Do niyam, ek parivaar! 👨👩👧 Chapter 4 mein 11 se guna seekha tha. Aaj dekhenge ki Trachtenberg ka × 11 niyam aur × 12 niyam bhai-bahan hain: farak sirf ek shabd ka, “digit” ya “double”. Phir seekhenge jawab check karne ke 3 aasaan tareeke, taaki galti turant pakdi jaaye.
⏱ Before Timer: Pehle Purane Tareeke Se
Pehle school wala tareeka. Dono guna alag-alag, do-do line mein:
- 76 × 11 = 76 × 10 + 76 × 1 = 760 + 76 = 836.
- 76 × 12 = 76 × 10 + 76 × 2 = 760 + 152 = 912.
- Dono ka jod: 836 + 912 = 1748.
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
76 × 11 + 76 × 12 = 1748
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: Post 11.1–11.2 ka × 12 niyam, Chapter 4 ka × 11, aur carry. Neeche ke boxes mein double aur Trachtenberg ke shabd dobara. Naye shabd:
📖 Shabd jo is post mein aayenge
- × 11 niyam (Trachtenberg): aage 0, right se left, har digit + padosi. Chapter 4 wala hi kaam, bas right se chalte hain.
- × 12 niyam: aage 0, right se left, har digit ka double + padosi.
- Check: jawab sahi hai ya nahi, yeh dekhne ka doosra tareeka.
- Andaaza: mote taur pe jawab ka size: × 11 ya × 12 ka jawab, number × 10 se thoda bada.
- Ikai check: jawab ka sabse right wala digit sirf number ki ikai se banta hai (ikai ka koi padosi nahi), to use pehle se jaan sakte ho.
📏 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.
✌️ 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: 76 → 076. Isse number ki keemat nahi badalti: 076 mein “0 saikda” hai, yaani koi saikda nahi, to 076 aur 76 ek hi hain (jaise ₹076 aur ₹76 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. 076 mein: 6 (ikai) ke right mein kuch nahi, to uska padosi nahi hai (0 maano). 7 ka padosi 6. Aage wale 0 ka padosi 7. 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 “digit (× 11) ya double (× 12), phir padosi jodo”, right se left, carry ke saath.
🔁 Chhota sa warm-up
- 76 × 10 = ?
- 760 + 76 = ?
- 6 ka double ka aakhri digit?
Answers dekho
- 760
- 836
- 2 (6 + 6 = 12)
Pehle do mila kar 76 × 11 ho gaya, aur teesra × 12 ki ikai hai!
— 📌 charumam.com
🎬 Kahani: Do Dukaan, Do Daam
Ek dukaan pe kaapi ₹11 ki hai, doosri pe ₹12 ki. Riya ko 76 kaapiyan chahiye. Papa ne poocha: “Dono dukaan pe kitna kharcha?”
Riya ne ek hi kaagaz pe 076 likha. “₹11 wali: digit + padosi: 6, phir 7 + 6 = 13, phir 0 + 7 + 1 = 8: ₹836. ₹12 wali: double + padosi: 12, phir 14 + 6 + 1 = 21, phir 0 + 7 + 2 = 9: ₹912.”
Papa: “Farak ₹76, kyunki har kaapi pe ₹1 ka farak!” 912 − 836 = 76 ✅. Riya muskurayi: “Dono niyam bhai-bahan hain.” 🔀
— 📌 charumam.com
🪄 Trick Step by Step: Ek Sawaal Poochho: Digit Ya Double?
💡 Is post ki ek badi baat: × 11: har digit + padosi. × 12: har digit ka double + padosi. Dono mein aage 0, right se left, carry aage, aakhri kadam poora. Aur check: andaaza, ikai check, aur × 12 = × 11 + number.
Step 1: Pehchano: 11 ya 12?
11 = 10 + 1 → digit ek baar (digit khud). 12 = 10 + 2 → digit do baar (double). Padosi dono mein, kyunki “10” dono mein hai.
Step 2: × 11 niyam: digit + padosi
Sawaal: 253 × 11.
