Chapter 7 mein dono numbers base ke ek hi taraf the: dono kam (8 × 7) ya dono zyada (12 × 13). Aur Post 7.4 mein 12 × 8 ko “dhokha” kaha tha. Aaj usi ka tareeka: jab ek number base se upar aur doosra neeche ho. Left hissa wahi tirchha niyam, par kona jodna nahi, ghatana hota hai. Aur ghatane ke liye ek purana dost wapas aayega: udhaar.
⏱ Before Timer: Pehle Purane Tareeke Se
Pehle seedha tareeka. 12 × 8 matlab 8 ko 12 baar, ya 12 ko 8 baar. Todo: 12 = 10 + 2. To 10 ko 8 baar = 80, aur 2 ko 8 baar = 16. Kul 80 + 16 = 96.
Yeh to aasaan tha! Par 13 × 7, 16 × 7, ya 103 × 97 mein yahi todna lamba ho jaata hai. Base method ek hi tareeke se sab karega.
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
12 × 8 = 96
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 7 ka base method, 10 se ghatana, aur udhaar (1 dahai = 10 ikai). Naya: minus wala kona. Sab neeche ke boxes mein, khaas kar laal “Minus wala right hissa” box. Naye shabd:
📖 Shabd jo is post mein aayenge
- Ek upar, ek neeche: ek number base se zyada (12), doosra kam (8).
- Left hissa (mix wala): upar wale number mein se neeche wale ka kitna kam ghatao: 12 − 2 = 10. Ya neeche wale mein upar wale ka kitna zyada jodo: 8 + 2 = 10. Dono same!
- Minus kona: kona = kitna zyada × kitna kam (2 × 2 = 4), par isse ghatana hai. Hum likhte hain −4 (“minus chaar”).
- Udhaar: left se 1 dahai (base 10 mein) le kar right mein 10 banana, taaki minus wala kona ghata sakein: 10 | −4 → 9 | 6.
- 10 se doori: kitna kam ya kitna zyada, dono ke liye ek shabd.
📏 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.
🔟 10 se guna = peeche ek 0 lagao
GUNA box mein dekha: 3 × 10 matlab 10 ko 3 baar jodo: 10 + 10 + 10 = 30. Isi tarah 46 × 10 matlab 46 ko 10 baar jodna. 46 mein 4 dahai aur 6 ikai hain. 10 baar lene pe 6 ikai ban jaati hain 6 dahai (60), aur 4 dahai ban jaati hain 4 saikde (400). Kul 400 + 60 = 460.
Aasaan niyam: kisi poore number ko 10 se guna karna ho to uske peeche (right mein) ek 0 likh do. Har digit ek jagah left khisak jaata hai, aur ikai ki khaali jagah mein 0 aa jaata hai.
| Number | × 10 | Kya hua |
|---|---|---|
| 7 | 70 | 7 ikai → 7 dahai |
| 46 | 460 | 4 dahai → 4 saikde, 6 ikai → 6 dahai |
| 128 | 1280 | har digit ek jagah left, peeche 0 |
| 365 | 3650 | har digit ek jagah left, peeche 0 |
🎯 Base kya hai? (Paas wala round number)
Round number woh hai jiske peeche (right mein) sirf zero hon: 10, 100, 1000. Aaj ki trick mein inhe base kehte hain. Base matlab “ghar” ya “nishaana”: woh round number jiske paas hamare dono numbers baithe hain.
- 8 aur 7: dono 10 ke paas hain (10 se thoda kam). Base = 10.
- 12 aur 13: dono 10 ke paas hain (10 se thoda zyada). Base = 10.
- 97 aur 96: dono 100 ke paas. Base = 100.
- 104 aur 107: dono 100 ke paas (thoda zyada). Base = 100.
- 997 aur 996: dono 1000 ke paas. Base = 1000.
Base kaise chunein? Number ko dekho aur socho: “Yeh 10, 100 ya 1000 mein se kiske sabse paas hai?” 8 to 10 se sirf 2 door hai, isliye 10. 97 to 100 se sirf 3 door, isliye 100. Dono numbers ka base ek hi hona chahiye.
| Base | Base mein kitne zero | Jawab ka left hissa kya ginta hai |
|---|---|---|
| 10 | 1 zero | kitne dahai (10 wale) |
| 100 | 2 zero | kitne saikde (100 wale) |
| 1000 | 3 zero | kitne hazaar (1000 wale) |
“Base mein kitne zero” wali baat yaad rakhna. Aage “right hissa kitne digit ka” isi se tay hoga.
