Post 6.1 mein 35² = 1225 seekha. Kya yahi trick 3-digit numbers (105, 115, 125 …) pe bhi chalti hai? Haan! Bas aakhri 5 hatane pe ab do digit bachte hain (jaise 11), aur humein 11 × 12 jaisa guna chahiye. Pehle woh guna seekhenge, phir trick, phir chitra se pakka karenge ki yeh har baar kyun chalti hai.
⏱ Before Timer: Pehle Purane Tareeke Se
Pehle seedha tareeka. 115 × 115 matlab 115 ko 115 baar jodna. 115 = 100 + 10 + 5, to 115 ko 100 baar, 10 baar aur 5 baar alag-alag, phir jodo:
- 115 ko 100 baar = 11500 (peeche 00).
- 115 ko 10 baar = 1150 (peeche 0).
- 115 ko 5 baar = 1150 ka aadha (Post 5.1). 1150 = 1000 + 150 → 500 + 75 = 575.
- Teeno jodo: 11500 + 1150 = 12650, phir 12650 + 575 = 13225.
Bade numbers, lambe jod. Trick: 11 × 12 aur 25!
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
115 × 115 = 13225
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
Aaj ke liye: square (²) ka matlab (Post 6.1), 100 se guna (peeche 00), aur naya: 10 se 19 tak ke number × unka agla number (10 × 11, 11 × 12 …). Sab neeche ke boxes mein. Naye shabd:
📖 Shabd jo is post mein aayenge
- 3-digit number 5 pe khatam: 105, 115, 125, … 195 (aur aage 205, 215 …).
- Aakhri 5 hatao (3-digit): 115 se 5 hatao to 11 bachta hai (poora 11, sirf 1 nahi). 125 → 12. 105 → 10.
- Agla number: 11 ka agla 12, 19 ka agla 20.
- Number × agla (10 se 19): 12 × 13 jaisa guna: 12 ko 10 baar (120) aur 12 ko 3 baar (36), jodo: 156.
- Das hazaar (5-digit): 13225 jaise 5 digit ke number. Right se 5vi jagah das hazaar (10000). 13225 = 132 saikde + 25.
- Aayat (rectangle): chaar kone wala aakaar jiski aamne-saamne ki bhujaayein barabar hon; ismein tile = lambai × chaudai.
📏 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 |
💯 100 se guna = peeche do 0 lagao
100 = 10 × 10 (10 ko 10 baar jodo: 10, 20, 30, … 100). Isliye 100 se guna karna matlab do baar 10 se guna. Har baar 10 se guna pe peeche ek 0 lagta hai (upar wala box), to 100 se guna pe do 0:
36 → (× 10) 360 → (× 10) 3600
Place value se: 36 × 100 matlab 36 saikde (36 sau). 36 sau = 3600 (padho: teen hazaar chhe sau).
| Number | × 100 |
|---|---|
| 7 | 700 |
| 36 | 3600 |
| 37 | 3700 |
| 124 | 12400 (padho: barah hazaar chaar sau) |
Bade number padhna: 12400 mein right se paanchvi jagah Das hazaar (10000 wale) hai. 12400 = 1 das hazaar + 2 hazaar + 4 saikde, yaani 10000 + 2000 + 400.
🟦 Square kya hai? Chhota ² ka matlab
Kisi number ko usi number se guna karna us number ka square (Hindi mein “varg”) kehlata hai. Ise number ke upar-right mein chhota ² likh kar dikhate hain. 5² padho “5 ka square” ya “5 square”, matlab 5 × 5 = 25. Dhyaan: 5² ka matlab 5 × 2 nahi hai (woh 10 hota)!