253 × 11: number ke aage 0 lagao → 0253. Ab right se left (ikai se shuru) har digit pe niyam:
| Kadam | Jagah | Digit | Kaam | Likho | Carry aage |
|---|---|---|---|---|---|
| 1 | Ikai | 3 | digit 3; padosi nahi (ikai ke right mein koi digit nahi). Kul: 3 | 3 | 0 |
| 2 | Dahai | 5 | digit 5; padosi 3 jodo. Jod: 5 + 3 = 8 | 8 | 0 |
| 3 | Saikda | 2 | digit 2; padosi 5 jodo. Jod: 2 + 5 = 7 | 7 | 0 |
| 4 | Hazaar | 0 (aage wala) | aage wala 0: digit 0; padosi 2 jodo. Jod: 0 + 2 = 2 | 2 | — |
Ab “Likho” wale digit neeche se upar, yaani left se right padho: 2 | 7 | 8 | 3 → 2783.
Chapter 4 mein yahi left se kiya tha: 2 | 2 + 5 | 5 + 3 | 3 = 2 | 7 | 8 | 3 = 2783. Wahi jawab! Trachtenberg bas right se chalte hain, taaki carry aasaan ho.
Step 3: × 12 niyam: double + padosi
Wahi number, 253 × 12:
253 × 12: number ke aage 0 lagao → 0253. 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 | 5 | 5 ka double: 5 + 5 = 10; padosi 3 jodo. Jod: 10 + 3 = 13 | 3 | 1 |
| 3 | Saikda | 2 | 2 ka double: 2 + 2 = 4; padosi 5 jodo; haath ka carry 1. Jod: 4 + 5 + 1 = 10 | 0 | 1 |
| 4 | Hazaar | 0 (aage wala) | aage wala 0: double 0 + 0 = 0; padosi 2 jodo; haath ka carry 1. Jod: 0 + 2 + 1 = 3 | 3 | — |
Ab “Likho” wale digit neeche se upar, yaani left se right padho: 3 | 0 | 3 | 6 → 3036.
Step 4: Check 1: andaaza
253 × 11 aur 253 × 12 dono 2530 (× 10) se thoda bade honge. 2783 aur 3036 ✅. Agar jawab 2530 se chhota aaye, ya 25300 jitna bada, to galti hai.
Step 5: Check 2: ikai check
Ikai ka koi padosi nahi, to jawab ki ikai sirf number ki ikai se banti hai. × 11: jawab ki ikai = number ki ikai (253 → 3, 2783 bhi 3 pe khatam ✅). × 12: jawab ki ikai = ikai ke double ka aakhri digit (3 + 3 = 6, aur 3036 bhi 6 pe khatam ✅).
Step 6: Check 3: × 12 = × 11 + number
12 = 11 + 1. To 253 × 12 = 253 × 11 + 253 = 2783 + 253 = 3036 ✅. Ek niyam se doosre ko check kar lo!
Dono niyam aamne-saamne
| Baat | × 11 | × 12 |
|---|---|---|
| Kyun | 11 = 10 + 1 | 12 = 10 + 2 |
| Har digit pe | digit + padosi | double + padosi |
| Ikai pe | digit khud | digit ka double |
| Aakhri kadam (aage 0) | padosi + carry | padosi + carry |
| Sabse bada carry | 1 (9 + 9 + 1 = 19) | 2 (18 + 9 + 2 = 29) |
| Example | 76 × 11 = 836 | 76 × 12 = 912 |
🆘 Atak gaye? Aise socho
- Bhool gaye kaunsa niyam? 11 mein ek “1” (digit ek baar), 12 mein “2” (double).
- Padosi dono mein right wala. × 10 dono mein hai, aur × 10 se right wala digit is khaane mein aata hai.
- × 11 mein carry bada? Kabhi 1 se zyada nahi. × 12 mein 2 tak.
- Check karo: andaaza, ikai check, ya × 12 = × 11 + number.
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Dono niyam ek hi jad se: 11 = 10 + 1 aur 12 = 10 + 2.
- × 10 dono mein: har digit ek jagah left khisakta hai, to har khaane mein uska right wala padosi aata hai.
- × 1 (11 mein): har digit apni jagah pe ek baar. Isliye “digit + padosi”.
- × 2 (12 mein): har digit apni jagah pe do baar. Isliye “double + padosi”.
- Farak sirf “× 1” aur “× 2” ka, yaani har khaane mein ek digit extra. Poore number pe yeh extra milkar number khud hai: isliye × 12 = × 11 + number.
Wahi number, × 12
Ab chitra mein × 1 ki jagah × 2 (double) hai. Har khaana ek digit se bada, aur carry ke baad jawab 2783 + 253 = 3036.
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Tareeke Se Check)
Example 1: 89 × 11 = ?