📏 Base 10 se kitna kam, kitna zyada?
Har number ke liye ek chhota sawaal: “Base tak pahunchne mein kitna bacha?” ya “Base se kitna aage nikal gaya?”
- Kitna kam (number base se chhota ho) = base − number. Matlab number ko base banne ke liye kitna aur chahiye.
- Kitna zyada (number base se bada ho) = number − base. Matlab number base se kitna aage hai.
| Number | Base 10 se | Kaise nikala |
|---|---|---|
| 9 | 1 kam | 10 − 9 = 1 |
| 8 | 2 kam | 10 − 8 = 2 (8 se 9, 10: do kadam) |
| 7 | 3 kam | 10 − 7 = 3 |
| 6 | 4 kam | 10 − 6 = 4 |
| 12 | 2 zyada | 12 − 10 = 2 |
| 13 | 3 zyada | 13 − 10 = 3 |
| 15 | 5 zyada | 15 − 10 = 5 |
Chhota raasta: 10 se zyada wale (11 se 19) ka “kitna zyada” bas uska ikai digit hai: 13 → 3, 17 → 7. Kyunki 13 = 10 + 3.
🧱 Deewar ( | ), right hisse ke digit, aur carry
Is trick mein jawab do hisson mein banta hai: left hissa | right hissa. Beech ki khadi lakeer ( | ) ko “deewar” samjho. Yeh sirf hamare samjhne ke liye hai; asli jawab mein deewar nahi likhte, dono hisse saath-saath likh dete hain.
Base 10 mein 1 zero hai. Isliye right hisse ko jawab mein sirf 1 khaana (ikai) milta hai, aur left hissa dahai ginta hai.
- 5 | 6 matlab 5 dahai aur 6 ikai: 50 + 6 = 56.
- 15 | 6 matlab 15 dahai aur 6: 150 + 6 = 156.
Right hissa 10 ya usse bada aaye to? (Carry) Ikai ke khaane mein sirf ek digit baith sakta hai. Jaise 3 | 12: 12 mein ek poori dahai (10) aur 2 ikai hain. Woh ek dahai left wale dahai ke dher mein chali jaati hai: 3 + 1 = 4 dahai, aur ikai mein 2 bacha. Jawab 42. Place value se pakka: 3 dahai + 12 = 30 + 12 = 42 ✅. Yeh wahi “haath ka 1” (carry) hai jo jod mein seekha tha.
- 2 | 16 → 16 mein 1 dahai aur 6: 2 + 1 = 3 → 36 (20 + 16 = 36).
- 19 | 20 → 20 mein 2 dahai aur 0: 19 + 2 = 21 → 210 (190 + 20 = 210).
Pehle guna, phir jod (zaroori niyam): jab ek hi line mein × aur + dono hon, to pehle × (guna) karo, phir + (jod). Jaise 5 × 10 + 6: pehle 5 × 10 = 50, phir 50 + 6 = 56. (Agar pehle 10 + 6 = 16 karke 5 × 16 karte to 80 aata, jo galat hai.) Is post mein “left × base + right” isi niyam se padhna.
➖ Minus wala right hissa aur udhaar (base 10)
Is chapter mein ek number base se upar hoga aur doosra neeche. Tab right hissa (kitna zyada × kitna kam) jodna nahi, ghatana hota hai. Ise hum − (minus) nishaan se dikhate hain. “−4” ko bolo “minus 4”, matlab “4 nikaalna hai”. Yeh koi naya number nahi, bas ek yaad-dilaane wala nishaan hai: “isko ghatana mat bhoolna!”
Udhaar ki yaad (Post 3.1): ghatane mein jab kisi jagah se kuch nikaalna ho aur wahan kam ho, to left wali jagah se 1 bundle udhaar lete hain. Jaise 1 dahai = 10 ikai, 1 saikda = 100.
Example: 10 | −4 (yeh 12 × 8 mein aayega). Matlab: 10 dahai mein se 4 nikaalo.
- Seedha tareeka: 10 dahai = 100. 100 − 4 = 96.