Naam “square” kyun? Square ka ek matlab hai chaukor (jiski chaaron bhujaayein barabar hon). 5 line banao, har line mein 5 dots: dots ka ek chaukor ban jaata hai, aur kul dots 5 × 5 = 25. To 5² = 25 dots ka chaukor.
| Square | Matlab | Jawab |
|---|---|---|
| 3² | 3 × 3 | 9 |
| 5² | 5 × 5 | 25 |
| 10² | 10 × 10 | 100 |
| 35² | 35 × 35 | 1225 (Post 6.1) |
➕1 Bade number × agla number (10 se 19 tak)
3-digit square ke liye 10 × 11, 11 × 12, 12 × 13 … chahiye. Inhe tukdon mein todo (GUNA box wali soch): 13 = 10 + 3, to 12 × 13 matlab: 12 ko 10 baar, aur 12 ko 3 baar. 12 ko 10 baar = 120 (peeche 0). 12 ko 3 baar = 12 + 12 + 12 = 36. Kul 120 + 36 = 156.
| Number × agla | Tukdon mein | Jawab |
|---|---|---|
| 10 × 11 | 11 ke peeche 0 (10 se guna) | 110 |
| 11 × 12 | 110 + 22 (11 ko 10 baar, 11 ko 2 baar) | 132 |
| 12 × 13 | 120 + 36 | 156 |
| 13 × 14 | 130 + 52 (13 ko 4 baar: 13, 26, 39, 52) | 182 |
| 14 × 15 | 140 + 70 (14 ko 5 baar = 140 ka aadha) | 210 |
| 15 × 16 | 150 + 90 (15 ko 6 baar: 15, 30, 45, 60, 75, 90) | 240 |
| 16 × 17 | 160 + 112 (16 ko 7 baar) | 272 |
| 17 × 18 | 170 + 136 (17 ko 8 baar) | 306 |
| 18 × 19 | 180 + 162 (18 ko 9 baar) | 342 |
| 19 × 20 | 19 ko 2 baar = 38, phir peeche 0 | 380 |
Ch 4 se ek chhota raasta: 11 × 12 aur 12 × 11 same hain (jagah badlo, jawab wahi). 12 × 11 ki trick: 1 | 1 + 2 | 2 = 132. ✅
🔁 Chhota sa warm-up
- 11 × 12 = ?
- 132 × 100 = ?
- 13200 + 25 = ?
Answers dekho
- 132 (110 + 22)
- 13200
- 13225
Teeno aa gaye? 115² = 13225 tayyar!
— 📌 charumam.com
🎬 Kahani: Sara Ka Bada Chaukor Hall
School ka naya hall chaukor tha: 115 tile lamba, 115 tile chauda. Principal ne poocha: “Kul kitni tile lagengi?”
Sara ne Post 6.1 yaad kiya: “Aakhri 5 hatao… 1 bacha? 1 × 2 = 2, to 225?” Phir socha: “225 tile? Itne bade hall mein? Nahi!” 🤔
CharuMam: “Shabaash, andaaza laga kar galti pakdi! 115 se 5 hatao to 11 bachta hai, sirf 1 nahi. 11 × 12 = 132. Peeche 25: 13225 tile.”
Sara: “Aur yeh 11 × 12 jaldi kaise?” CharuMam: “Wahi aaj seekhenge!” 📐
— 📌 charumam.com
🪄 Trick Step by Step: Aakhri 5 Hatao (Jo Bacha Poora Lo), Agle Se Guna, Peeche 25
💡 Is post ki ek badi baat: 3-digit number jo 5 pe khatam ho: aakhri 5 hatao, jo poora number bacha (jaise 11) use uske agle number (12) se guna karo, aur peeche 25 likho. 115²: 11 × 12 = 132 → 13225.
Step 1: Aakhri 5 hatao, jo bacha poora lo
Sawaal: 115². 5 hatao: 11 bacha (gyaarah). Dhyaan: 1 nahi, 11!
Step 2: Agla number
11 ka agla 12 (11 + 1 = 12).
Step 3: Guna karo (tukdon mein)
11 × 12: 11 ko 10 baar = 110, 11 ko 2 baar = 22. Jodo: 110 + 22 = 132. (Ya Ch 4: 12 × 11 → 1 | 1 + 2 | 2 = 132.)
Step 4: Peeche 25 likho
132 ke peeche 25: 13225. Yaani 132 saikde + 25 = 13200 + 25 = 13225. To 115² = 13225.