Aage 0: 089. Ikai 9: digit khud, padosi nahi → 9 likho. Dahai 8: padosi 9; 8 + 9 = 17 → 7 likho, carry 1. Aage wala 0: padosi 8, carry 1; 0 + 8 + 1 = 9 → 9 likho (aakhri kadam). Jawab: 9 | 7 | 9 → 979.
✔️ Normal tareeke se check: 89 × 10 = 890, 890 + 89 = 979 ✅. Ikai check: 89 ki ikai 9, jawab bhi 9 pe khatam ✅
Example 2: 89 × 12 = ?
Aage 0: 089. Ikai 9: double 9 + 9 = 18 → 8 likho, carry 1. Dahai 8: double 8 + 8 = 16, padosi 9, carry 1; 16 + 9 + 1 = 26 → 6 likho, carry 2. Aage wala 0: padosi 8, carry 2; 0 + 8 + 2 = 10 → 10 likho (aakhri kadam). Jawab: 10 | 6 | 8 → 1068.
✔️ Normal tareeke se check: × 11 + number: 979 + 89 = 1068 ✅. Ikai check: 9 + 9 = 18, aakhri digit 8 ✅
Example 3: 4321 × 11 = ?
Aage 0: 04321. Ikai 1: digit khud, padosi nahi → 1 likho. Dahai 2: padosi 1; 2 + 1 = 3 → 3 likho. Saikda 3: padosi 2; 3 + 2 = 5 → 5 likho. Hazaar 4: padosi 3; 4 + 3 = 7 → 7 likho. Aage wala 0: padosi 4; 0 + 4 = 4 → 4 likho (aakhri kadam). Jawab: 4 | 7 | 5 | 3 | 1 → 47531.
✔️ Normal tareeke se check: 4321 × 10 = 43210, 43210 + 4321 = 47531 ✅
Example 4: 909 × 12 = ?
Aage 0: 0909. Ikai 9: double 9 + 9 = 18 → 8 likho, carry 1. Dahai 0: double 0 + 0 = 0, padosi 9, carry 1; 0 + 9 + 1 = 10 → 0 likho, carry 1. Saikda 9: double 9 + 9 = 18, padosi 0, carry 1; 18 + 0 + 1 = 19 → 9 likho, carry 1. Aage wala 0: padosi 9, carry 1; 0 + 9 + 1 = 10 → 10 likho (aakhri kadam). Jawab: 10 | 9 | 0 | 8 → 10908.
✔️ Normal tareeke se check: 909 × 10 = 9090, 909 × 2 = 1818, 9090 + 1818 = 10908 ✅
— 📌 charumam.com
🕉️ Vedic Maths Connection
Vedic Maths ka Urdhva-Tiryagbhyam (seedha aur tirchha) 11 aur 12 dono pe ek hi tarah chalta hai. 11 = “1 | 1”: har khaane mein digit × 1 + padosi × 1. 12 = “1 | 2”: har khaane mein digit × 2 + padosi × 1. Trachtenberg ne isi ko aasaan shabdon mein bola. Aur Chapter 4 ka “beech mein digits ka jod” bhi yahi hai, bas left se. Teen raaste, ek hi sach: place value.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| × 11 (koi bhi number) | ✅ Haan | Digit + padosi; ya Chapter 4 ka left se tareeka. |
| × 12 (koi bhi number) | ✅ Haan | Double + padosi. |
| Ek hi number × 11 aur × 12 dono | ✅ Bahut tez | Pehle × 11, phir number jodo: × 12. |
| × 21, × 22 | ❌ Is niyam se nahi | 21 = 20 + 1, alag cheez. School tareeka (× 20 + × 1). |
| × 110, × 120 | ✅ Haan | × 11 ya × 12, phir peeche 0. |
| Jawab check | ✅ Hamesha | Andaaza + ikai check: 5 second. |
— 📌 charumam.com
❌→✅ Common Mistakes
Galti 1: × 12 mein double bhool gaye
Sawaal: 76 × 12
❌ Galat: 836 (yeh × 11 ka jawab hai).
✅ Sahi: double + padosi: 912. Check: 836 + 76 = 912 ✅
Galti 2: × 11 mein double kar diya
Sawaal: 253 × 11
❌ Galat: 3036 (yeh × 12 ka jawab hai).
✅ Sahi: digit + padosi: 2783. Ikai check: 253 ki ikai 3, 2783 ki ikai 3 ✅.