- Udhaar tareeka: 10 dahai mein se 1 dahai udhaar lo. Left mein 9 dahai bache. Woh 1 dahai = 10 ikai. Ab 10 − 4 = 6. Jawab 9 | 6 = 96. Dono tareeke se same ✅.
Right bada ho to: 14 | −12 (yeh 16 × 8 mein aayega): 14 dahai = 140. 140 − 12 = 128 (Post 3.3 ki tarah: 140 − 10 = 130, phir 130 − 2 = 128).
🔁 Chhota sa warm-up
- 12 − 2 = ?
- 10 − 4 = ?
- 90 + 6 = ?
Answers dekho
- 10
- 6
- 96
Teeno aa gaye? Yahi milkar 12 × 8 = 96 banta hai!
— 📌 charumam.com
🎬 Kahani: Dukaan Ka Hisaab: 12 Packet, 8 Rupaye
Isha ki mummy ne 12 packet biscuit liye, har packet ₹8 ka. Dukaandaar ne kaha “₹96”. Isha ne socha: “Base method se check karti hoon!” 12 → 2 zyada, 8 → 2 kam. Left: 12 + 2? Ya 12 − 2? Right: 2 × 2 = 4… 104? 144? Kuch gadbad!
CharuMam ne kaha: “Jab ek upar, ek neeche ho, to dhyaan se. Left mein upar wale mein se neeche wale ka kam ghatao: 12 − 2 = 10. Aur kona 4 nikaalna hai: 10 dahai = 100, 100 − 4 = 96.”
Isha: “Dukaandaar sahi tha!” 😊
Aaj seekhenge yeh minus kona kahan se aata hai (chitra mein ek khaali jagah bachti hai!), aur udhaar se jaldi kaise likhte hain. ⚖️
— 📌 charumam.com
🪄 Trick Step by Step: Ek Upar Ek Neeche: Left Tirchha, Kona Minus
💡 Is post ki ek badi baat: Ek number base se upar, ek neeche: left = upar wala − neeche wale ka kitna kam (ya neeche wala + upar wale ka kitna zyada); kona = kitna zyada × kitna kam, aur ise ghatana hai. 12 × 8: 10 | −4 → 96.
Step 1: Base aur taraf dekho
Sawaal: 12 × 8. Base 10. 12 base se upar, 8 neeche. Chapter 7 se alag type: dhyaan!
Step 2: Dooriyan likho
12 → 2 zyada (+2). 8 → 2 kam (−2).
Step 3: Left: tirchha
Upar wale mein se neeche wale ka kam: 12 − 2 = 10. Doosri taraf: neeche wale mein upar wale ka zyada: 8 + 2 = 10. Same ✅. (Yaad: tirchha kadam hamesha doosre number ki doori se.)
Step 4: Kona: guna, phir minus
2 × 2 = 4. Ek zyada, ek kam hai, to kona ghatana hai: −4. Ab hamare paas: 10 | −4, matlab “10 dahai, usme se 4 nikaalo”.
Step 5: Udhaar se theek karo
Right mein minus nahi likh sakte. To left se 1 dahai udhaar lo: left 10 − 1 = 9. Woh 1 dahai = 10 ikai; usme se 4 nikaalo: 10 − 4 = 6. Jawab 9 | 6 = 96 ✅. (Seedha bhi: 10 dahai = 100, 100 − 4 = 96. Same!)