105² se 195² tak
| Square | 5 hatao | × agla | Peeche 25 |
|---|---|---|---|
| 105² | 10 | 10 × 11 = 110 | 11025 |
| 115² | 11 | 11 × 12 = 132 | 13225 |
| 125² | 12 | 12 × 13 = 156 | 15625 |
| 135² | 13 | 13 × 14 = 182 | 18225 |
| 145² | 14 | 14 × 15 = 210 | 21025 |
| 155² | 15 | 15 × 16 = 240 | 24025 |
| 165² | 16 | 16 × 17 = 272 | 27225 |
| 175² | 17 | 17 × 18 = 306 | 30625 |
| 185² | 18 | 18 × 19 = 342 | 34225 |
| 195² | 19 | 19 × 20 = 380 | 38025 |
Dhyaan (0 wale): 105²: 110 ke peeche 25 = 11025 (110 ka 0 rehta hai; 1125 galat). 145²: 210 → 21025. 195²: 380 → 38025.
Bada number padhna: 13225 = 1 das hazaar + 3 hazaar + 2 saikde + 2 dahai + 5 ikai = 10000 + 3000 + 200 + 20 + 5. Padho: “terah hazaar do sau pachchees”.
🆘 Atak gaye? Aise socho
- Kitna bacha? 115 se aakhri 5 hatao: 11 (do digit), 1 nahi. Andaaza: 115² to 100 × 100 = 10000 se bada hai, 225 nahi ho sakta.
- 11 × 12 jaisa guna? Tukde: 11 ko 10 baar + 11 ko 2 baar. Ya upar ki table.
- 0 wale jawab (110, 210, 380)? 0 mat chhodo: 110 | 25 = 11025.
- Check: 100 baar + 10 baar + 5 baar wala seedha tareeka, ya chitra wali soch: 110 × 120 + 25.
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Post 6.1 ka chitra yaad karo: chaukor ko 4 hisson mein kaata, ek patti hilayi, aur ek aayat + chhota kona bana. Yahi har 5 pe khatam number ke saath hota hai, number chhota ho ya bada.
115 × 115 ke liye: chaukor ko 110 aur 5 mein kaato. Bada chaukor 110 × 110, do patli pattiyaan (110 × 5), aur kona 5 × 5. Ek patti utha kar doosri ke neeche lagao: ab aayat ki chaudai 110, lambai 110 + 5 + 5 = 120. Kona alag: 25.
Aayat ki tile: 110 × 120. 110 ko 12 baar = 1320 (11 × 12 = 132, peeche 0). Phir 120 = 12 ka 10 guna, to 1320 ka 10 guna = 13200. Chhota raasta: 11 × 12 = 132, aur dono (110 aur 120) ke 0 milakar peeche 00.
Kul: 13200 + 25 = 13225. Dekha? “Agla number” (12) isliye aata hai kyunki patti hilane se lambai 10 badh jaati hai (110 se 120). Aur 25 hamesha woh chhota 5 × 5 kona hai. Isliye trick pakki hai: kabhi fail nahi hoti (jab tak number 5 pe khatam hai aur square hai).
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Tareeke Se Check)
Example 1: 105² = ?
5 hatao: 10. 10 × 11 = 110. Peeche 25: 11025.
✔️ Normal tareeke se check: Seedha: 105 ko 100 baar = 10500. 105 ko 5 baar = 1050 ka aadha = 525. Kul 10500 + 525 = 11025 ✅
Example 2: 125² = ?
12 × 13 = 156 (120 + 36). Peeche 25: 15625.
✔️ Normal tareeke se check: Seedha: 125 ko 100 baar = 12500. 125 ko 20 baar = 1250 + 1250 = 2500. 125 ko 5 baar = 1250 ka aadha = 625. Kul 12500 + 2500 = 15000, 15000 + 625 = 15625 ✅
Example 3: 135² = ?
13 × 14 = 182 (130 + 52). Peeche 25: 18225.
✔️ Normal tareeke se check: Seedha: 135 ko 100 baar = 13500. 135 ko 30 baar = 1350 ko 3 baar: 1350, 2700, 4050. 135 ko 5 baar = 1350 ka aadha = 675. Kul 13500 + 4050 = 17550, 17550 + 675 = 18225 ✅
Example 4: 145² = ?
14 × 15 = 210 (140 + 70). Peeche 25: 21025.