Galti 3: Ikai check ulta
Sawaal: kisi ne 89 × 12 ka jawab 1069 likha. Sahi hai?
❌ Galat: “ikai 9 hai, to jawab 9 pe khatam, theek hai.”
✅ Sahi: × 12 mein ikai ka double: 9 + 9 = 18 → 8. To jawab 8 pe khatam hona chahiye: 1068.
Galti 4: × 11 mein carry 2 le liya
Sawaal: 89 × 11
❌ Galat: 8 + 9 = 17, 7 likho aur 2 carry.
✅ Sahi: 17 = 1 dahai + 7, to 1 carry. Jawab 979.
🧠 Rule of thumb: 11 → digit + padosi, 12 → double + padosi. Hamesha andaaza aur ikai check.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 mix sawaal (× 11 aur × 12). Pehle dekho kaunsa niyam! Target: 6 minute.
- 34 × 11 = ?
- 34 × 12 = ?
- 57 × 11 = ?
- 57 × 12 = ?
- 68 × 11 = ?
- 68 × 12 = ?
- 123 × 11 = ?
- 123 × 12 = ?
- 345 × 11 = ?
- 345 × 12 = ?
- 909 × 11 = ?
- 808 × 12 = ?
- 2134 × 11 = ?
- 2134 × 12 = ?
- 99 × 11 = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- 374
- 408
- 627
- 684
- 748
- 816
- 1353
- 1476
- 3795
- 4140
- 9999
- 9696
- 23474
- 25608
- 1089
Kaise kiya (kuch): 57 × 12 = 57 × 11 + 57 = 627 + 57 = 684. 345 × 12 = 3795 + 345 = 4140.
Aage 0: 0345. Ikai 5: digit khud, padosi nahi → 5 likho. Dahai 4: padosi 5; 4 + 5 = 9 → 9 likho. Saikda 3: padosi 4; 3 + 4 = 7 → 7 likho. Aage wala 0: padosi 3; 0 + 3 = 3 → 3 likho (aakhri kadam). Jawab: 3 | 7 | 9 | 5 → 3795.
— 📌 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.
- 12 × 11 = ?
- 12 × 12 = ?
- 21 × 11 = ?
- 21 × 12 = ?
- 40 × 11 = ?
- 40 × 12 = ?
- 15 × 11 = ?
- 15 × 12 = ?
- 81 × 11 = ?
- 81 × 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
- 132
- 144
- 231
- 252
- 440
- 480
- 165
- 180
- 891
- 972
Dhyaan: 40 × 11: 4 × 11 = 44, peeche 0 → 440. 40 × 12: 4 × 12 = 48, peeche 0 → 480. 81 × 12 = 891 + 81 = 972.
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. 76 × 12 = ?
Jawab aur kaaran dekho
Sahi jawab: (c) 912
- (a) galat ❌: 702: carry nahi jode.
- (b) galat ❌: 12: sirf ikai ka kaam kiya, baaki kadam bhool gaye.
- (c) sahi ✅: 076: 12 (2, c1); 14 + 6 + 1 = 21 (1, c2); 0 + 7 + 2 = 9 → 912.
- (d) galat ❌: 836: double nahi kiya; yeh 76 × 11 hai.
Q2. 253 × 11 = ?
Jawab aur kaaran dekho
Sahi jawab: (b) 2783
- (a) galat ❌: 2753: dahai mein padosi 3 nahi, wahi 5 rakh diya (5 + 0).
- (b) sahi ✅: 0253: 3; 5 + 3 = 8; 2 + 5 = 7; 0 + 2 = 2 → 2783.
- (c) galat ❌: 3036: double kar diya; yeh 253 × 12 hai.
- (d) galat ❌: 783: aage wale 0 ka kadam bhool gaye.
Q3. × 11 aur × 12 ke niyam mein kya same hai?
Jawab aur kaaran dekho
Sahi jawab: (a) Dono mein padosi jodte hain
- (a) sahi ✅: Dono mein × 10 hai, to padosi dono mein judta hai.
- (b) galat ❌: Double sirf × 12 mein (12 = 10 + 2). × 11 mein digit ek baar.
- (c) galat ❌: Carry dono mein aata hai: 89 × 11 mein 8 + 9 = 17, to 7 likho aur 1 carry (jawab 979). Carry bhool gaye to galat jawab aayega.
- (d) galat ❌: Trachtenberg dono mein right se chalte hain (Chapter 4 wala × 11 left se tha).