Aur jode (ek upar, ek neeche)
| Sawaal | Base se doori | Left (tirchha) | Kona (right) | Jawab |
|---|---|---|---|---|
| 12 × 9 | 2 zyada, 1 kam | 12 − 1 = 11 | 2 × 1 = 2, ghatao | 11 | −2 → 10 | 8 → 108 |
| 11 × 9 | 1 zyada, 1 kam | 11 − 1 = 10 | 1 × 1 = 1, ghatao | 10 | −1 → 9 | 9 → 99 |
| 13 × 9 | 3 zyada, 1 kam | 13 − 1 = 12 | 3 × 1 = 3, ghatao | 12 | −3 → 11 | 7 → 117 |
| 11 × 8 | 1 zyada, 2 kam | 11 − 2 = 9 | 1 × 2 = 2, ghatao | 9 | −2 → 8 | 8 → 88 |
| 13 × 8 | 3 zyada, 2 kam | 13 − 2 = 11 | 3 × 2 = 6, ghatao | 11 | −6 → 10 | 4 → 104 |
| 14 × 9 | 4 zyada, 1 kam | 14 − 1 = 13 | 4 × 1 = 4, ghatao | 13 | −4 → 12 | 6 → 126 |
| 12 × 7 | 2 zyada, 3 kam | 12 − 3 = 9 | 2 × 3 = 6, ghatao | 9 | −6 → 8 | 4 → 84 |
| 13 × 7 | 3 zyada, 3 kam | 13 − 3 = 10 | 3 × 3 = 9, ghatao | 10 | −9 → 9 | 1 → 91 |
| 14 × 8 | 4 zyada, 2 kam | 14 − 2 = 12 | 4 × 2 = 8, ghatao | 12 | −8 → 11 | 2 → 112 |
| 15 × 9 | 5 zyada, 1 kam | 15 − 1 = 14 | 5 × 1 = 5, ghatao | 14 | −5 → 13 | 5 → 135 |
Dekho: “Left | −kona” se seedha bhi nikal sakte ho: 13 × 7 → 10 | −9: 10 dahai = 100, 100 − 9 = 91. Udhaar wala tareeka (9 | 1) bas isi ka jaldi likhne ka roop hai.
Kona 10 se bada? Seedha ghatao
16 × 8: 6 zyada, 2 kam. Left: 16 − 2 = 14. Kona: 6 × 2 = 12, minus: −12. 14 | −12 matlab 14 dahai (140) mein se 12 nikaalo: 140 − 12 = 128. (140 − 10 = 130, phir 130 − 2 = 128.)
Udhaar se bhi: 12 nikaalne ke liye 2 dahai udhaar: left 14 − 2 = 12, right 20 − 12 = 8 → 12 | 8 = 128. Same! Jo aasaan lage.
Teeno type, ek nazar mein
| Type | Example | Left | Kona | Jawab |
|---|---|---|---|---|
| Dono kam | 8 × 7 | 8 − 3 = 5 | 6, jodo | 56 |
| Dono zyada | 12 × 13 | 12 + 3 = 15 | 6, jodo | 156 |
| Ek upar, ek neeche | 12 × 8 | 12 − 2 = 10 | 4, ghatao | 96 |
Ek line mein: dono doori ek taraf (dono kam ya dono zyada) → kona jodo. Alag taraf → kona ghatao.
🆘 Atak gaye? Aise socho
- Left mein kya? Upar wale number mein se neeche wale ka kam. Ya neeche wale mein upar wale ka zyada. Dono se same aaye to sahi.
- Kona jodun ya ghataun? Ek upar ek neeche → hamesha ghatao.
- Minus kona kaise likhun? Left ko dahai bana ke ghatao (10 dahai = 100, 100 − 4), ya 1 dahai udhaar (9 | 6).
- Check: 12 × 8 to 10 × 10 = 100 ke paas, thoda kam hona chahiye. 96 ✅
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Tiles se samjho. 12 × 8 matlab 12 column, har column mein 8 tile. (Neeche chitra.)
- Har column ko 10 ka banana hai, to 2 tile aur chahiye (2 hai 8 ka kitna kam).
- Right se 2 column kaato (naarangi). Left mein 12 − 2 = 10 neele column bache. Yahi left hissa!
- Naarangi patti (2 column × 8 tile) ko ghuma kar neele ke upar rakho. Ab woh 8 lambi aur 2 oonchi.
- Par neela hissa 10 chauda hai, aur patti sirf 8 lambi! To upar right mein 2 × 2 = 4 tile ki jagah khaali reh gayi (dotted). Woh 2 hai 12 ka kitna zyada, aur 2 hai 8 ka kitna kam.
- Agar khaali jagah bhi bhari hoti, to poora dabba 10 column × 10 = 100 hota. Par 4 jagah khaali hai, to asli tiles: 100 − 4 = 96.
Isliye kona minus! Chapter 7 mein kona baahar nikla hua extra tukda tha (jodna). Yahan woh ek khaali jagah hai (ghatana).
Ek aur: 14 × 8
14 → 4 zyada, 8 → 2 kam. 2 column kaato, upar rakho: 12 column × 10 = 120 jagah. Patti 8 lambi, neela 12 chauda → 4 × 2 = 8 khaali. 120 − 8 = 112. Base method: 14 − 2 = 12 | −8 → 11 | 2 → 112 ✅.
