2 × 6 = ?
- 5 × 6 = ?
- 8 × 6 = ?
- 12 × 6 = ?
- 15 × 6 = ?
- 24 × 6 = ?
- 37 × 6 = ?
- 46 × 6 = ?
- 53 × 6 = ?
- 68 × 6 = ?
- 79 × 6 = ?
- 86 × 6 = ?
- 135 × 6 = ?
- 246 × 6 = ?
- 999 × 6 = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- 12
- 30
- 48
- 72
- 90
- 144
- 222
- 276
- 318
- 408
- 474
- 516
- 810
- 1476
- 5994
Kaise kiya (kuch):
Aage 0: 079. Ikai 9: 9 taaq → +5; 9 + 5 = 14 → 4 likho, carry 1. Dahai 7: 7 taaq → +5, padosi 9 ka aadha 4, carry 1; 7 + 5 + 4 + 1 = 17 → 7 likho, carry 1. Aage wala 0: padosi 7 ka aadha 3, carry 1; 0 + 3 + 1 = 4 → 4 likho (aakhri kadam). Jawab: 4 | 7 | 4 → 474.
Aage 0: 0999. Ikai 9: 9 taaq → +5; 9 + 5 = 14 → 4 likho, carry 1. Dahai 9: 9 taaq → +5, padosi 9 ka aadha 4, carry 1; 9 + 5 + 4 + 1 = 19 → 9 likho, carry 1. Saikda 9: 9 taaq → +5, padosi 9 ka aadha 4, carry 1; 9 + 5 + 4 + 1 = 19 → 9 likho, carry 1. Aage wala 0: padosi 9 ka aadha 4, carry 1; 0 + 4 + 1 = 5 → 5 likho (aakhri kadam). Jawab: 5 | 9 | 9 | 4 → 5994.
— 📌 charumam.com
⏰ 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 × 6 = ?
- 6 × 6 = ?
- 10 × 6 = ?
- 11 × 6 = ?
- 13 × 6 = ?
- 20 × 6 = ?
- 22 × 6 = ?
- 25 × 6 = ?
- 44 × 6 = ?
- 50 × 6 = ?
| 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
- 18
- 36
- 60
- 66
- 78
- 120
- 132
- 150
- 264
- 300
Dhyaan:
Aage 0: 013. Ikai 3: 3 taaq → +5; 3 + 5 = 8 → 8 likho. Dahai 1: 1 taaq → +5, padosi 3 ka aadha 1; 1 + 5 + 1 = 7 → 7 likho. Aage wala 0: padosi 1 ka aadha 0; 0 + 0 = 0 → aage ka 0, nahi likhte. Jawab: 0 | 7 | 8 → 78.
20 × 6: 2 × 6 = 12, peeche 0 → 120. 50 × 6: 5 × 6 = 30, peeche 0 → 300.
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. 357 × 6 = ?
Jawab aur kaaran dekho
Sahi jawab: (a) 2142
- (a) sahi ✅: 0357: 7 + 5 = 12 (2, c1); 5 + 5 + 3 + 1 = 14 (4, c1); 3 + 5 + 2 + 1 = 11 (1, c1); 0 + 1 + 1 = 2 → 2142.
- (b) galat ❌: 1587: taaq digit pe 5 jodna bhool gaye.
- (c) galat ❌: 4482: padosi ka aadha nahi, poora padosi joda.
- (d) galat ❌: 1032: carry nahi jode.
Q2. × 6 niyam mein “5 jodo” kab?
Jawab aur kaaran dekho
Sahi jawab: (d) Jab digit khud taaq ho
- (a) galat ❌: Padosi taaq ho to sirf uske aadhe ka ½ chhodte hain; 5 digit khud ko dekh kar.
- (b) galat ❌: Har kadam pe nahi: 24 × 6 mein kahin 5 nahi judta (2 aur 4 juft).
- (c) galat ❌: Ulta: juft digit pe 5 nahi judta.
- (d) sahi ✅: Haan: jis digit pe kaam ho raha hai, woh khud taaq (1, 3, 5, 7, 9) ho.
Q3. 9 ka aadha (½ chhod kar) kitna?
Jawab aur kaaran dekho
Sahi jawab: (c) 4
- (a) galat ❌: 4.5 asli aadha hai, par niyam mein ½ chhod kar 4 lena hai.
- (b) galat ❌: 9 to digit khud hai, aadha nahi.
- (c) sahi ✅: 9 ka aadha 4½; ½ chhoda → 4. (Pehle 1 kam: 8, phir aadha 4.)
- (d) galat ❌: 5: ½ ko upar round kar diya; niyam mein ½ chhodte hain.
Q4. 24 × 6 = ?