Q4. 89 × 12 ka jawab kis digit pe khatam hoga (ikai check)?
Jawab aur kaaran dekho
Sahi jawab: (d) 8
- (a) galat ❌: 9 to × 11 ka ikai check hai (89 × 11 = 979).
- (b) galat ❌: 2: 9 + 2 kar diya; double chahiye tha.
- (c) galat ❌: 6: koi niyam nahi; 1068 to 8 pe khatam.
- (d) sahi ✅: Ikai 9 ka double 18, aakhri digit 8. (89 × 12 = 1068.)
Q5. 57 × 11 = 627. To 57 × 12 = ?
Jawab aur kaaran dekho
Sahi jawab: (c) 684
- (a) galat ❌: 628: 627 + 1 kar diya.
- (b) galat ❌: 1254: 627 ka double kar diya; yeh 57 × 22 hai.
- (c) sahi ✅: × 12 = × 11 + number: 627 + 57 = 684.
- (d) galat ❌: 638: 627 + 11 kar diya; number (57) jodna tha.
Q6. Kaunsa jawab andaaze se hi galat dikhta hai: 345 × 12 = ?
Jawab aur kaaran dekho
Sahi jawab: (b) 414
- (a) Yeh option nahi chunna: yeh theek hai. Yeh bhi theek: 4140, 3450 se zyada hai.
- (b) sahi ✅: 414 to 3450 (345 × 10) se bhi chhota! × 12 ka jawab × 10 se bada hona chahiye. Galat.
- (c) Yeh option nahi chunna: yeh theek hai. 4140 sahi jawab hai (3795 + 345).
- (d) Yeh option nahi chunna: yeh theek hai. Yeh andaaza theek hai: 4140 sach mein 4000 se thoda zyada hai.
Q7. × 11 niyam mein sabse bada carry?
Jawab aur kaaran dekho
Sahi jawab: (a) 1
- (a) sahi ✅: Sabse bada jod 9 + 9 + 1 = 19, to carry 1.
- (b) galat ❌: 2 sirf × 12 mein (jod 29 tak).
- (c) galat ❌: 0 nahi: 89 × 11 mein 8 + 9 = 17 aata hai, carry 1.
- (d) galat ❌: 11 kabhi nahi; carry jod ka pehla digit hai.
Q8. 81 × 12 = ?
Jawab aur kaaran dekho
Sahi jawab: (d) 972
- (a) galat ❌: 891: double nahi kiya; yeh 81 × 11 hai.
- (b) galat ❌: 962: dahai mein padosi 1 jodna bhool gaye (16 → 6 likha, 7 hona tha).
- (c) galat ❌: 162: sirf double kiya (81 × 2), × 10 wala hissa (padosi) bhool gaye.
- (d) sahi ✅: 081: 2; 16 + 1 = 17 (7, c1); 0 + 8 + 1 = 9 → 972.
Q9. Kaunsa sawaal × 11 ya × 12 niyam se seedha NAHI hota?
Jawab aur kaaran dekho
Sahi jawab: (c) 34 × 21
- (a) galat ❌: 34 × 12 = 408, double + padosi.
- (b) galat ❌: 340 × 12: 34 × 12 = 408, peeche 0 → 4080.
- (c) sahi ✅: 21 na 11 hai na 12; 34 × 21 = 714 (680 + 34), school tareeka.
- (d) galat ❌: 34 × 11 = 374, digit + padosi.
Q10. 99 × 11 = ?
Jawab aur kaaran dekho
Sahi jawab: (b) 1089
- (a) galat ❌: 99: number hi likh diya, × 11 kiya hi nahi.
- (b) sahi ✅: 099: 9; 9 + 9 = 18 (8, c1); 0 + 9 + 1 = 10 → 1089.
- (c) galat ❌: 1188: double kar diya; yeh 99 × 12 hai.
- (d) galat ❌: 989: dahai ka carry 1 aakhri kadam mein nahi joda (0 + 9 = 9 likha, 10 hona tha).
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🧠 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. 76 × 11 + 76 × 12 ko ek hi guna mein likho: 76 ko kis number se guna? Phir jawab nikaalo.
Solution dekho
76 ko 11 baar aur 12 baar, yaani kul 11 + 12 = 23 baar: 76 × 23. Jawab: 836 + 912 = 1748.