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Tareeke Se Check)
Example 1: 13 × 7 = ?
3 zyada, 3 kam. Left: 13 − 3 = 10 (ya 7 + 3 = 10). Kona: 3 × 3 = 9, minus. 10 | −9 → udhaar: 9 | 1 → 91.
✔️ Normal tareeke se check: 7 ko 13 baar = 7 ko 10 baar + 7 ko 3 baar = 70 + 21 = 91 ✅
Example 2: 14 × 9 = ?
4 zyada, 1 kam. Left: 14 − 1 = 13. Kona: 4 × 1 = 4, minus. 13 | −4 → 12 | 6 → 126.
✔️ Normal tareeke se check: 14 ko 10 baar − 14 = 140 − 14 = 126 ✅
Example 3: 11 × 8 = ?
1 zyada, 2 kam. Left: 11 − 2 = 9 (ya 8 + 1 = 9). Kona: 1 × 2 = 2, minus. 9 | −2 → 8 | 8 → 88.
✔️ Normal tareeke se check: Ch 4 ki × 11 trick: 8 × 11 = 88 ✅
Example 4: 16 × 7 = ? (bada kona)
6 zyada, 3 kam. Left: 16 − 3 = 13. Kona: 6 × 3 = 18, minus. 13 dahai = 130, 130 − 18 = 112. (Udhaar: 2 dahai → 11 | 2.)
✔️ Normal tareeke se check: 7 ko 16 baar = 70 + 42 = 112 ✅ (7 ko 6 baar = 42)
— 📌 charumam.com
🕉️ Vedic Maths Connection
Vedic Maths ki kitabein is type ko bhi Nikhilam mein rakhti hain, aur minus kona dikhane ke liye ek “vinculum” (number ke upar ek chhoti line) ka use karti hain: jaise 4 ke upar line matlab −4. Hum line ki jagah seedha “−” likhte hain aur udhaar se theek karte hain. Wajah wahi: chitra ki khaali jagah.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| Ek 10 se thoda upar, ek thoda neeche (12 × 8) | ✅ Haan | Left tirchha, kona ghatao. |
| Kona chhota (12 × 9, 13 × 8) | ✅ Bahut aasaan | Udhaar se seedha. |
| Kona bada (16 × 7, 17 × 8) | ⚠️ Chalega | Left dahai mein se kona seedha ghatao. |
| Dono ek taraf (8 × 7, 12 × 13) | ➡️ Chapter 7 | Kona jodo. |
| Base se door (19 × 2) | ❌ Nahi | Doori 9 aur 8; seedha 19 × 2 = 38 aasaan. |
| Alag base (12 × 98) | ❌ Nahi | Seedha todo. |
— 📌 charumam.com
❌→✅ Common Mistakes
Galti 1: Kona jod diya
Sawaal: 12 × 8
❌ Galat: 10 | 4 → 104.
✅ Sahi: ek upar ek neeche → kona ghatao: 10 | −4 → 96. Seedhe tod se check: 10 ko 8 baar (80) aur 2 ko 8 baar (16), kul 96. Dhyaan: yahan andaaza (80 aur 120 ke beech: 10 × 8 = 80 aur 12 × 10 = 120) 104 ko nahi pakad paata, isliye niyam yaad rakho: alag taraf → minus, aur tod se check karo.
Galti 2: Left mein upar wale ka zyada jod diya
Sawaal: 12 × 8
❌ Galat: 12 + 2 = 14 → 14 | −4 → 136.
✅ Sahi: tirchha: 12 mein se 8 ki doori (2 kam) → 12 − 2 = 10. Ya 8 mein 12 ki doori (2 zyada) jodo → 10.
Galti 3: Udhaar ke baad left kam karna bhool gaye
Sawaal: 13 × 7
❌ Galat: 10 | −9 → right 10 − 9 = 1, par left 10 hi rakha → 101.
✅ Sahi: udhaar diya to left 1 kam: 9 | 1 → 91.
Galti 4: Bada kona: galat udhaar
Sawaal: 16 × 7
❌ Galat: 13 | −18, sirf 1 dahai udhaar → 12 | −8 phir bhi minus!