✔️ Normal tareeke se check: Seedha: 145 ko 100 baar = 14500. 145 ko 40 baar = 1450 ko 4 baar: 1450, 2900, 4350, 5800. 145 ko 5 baar = 725. Kul 14500 + 5800 = 20300, 20300 + 725 = 21025 ✅
— 📌 charumam.com
🕉️ Vedic Maths Connection
Yeh wahi “Ekadhikena Purvena” (pichle se ek zyada) wali soch hai jo Post 6.1 mein dekhi: aakhri 5 se pehle jo number hai (11), use uske “ek zyada” (12) se guna karo. 3-digit pe bhi bilkul wahi.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| 105² se 195² | ✅ Haan | Bas 10 × 11 se 19 × 20 tak ke guna chahiye (table upar). |
| 205², 215², … (aur bade) | ✅ Chalegi | 20 × 21, 21 × 22 jaise guna thode bade; tukdon mein karo. |
| 2-digit (35², 85²) | ✅ Post 6.1 | Wahi trick. |
| Number 5 pe khatam nahi (116²) | ❌ Nahi | Trick sirf 5 pe khatam numbers ke square ke liye. |
| 115 × 125 (do alag number) | ❌ Nahi | Square nahi; is trick se nahi. |
| Teacher ne steps maange | ❌ Answer sheet pe nahi | Sheet pe seedha tareeka, rough mein trick. |
— 📌 charumam.com
❌→✅ Common Mistakes
Galti 1: Sirf pehla digit liya
Sawaal: 115²
❌ Galat: 5 hatao → “1”, 1 × 2 = 2 → 225.
✅ Sahi: 5 hatao → 11, 11 × 12 = 132 → 13225.
Galti 2: Agle ki jagah wahi number
Sawaal: 115²
❌ Galat: 11 × 11 = 121 → 12125.
✅ Sahi: 11 × 12 = 132 → 13225.
Galti 3: 0 chhoot gaya
Sawaal: 105²
❌ Galat: 10 × 11 = 110, phir “1125”.
✅ Sahi: 110 ke peeche 25 = 11025.
Galti 4: 12 × 13 galat
Sawaal: 125²
❌ Galat: 12 × 13 ko 136 maan liya → 13625.
✅ Sahi: 12 ko 10 baar = 120, 12 ko 3 baar = 36, jodo 156 → 15625.
🧠 Rule of thumb: 5 hatao (jo poora bacha, jaise 11), × agla, peeche 25. 0 wale jawab ka 0 mat chhodo.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 sawaal: pehle 5 “× agla” wale guna, phir 10 squares. Target: 5 minute.
- 10 × 11 = ?
- 11 × 12 = ?
- 12 × 13 = ?
- 13 × 14 = ?
- 14 × 15 = ?
- 105² = ?
- 115² = ?
- 125² = ?
- 135² = ?
- 145² = ?
- 155² = ?
- 165² = ?
- 175² = ?
- 185² = ?
- 195² = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- 110
- 132
- 156
- 182
- 210
- 11025
- 13225
- 15625
- 18225
- 21025
- 24025
- 27225
- 30625
- 34225
- 38025
Kaise kiya (kuch): 165²: 16 × 17 = 160 + 112 = 272 → 27225. 175²: 17 × 18 = 170 + 136 = 306 → 30625. 185²: 18 × 19 = 180 + 162 = 342 → 34225.
— 📌 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.
- 11 × 12 = ?
- 12 × 13 = ?
- 15 × 16 = ?
- 17 × 18 = ?
- 19 × 20 = ?
- 105² = ?
- 125² = ?
- 145² = ?
- 165² = ?
- 185² = ?
| 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
- 156
- 240
- 306
- 380
- 11025
- 15625
- 21025
- 27225
- 34225
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. 115² = ?
Jawab aur kaaran dekho
Sahi jawab: (b) 13225
- (a) galat ❌: 1325 mein 132 ka 2 aur 25 ka 2 mila diye; poore 132 ke peeche 25 likhna tha.
- (b) sahi ✅: 11 × 12 = 132, peeche 25: 13225.
- (c) galat ❌: 225 tab aata jab 115 se sirf “1” liya; 11 lena tha.
- (d) galat ❌: 12125 mein 11 × 11 kiya; agla number 12 lena tha.
Q2. 125² mein aakhri 5 hatane ke baad kya karte hain?
Jawab aur kaaran dekho
Sahi jawab: (a) 12 × 13
- (a) sahi ✅: 12 ko uske agle 13 se guna: 156 → 15625.