Jawab aur kaaran dekho
Sahi jawab: (b) 144
- (a) galat ❌: 154: dahai mein padosi 4 ka aadha 2 ki jagah 3 le liya (2 + 3 = 5).
- (b) sahi ✅: 024: 4; 2 + 2 = 4; 0 + 1 = 1 → 144.
- (c) galat ❌: 264: padosi poora joda (aadha nahi): 4; 2 + 4 = 6; 0 + 2 = 2.
- (d) galat ❌: 44: aage wale 0 ka kadam bhool gaye.
Q5. 7 × 6 = ?
Jawab aur kaaran dekho
Sahi jawab: (a) 42
- (a) sahi ✅: 07: 7 + 5 = 12 (2, c1); 0 + 3 + 1 = 4 → 42.
- (b) galat ❌: 37: taaq pe 5 bhool gaye: 7 → 7; 0 + 3 = 3 → 37.
- (c) galat ❌: 52: 7 ka aadha 4 le liya (½ round kar diya): 0 + 4 + 1 = 5.
- (d) galat ❌: 82: padosi 7 poora joda: 0 + 7 + 1 = 8.
Q6. 99 × 6 = ?
Jawab aur kaaran dekho
Sahi jawab: (d) 594
- (a) galat ❌: 539: taaq pe 5 bhool gaye.
- (b) galat ❌: 484: carry nahi jode.
- (c) galat ❌: 1144: padosi poora joda (aadha nahi).
- (d) sahi ✅: 099: 9 + 5 = 14 (4, c1); 9 + 5 + 4 + 1 = 19 (9, c1); 0 + 4 + 1 = 5 → 594.
Q7. 6 = 1 + 5. Is tod se niyam ka kaunsa hissa “× 1” se aata hai?
Jawab aur kaaran dekho
Sahi jawab: (c) Digit khud
- (a) galat ❌: Taaq pe 5 bhi × 5 wale hisse se (toota hua note).
- (b) galat ❌: Aage 0 bas extra khaane ke liye hai, kisi guna se nahi.
- (c) sahi ✅: × 1 matlab har digit apne khaane mein waisa hi: “digit khud”.
- (d) galat ❌: Padosi ka aadha × 5 wale hisse se aata hai (× 10 ka aadha).
Q8. 1234 × 6 = ?
Jawab aur kaaran dekho
Sahi jawab: (b) 7404
- (a) galat ❌: 17404: aakhri kadam pe padosi 1 ka aadha 0 hota hai, 1 nahi.
- (b) sahi ✅: 01234: 4; 3 + 5 + 2 = 10 (0, c1); 2 + 1 + 1 = 4; 1 + 5 + 1 = 7; aage 0: 1 ka aadha 0 → 7404.
- (c) galat ❌: 2354: taaq digit (3 aur 1) pe 5 bhool gaye.
- (d) galat ❌: 7304: carry nahi joda (saikde mein 2 + 1 = 3 likha).
Q9. 503 × 6 mein dahai wale 0 ka kadam?
Jawab aur kaaran dekho
Sahi jawab: (a) 0 + 1 = 1
- (a) sahi ✅: 0 juft (5 nahi), padosi 3 ka aadha 1 → 1. (Ikai se carry nahi tha: 3 + 5 = 8.) Jawab 3018.
- (b) galat ❌: 6: 0 pe 5 jod diya, par 0 juft hai.
- (c) galat ❌: 3: padosi 3 poora joda, aadha nahi.
- (d) galat ❌: 0: padosi ka aadha bhool gaye.
Q10. Kaunsa jawab juft digit wale number ka hai, jisme kahin 5 nahi judta?
Jawab aur kaaran dekho
Sahi jawab: (d) 2468 × 6 = 14808
- (a) galat ❌: 357 mein 3, 5, 7 taaq: har kadam 5 judta hai.
- (b) galat ❌: 99 mein dono 9 taaq: 5 judta hai.
- (c) galat ❌: 135 mein 1, 3, 5 taaq: 5 judta hai. (135 × 6 = 810 sahi hai, par sawaal juft digit wale ka tha.)
- (d) sahi ✅: 2, 4, 6, 8 sab juft: har kadam bas digit + aadha padosi → 14808.
— 📌 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. Bonus niyam: × 5 ke liye Trachtenberg kehte hain “sirf padosi ka aadha, aur digit taaq ho to 5” (digit khud nahi). 48 × 5 aur 37 × 5 karke dekho.
Solution dekho
48 × 5: 048 → ikai 8 juft: 0; dahai 4 juft, padosi 8 ka aadha 4: 4; aage 0: padosi 4 ka aadha 2 → 240. 37 × 5: 037 → ikai 7 taaq: 5; dahai 3 taaq, padosi 7 ka aadha 3: 5 + 3 = 8; aage 0: padosi 3 ka aadha 1 → 185.