✅ Sahi: seedha: 13 dahai = 130, 130 − 18 = 112. (Ya 2 dahai udhaar: 11 | 2.)
🧠 Rule of thumb: Alag taraf (ek upar, ek neeche) → left tirchha, kona MINUS, phir udhaar ya seedha ghatana.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 sawaal, sab ek upar ek neeche (base 10). Target: 4 minute.
- 11 × 9 = ?
- 12 × 9 = ?
- 13 × 9 = ?
- 14 × 9 = ?
- 12 × 8 = ?
- 13 × 8 = ?
- 14 × 8 = ?
- 12 × 7 = ?
- 13 × 7 = ?
- 11 × 7 = ?
- 15 × 9 = ?
- 16 × 9 = ?
- 15 × 8 = ?
- 17 × 9 = ?
- 16 × 7 = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- 99
- 108
- 117
- 126
- 96
- 104
- 112
- 84
- 91
- 77
- 135
- 144
- 120
- 153
- 112
Kaise kiya (kuch): 15 × 8: 15 − 2 = 13 | −10 → 130 − 10 = 120. 17 × 9: 17 − 1 = 16 | −7 → 15 | 3 → 153. 12 × 7: 12 − 3 = 9 | −6 → 8 | 4 → 84.
— 📌 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 × 9 = ?
- 11 × 8 = ?
- 13 × 9 = ?
- 14 × 7 = ?
- 12 × 6 = ?
- 18 × 9 = ?
- 13 × 6 = ?
- 15 × 7 = ?
- 11 × 6 = ?
- 19 × 9 = ?
| 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
- 108
- 88
- 117
- 98
- 72
- 162
- 78
- 105
- 66
- 171
Dhyaan: 12 × 6: 2 zyada, 4 kam → 12 − 4 = 8 | −8 → 80 − 8 = 72. 19 × 9: 18 | −9 → 17 | 1 → 171.
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. 12 × 8 mein left hissa?
Jawab aur kaaran dekho
Sahi jawab: (a) 10
- (a) sahi ✅: 12 − 2 = 10 (2 hai 8 ka kitna kam), ya 8 + 2 = 10 (2 hai 12 ka kitna zyada).
- (b) galat ❌: 14: upar wale mein uska apna zyada jod diya: 12 + 2 = 14.
- (c) galat ❌: 6: neeche wale mein se uska apna kam ghata diya: 8 − 2 = 6.
- (d) galat ❌: 4: yeh to kona hai: 2 × 2 = 4.
Q2. Ek upar, ek neeche ho to kona?
Jawab aur kaaran dekho
Sahi jawab: (d) Ghatate hain (minus)
- (a) galat ❌: Jodne se 12 × 8 mein 10 | 4 → 104 aata, par 12 × 8 = 96.
- (b) galat ❌: Chhod dene se 12 × 8 mein 10 | 0 → 100 aata, par 12 × 8 = 96.
- (c) galat ❌: Kona left mein jodne se 10 + 4 = 14 aata; kona right mein hi ghatate hain.
- (d) sahi ✅: Chitra mein kona ek khaali jagah hai, isliye ghatate hain: 12 × 8 → 10 | −4 → 96.
Q3. 10 | −4 = ?
Jawab aur kaaran dekho
Sahi jawab: (c) 96
- (a) galat ❌: 60: 4 ki jagah 4 dahai (40) nikaal di: 100 − 40 = 60.
- (b) galat ❌: 94: udhaar liya par right mein 10 − 4 ki jagah 4 hi likh diya: 9 | 4 → 94.
- (c) sahi ✅: 10 dahai = 100, 100 − 4 = 96 (ya udhaar: 9 | 6 → 96).
- (d) galat ❌: 104: ghatane ki jagah jod diya: 10 | 4 → 104.
Q4. 13 × 7 = ?
Jawab aur kaaran dekho
Sahi jawab: (b) 91
- (a) galat ❌: 99: udhaar se left 9 kiya, par right mein 10 − 9 ki jagah 9 hi likh diya: 9 | 9 → 99.
- (b) sahi ✅: 3 zyada, 3 kam: 13 − 3 = 10 | −9 → udhaar: 9 | 1 → 91.
- (c) galat ❌: 101: right mein 10 − 9 = 1 kiya par left 1 kam karna bhool gaye: 10 | 1 → 101.