- (b) galat ❌: 12 × 12 mein agla number nahi liya.
- (c) galat ❌: 125 se 5 hatao to 12 bachta hai, sirf 1 nahi.
- (d) galat ❌: 5 to hata diya; × 5 nahi karna.
Q3. 12 × 13 = ?
Jawab aur kaaran dekho
Sahi jawab: (d) 156
- (a) galat ❌: 136 mein 12 × 3 ko 16 maan liya; 12 + 12 + 12 = 36.
- (b) galat ❌: 144 = 12 × 12, agla number nahi liya.
- (c) galat ❌: 12 × 13 karna bhool gaye, bas 12 ke baad 5 likh diya = 125.
- (d) sahi ✅: 12 ko 10 baar = 120, 12 ko 3 baar = 36; 120 + 36 = 156.
Q4. 105² = ?
Jawab aur kaaran dekho
Sahi jawab: (c) 11025
- (a) galat ❌: 10025 mein 10 × 10 = 100 kiya; agla 11 lena tha.
- (b) galat ❌: 11125 mein 110 ki jagah 111 likh diya.
- (c) sahi ✅: 10 × 11 = 110, peeche 25: 11025.
- (d) galat ❌: 1125 mein 110 ka 0 chhoot gaya.
Q5. Chitra mein 115 × 115 ka chaukor kis ke barabar banta hai?
Jawab aur kaaran dekho
Sahi jawab: (b) 110 × 120 ka aayat + 5 × 5 ka kona
- (a) galat ❌: 120 × 120 = 14400, bahut bada.
- (b) sahi ✅: Patti hilane se lambai 120 ho jaati hai: 13200 + 25 = 13225.
- (c) galat ❌: Dono pattiyaan bhi chahiye; sirf 5 jodna galat.
- (d) galat ❌: 115 × 2 = 230, bahut chhota.
Q6. 145² = ?
Jawab aur kaaran dekho
Sahi jawab: (a) 21025
- (a) sahi ✅: 14 × 15 = 210, peeche 25: 21025.
- (b) galat ❌: 2125 mein 210 ka 0 chhoot gaya.
- (c) galat ❌: 19625 mein 14 × 14 = 196 kiya; agla 15 lena tha.
- (d) galat ❌: 21125 mein 210 ki jagah 211 likha.
Q7. Kis number ka square 15625 hai?
Jawab aur kaaran dekho
Sahi jawab: (d) 125
- (a) galat ❌: 135² = 18225 (13 × 14 = 182).
- (b) galat ❌: 115² = 13225 (11 × 12 = 132).
- (c) galat ❌: 156 to 12 × 13 ka jawab hai; number nahi.
- (d) sahi ✅: 25 hatao: 156 = 12 × 13. To number 12 ke baad 5: 125.
Q8. 11 × 12 ko Ch 4 ki × 11 trick se kaise karein?
Jawab aur kaaran dekho
Sahi jawab: (c) 12 × 11: 1 | 1 + 2 | 2 = 132
- (a) galat ❌: Beech mein digits ka jod (3) likhna tha, 1 nahi.
- (b) galat ❌: 11 × 11 alag sawaal hai; humein 11 × 12 chahiye.
- (c) sahi ✅: Jagah badlo: 12 × 11. Left 1, beech 1 + 2 = 3, right 2: 132.
- (d) galat ❌: Yeh jod hai, guna nahi.
Q9. 195² = ?
Jawab aur kaaran dekho
Sahi jawab: (b) 38025
- (a) galat ❌: 39025 mein 19 × 20 ko 390 maan liya; 19 ko 2 baar = 38, peeche 0 = 380.
- (b) sahi ✅: 19 × 20 = 380, peeche 25: 38025.
- (c) galat ❌: 3825 mein 380 ka 0 chhoot gaya.
- (d) galat ❌: 36125 mein 19 × 19 = 361 kiya; agla 20 lena tha.
Q10. 115² ka andaaza: jawab kitne ke aas-paas?
Jawab aur kaaran dekho
Sahi jawab: (a) 10000 se thoda zyada (100 × 100 se bada)
- (a) sahi ✅: 115, 100 se bada hai, to 115² bhi 10000 se bada. Sahi jawab 13225.
- (b) galat ❌: 225 to 15² hai; bahut chhota.