Check (Chapter 5): 480 ka aadha 240 ✅, 370 ka aadha 185 ✅. Yeh × 6 ka hi niyam hai, bas “× 1” wala digit hata do!
Q2. Ek peti mein 6 aam. 86 peti. Kitne aam?
Solution dekho
86 × 6: 086 → ikai 6 juft: 6; dahai 8 juft, padosi 6 ka aadha 3: 8 + 3 = 11 → 1, carry 1; aage 0: padosi 8 ka aadha 4, carry 1: 4 + 1 = 5. Jawab 516.
Check: 86 × 5 = 430, 430 + 86 = 516 ✅
Q3. 555 × 6 = ? Har digit taaq hai: dhyaan se!
Solution dekho
0555: ikai 5 + 5 = 10 → 0, carry 1. Dahai: digit 5, taaq ka 5, padosi 5 ka aadha 2, carry 1: 5 + 5 + 2 + 1 = 13 → 3, carry 1. Saikda: wahi, 5 + 5 + 2 + 1 = 13 → 3, carry 1. Aage 0: 2 + 1 = 3. Jawab 3330.
Check: 555 × 5 = 2775, 2775 + 555 = 3330 ✅
Q4. Kis ek-digit number ko 6 se guna karne pe jawab ki ikai wahi digit aata hai? Niyam se socho.
Solution dekho
Ikai pe: digit + (taaq ho to 5). Jawab ki ikai digit hi rahe, iske liye 5 nahi judna chahiye, yaani digit juft: 0, 2, 4, 6, 8. Check: 2 × 6 = 12, 4 × 6 = 24, 6 × 6 = 36, 8 × 6 = 48 ✅ (sab ki ikai wahi digit).
— 📌 charumam.com
🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | Guna ke sawaal, 6 cheezon ki peti/plate, pahaada check. |
| Maths Olympiad | Pattern (kaunse digit pe ikai wahi), × 5 bonus niyam. |
| NTSE / MAT | Mental maths aur number series. |
| SSC / Bank (aage) | Simplification mein × 6 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: “digit, aadha padosi, taaq pe paanch.” Shuru mein har digit ke neeche J/T likho. 3 din copy pe, phir mann mein. Agle post mein × 7: bas “digit” ki jagah “double”! 6️⃣
— 📌 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: 0357. Ikai 7 (taaq): 7 + 5 = 12 → 2, carry 1. Dahai 5 (taaq), padosi 7 ka aadha 3: 5 + 5 + 3 + 1 = 14 → 4, carry 1. Saikda 3 (taaq), padosi 5 ka aadha 2: 3 + 5 + 2 + 1 = 11 → 1, carry 1. Aage 0: padosi 3 ka aadha 1, carry 1: 0 + 1 + 1 = 2. Jawab 2142 ✅
| 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. ½ chhodna galat nahi hai?
Nahi. Woh ½ “taaq pe 5” mein pehle hi gin liya jaata hai (“× 5 = × 10 ka aadha” box ke toote note jaisa). Isliye dono saath mein bilkul sahi jawab dete hain.
2. 5 digit ko dekh kar ya padosi ko?
Digit khud ko (jis pe kaam ho raha hai). Padosi ka sirf aadha lete hain.
3. Carry kitna tak?
× 6 mein carry sirf 1 tak: sabse bada jod 9 + 5 + 4 + 1 = 19.
4. Aakhri kadam pe 0 aaya to?
Aage ka 0 nahi likhte: 1234 × 6 = 07404 → 7404.
5. Agle post mein kya hai?
Post 12.2: × 7. Niyam: double + padosi ka aadha + taaq pe 5.
— 📌 charumam.com
🟢 Recap: Aaj Kya Seekha
- 6 = 1 + 5, aur × 5 matlab × 10 ka aadha.
- × 6: digit + padosi ka aadha (½ chhodo) + 5 agar digit khud taaq.
- Aage 0, right se left, ikai ka padosi nahi.
- Juft: 0, 2, 4, 6, 8. Taaq: 1, 3, 5, 7, 9.
- × 6 mein carry sirf 1 tak.
- Check: × 5 + number (24 × 6 = 120 + 24).
— charumam.com 🎯
👉 Aage Kya?
Ab × 7 ki baari: Post 12.2 mein niyam bas ek shabd badlega: “digit” ki jagah “double”. 34 × 7 = 238 aur 625 × 7 = 4375!
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
- ▶️ YouTube (subscribe): @charumamofficial
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— 📌 charumam.com