- (d) galat ❌: 109: kona ghatane ki jagah jod diya: 10 | 9 → 109.
Q5. 14 × 9 = ?
Jawab aur kaaran dekho
Sahi jawab: (a) 126
- (a) sahi ✅: 4 zyada, 1 kam: 14 − 1 = 13 | −4 → udhaar: 12 | 6 → 126.
- (b) galat ❌: 136: right mein 10 − 4 = 6 kiya par left 1 kam karna bhool gaye: 13 | 6 → 136.
- (c) galat ❌: 134: kona ghatane ki jagah jod diya: 13 | 4 → 134.
- (d) galat ❌: 76: 9 mein se uska apna hi kitna kam ghata diya: 9 − 1 = 8 | −4 → udhaar: 7 | 6 → 76.
Q6. Chitra mein 12 × 8 ke upar right mein khaali jagah kitni?
Jawab aur kaaran dekho
Sahi jawab: (d) 2 × 2 = 4
- (a) galat ❌: 2 × 8 = 16 to naarangi patti ki tiles hain (2 column, har ek mein 8).
- (b) galat ❌: 10 × 10 = 100 poora dabba hai (khaali jagah ko milakar).
- (c) galat ❌: 12 × 2 = 24 chitra mein kahin nahi dikhta; khaali jagah sirf 2 chaudi aur 2 oonchi hai.
- (d) sahi ✅: Patti 8 lambi, neela hissa 10 chauda: 2 chaudi aur 2 oonchi jagah khaali.
Q7. 16 × 7 = ?
Jawab aur kaaran dekho
Sahi jawab: (c) 112
- (a) galat ❌: 122: 18 mein se sirf 8 ghataya: 130 − 8 = 122.
- (b) galat ❌: 22: 7 mein se uska apna hi kitna kam ghata diya: 7 − 3 = 4, phir 40 − 18 = 22.
- (c) sahi ✅: 6 zyada, 3 kam: 16 − 3 = 13 | −18 → 13 dahai = 130, 130 − 18 = 112.
- (d) galat ❌: 148: kona ghatane ki jagah jod diya: 130 + 18 = 148.
Q8. Kis sawaal mein aaj wali (ek upar ek neeche) trick?
Jawab aur kaaran dekho
Sahi jawab: (b) 13 × 8
- (a) galat ❌: 12 base 10 ke paas, 98 base 100 ke paas: alag base, seedha karo.
- (b) sahi ✅: 13 upar, 8 neeche: 13 − 2 = 11 | −6 → udhaar: 10 | 4 → 104.
- (c) galat ❌: 8 aur 7 dono 10 se neeche: yeh Post 7.1 ka type hai (kona jodo).
- (d) galat ❌: 12 aur 13 dono 10 se upar: yeh bhi Post 7.1 ka type hai.
Q9. 11 × 8 = ?
Jawab aur kaaran dekho
Sahi jawab: (a) 88
- (a) sahi ✅: 1 zyada, 2 kam: 11 − 2 = 9 | −2 → udhaar: 8 | 8 → 88.
- (b) galat ❌: 98: right mein 10 − 2 = 8 kiya par left 1 kam karna bhool gaye: 9 | 8 → 98.
- (c) galat ❌: 92: kona ghatane ki jagah jod diya: 9 | 2 → 92.
- (d) galat ❌: 82: udhaar se left 8 kiya par right mein 10 − 2 ki jagah 2 hi likha: 8 | 2 → 82.
Q10. Udhaar lene ke baad left ka kya hota hai?
Jawab aur kaaran dekho
Sahi jawab: (d) 1 kam ho jaata hai
- (a) galat ❌: Left ne dahai di hai, li nahi. Badhaane se 13 × 7 mein 11 | 1 → 111 aata, galat.
- (b) galat ❌: Waise rakhne se 13 × 7 mein 10 | 1 → 101 aata, jo 91 se 10 zyada hai.
- (c) galat ❌: Left 0 karne se 13 × 7 mein sirf 1 bachta; bas 1 kam hota hai.
- (d) sahi ✅: 1 dahai right mein chali gayi, to left 1 kam: 10 → 9 (13 × 7 → 9 | 1 → 91).
— 📌 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. 12 × 8, 13 × 7, 14 × 6, 15 × 5: pattern?