- (c) galat ❌: 1000 bhi bahut chhota; 100 × 100 hi 10000 hai.
- (d) galat ❌: 1 lakh bahut bada hai, 115² to 13225 hai.
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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. 205² = ?
Solution dekho
5 hatao: 20. Agla 21. 20 × 21 = 420 (21 ko 2 baar = 42, peeche 0). Peeche 25: 42025.
Check: 205 ko 200 baar = 41000 (205 ko 2 baar = 410, peeche 00). 205 ko 5 baar = 2050 ka aadha = 1025. 41000 + 1025 = 42025 ✅
Q2. Kis number ka square 24025 hai?
Solution dekho
25 hatao: 240. 240 = kaun × agla? 15 × 16 = 150 + 90 = 240 ✅. To 15 ke baad 5: 155.
Check: 155²: 15 × 16 = 240 → 24025 ✅
Q3. 1005² = ? (Bonus)
Solution dekho
5 hatao: 100. Agla 101. 100 × 101 = 10100 (101 ke peeche 00). Peeche 25: 1010025 (das lakh das hazaar pachchees).
Check: chitra wali soch: 1000 × 1010 + 25. 1000 × 1010 matlab 1010 ke peeche 000: 1010000. Phir 1010000 + 25 = 1010025 ✅
Q4. 115² aur 105² mein kitna farq?
Solution dekho
13225 − 11025 = 2200. Seedha ghatao: 13225 − 11000 = 2225, phir 2225 − 25 = 2200.
Check: 11025 + 2200 = 13225 ✅
— 📌 charumam.com
🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | Squares wale chapter mein 105², 125² jaise sawaal; bade chaukor ka area (bhuja × bhuja). |
| Maths Olympiad | Ulte sawaal (kis number ka square 15625?) aur pattern. |
| NTSE / MAT | Mental calculation mein bade squares. |
| SSC / Bank (aage) | Simplification mein 125², 175² jaise squares. |
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. Mental calculation = bina copy-pencil ke, mann mein hisaab karna. Class 6–8 = school ki chhathi se aathvi kaksha.
— 📌 charumam.com
👩🏫 CharuMam Ka Tip
Har bade jawab se pehle ek andaaza lagao: “115² to 10000 se bada hoga.” Isse Sara jaisi galti (225) turant pakad mein aa jaati hai. Andaaza = suraksha belt! 📐
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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)
Aakhri 5 hatao: 11. Agla 12. 11 × 12 = 132 (12 × 11: 1 | 1 + 2 | 2, Ch 4 trick). Peeche 25: 13225 ✅
| 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:
— 📌 charumam.com
❓ FAQs (Bachchon Ke Asli Sawaal)
1. 10 × 11 se 19 × 20 tak ratna padega?
Ratna zaroori nahi. Tukde karo: 16 × 17 = 160 + 112 = 272. Practice se khud yaad ho jaayega.
2. Kya 4-digit (1005²) bhi chalega?
Haan! 100 × 101 = 10100 → 1010025. Challenge mein dekho.
3. Chitra naap ke hisaab se kyun nahi?
5 ki patti 110 ke saamne bahut patli hogi, dikhegi nahi. Isliye chitra sirf samjhane ke liye bada-chhota banaya hai; tile ki ginti (110 × 120 + 25) wahi hai.
4. Agle post mein kya hai?
Post 6.3: bahan-trick! 43 × 47 jaise guna, jab dahai same ho aur ikaiyon ka jod 10.
— 📌 charumam.com
🟢 Recap: Aaj Kya Seekha
- 3-digit 5 pe khatam: 5 hatao, jo poora bacha (11) × agla (12), peeche 25.
- 115² = 13225 (11 × 12 = 132).
- 10 × 11 se 19 × 20: tukdon mein (12 × 13 = 120 + 36 = 156).
- Chitra: 115 × 115 = 110 × 120 + 25. Isliye trick hamesha chalti hai.
- 0 mat chhodo: 105² = 11025, 195² = 38025.
- Andaaza: 115², 10000 se bada.
— charumam.com 🎯
👉 Aage Kya?
Ab bahan-trick: Post 6.3 mein 43 × 47 = 2021, jab dahai same aur ikaiyon ka jod 10 ho!
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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