Check: 76 × 23 = 76 × 20 + 76 × 3 = 1520 + 228 = 1748 ✅ (yahan 1520 = 76 × 20 aur 228 = 76 × 3)
Q2. Agar × 11 = digit + padosi, × 12 = double + padosi, to × 13 ka niyam kya hoga? 21 × 13 karke dekho.
Solution dekho
13 = 10 + 3, to har digit teen baar (digit + digit + digit) + padosi. 021: ikai 1 + 1 + 1 = 3; dahai 2 + 2 + 2 = 6, padosi 1: 6 + 1 = 7; aage 0: padosi 2 → 2. Jawab 273.
Check: 210 + 63 = 273 ✅ (yahan 63 = 21 × 3). (Is niyam mein carry 3 tak aa sakta hai, dhyaan rahe.)
Q3. Ek number × 11 = 528. Wahi number × 12 = ?
Solution dekho
Number dhoondho: 528 ki ikai 8, to number ki ikai bhi 8. Andaaza: 40 × 11 = 440 aur 50 × 11 = 550; 528 inke beech hai, to number 40 aur 50 ke beech, aur ikai 8, yaani 48. Check karo 48 × 11: 048 → 8; 4 + 8 = 12 (2, c1); 0 + 4 + 1 = 5 → 528 ✅. Number 48. To 48 × 12 = 528 + 48 = 576.
Check: 480 + 96 = 576 ✅
Q4. Riya ne 234 × 12 ka jawab 2574 likha. Bina poora guna kiye 2 check se batao: sahi ya galat?
Solution dekho
Ikai check: 4 ka double 8, to jawab 8 pe khatam hona chahiye; 2574 to 4 pe khatam: galat. (Asal mein 2574 = 234 × 11.)
Sahi jawab: 2574 + 234 = 2808. Check: 2340 + 468 = 2808 ✅
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🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | Do dukaan ke daam, × 11 / × 12 ka hisaab, jawab check. |
| Maths Olympiad | Niyam ka pattern (× 13 kya hoga?) aur galti pakdo. |
| NTSE / MAT | Mental maths, number series. |
| SSC / Bank (aage) | Simplification aur jawab jaldi check karna. |
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: “11 mein ek, 12 mein do, padosi dono mein.” Aur har jawab pe 2 second: andaaza + ikai check. Galti wahin pakdi jaayegi. Agle post mein × 12 ke tricky sawaal! 🔀
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⏱ 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)
× 11 niyam (digit + padosi): 076 → 6; 7 + 6 = 13 → 3, carry 1; 0 + 7 + 1 = 8 → 836. × 12 niyam (double + padosi): 076 → 12 → 2, carry 1; 14 + 6 + 1 = 21 → 1, carry 2; 0 + 7 + 2 = 9 → 912. Jod: 836 + 912 = 1748 ✅. (Aur ek chhota raasta: 76 × 23 = 76 × 11 + 76 × 12, yeh 11.4 mein!)
| 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. 🚀
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🤖 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:
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Chapter 4 ka × 11 aur yeh × 11 alag hain kya?
Nahi, jawab same aata hai. Chapter 4 mein left se chalte the (2 | 2 + 5 | 5 + 3 | 3), yahan right se. Right se chalne mein carry aasaan hota hai.
2. Ikai check se kya poori galti pakdi jaati hai?
Nahi, sirf ikai ki galti. Isliye andaaza bhi saath mein lagao. Dono milkar zyadatar galtiyan pakad lete hain.
3. × 12 = × 11 + number kab kaam aata hai?
Jab ek hi number ka × 11 aur × 12 dono chahiye, ya × 12 ko check karna ho.
4. Agle post mein kya hai?
Post 11.4: × 12 ke tricky sawaal: darjan, ulte sawaal (□ × 12), dhokhe (× 21) aur jodi.
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🟢 Recap: Aaj Kya Seekha
- 11 = 10 + 1 → digit + padosi.
- 12 = 10 + 2 → double + padosi.
- Dono: aage 0, right se left, carry aage.
- Carry: × 11 mein 1 tak, × 12 mein 2 tak.
- Check 1: andaaza (× 10 se thoda bada).
- Check 2: ikai check. Check 3: × 12 = × 11 + number.
— charumam.com 🎯
👉 Aage Kya?
Chapter 11 ka aakhri post: Post 11.4 mein × 12 ke tricky sawaal: ulte sawaal (□ × 12 = 696), jodi (12 × 37 + 12 × 63), dhokhe aur darjan ki paheliyan!
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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