Solution dekho
Sab mein left = 10 (12 − 2, 13 − 3, 14 − 4, 15 − 5). Kona: 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25. To jawab: 100 − 4 = 96, 100 − 9 = 91, 100 − 16 = 84, 100 − 25 = 75.
Check: 14 × 6 = 60 + 24 = 84 ✅, 15 × 5 = 50 + 25 = 75 ✅. Seekh: dono numbers 10 se jitne door (ek upar, ek neeche), guna utna chhota!
Q2. □ × 9 = 117, □ 10 se zyada. □ = ?
Solution dekho
Ulta chalo. 117 ko deewar se todo: 11 | 7. Yeh udhaar ke baad ka roop hai (□ 10 se upar aur 9 neeche, to kona minus tha, isliye udhaar hua hoga): left ne 1 dahai di thi (to pehle left 12 tha), aur right 7 aaya tha 10 mein se kona ghata ke. 10 mein se kya ghatayein ki 7 bache? 3. To kona = 3. 9 → 1 kam, aur kona = (□ ka kitna zyada) × 1, to □ ka kitna zyada 3 → □ = 13. Left check: 13 − 1 = 12 ✅.
Check: 13 × 9 = 130 − 13 = 117 ✅
Q3. 17 packet, har packet ₹8. Kul?
Solution dekho
17 × 8: 7 zyada, 2 kam. Left: 17 − 2 = 15. Kona: 7 × 2 = 14, minus. 150 − 14 = ₹136.
Check: 8 ko 17 baar = 80 + 56 = 136 ✅
Q4. Kya 19 × 2 pe base 10 trick lagani chahiye?
Solution dekho
19 → 9 zyada, 2 → 8 kam. Left: 19 − 8 = 11. Kona: 9 × 8 = 72, minus: 110 − 72 = 38. Sahi to aaya, par 9 × 8 ka guna khud mushkil! Seedha: 19 × 2 = 38 (19 + 19). Seekh: trick tab jab dono base ke paas hon.
Check: 19 + 19 = 38 ✅
— 📌 charumam.com
🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | Dukaan/cost ke sawaal (12 packet × ₹8) aur guna check. |
| Maths Olympiad | Pattern (12 × 8, 13 × 7, 14 × 6) aur khaali jagah wala chitra. |
| NTSE / MAT | Mental maths: ek upar ek neeche pehchanna. |
| SSC / Bank (aage) | Simplification mein 13 × 7, 16 × 9 jaise guna. |
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: “ek taraf to jodo, alag taraf to ghatao.” Roz 5 sawaal 11–16 × 6–9 ke. Agle post mein yahi 100 ke paas: 103 × 97! ⚖️
— 📌 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)
Base 10. 12 → 2 zyada, 8 → 2 kam. Left: 12 − 2 = 10. Kona: 2 × 2 = 4, par minus: −4. 10 | −4 → udhaar → 9 | 6 = 96 ✅
| 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:
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Kona minus kyun hota hai?
Chitra mein: kaati patti upar rakhne pe ek kona khaali reh jaata hai (2 × 2). Poore dabbe mein se woh khaali jagah ghatani padti hai.
2. Udhaar zaroori hai?
Nahi. 10 | −4 ko seedha 100 − 4 = 96 bhi likh sakte ho. Udhaar bas jaldi likhne ka tareeka.
3. Left dono taraf se same kyun?
12 − 2 aur 8 + 2 dono “10 + 2 − 2” hi hain.
4. Agle post mein kya hai?
Post 8.2: base 100: 103 × 97 = 9991, jahan udhaar 1 saikde ka hoga.
— 📌 charumam.com
🟢 Recap: Aaj Kya Seekha
- Ek upar, ek neeche → alag taraf.
- Left: upar wala − neeche ka kam (12 − 2 = 10).
- Kona: doori × doori, MINUS (−4).
- Udhaar: 10 | −4 → 9 | 6 = 96.
- Bada kona: left dahai − kona (140 − 12 = 128).
- Chitra: khaali jagah = minus kona.
— charumam.com 🎯
👉 Aage Kya?
Ab base 100 pe yahi: Post 8.2 mein 103 × 97, 104 × 98 jaise guna, jahan udhaar 1 saikde ka hoga aur Chapter 3 ki trick phir kaam aayegi!
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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