Chapter 13 shuru! 🎉 Level 3 ka pehla chapter: squares. Chapter 7 mein base method se 97 × 96 jaise multiply kiye the. Aaj wahi trick ek khaas sawaal pe: jab dono number ek jaise hon, jaise 97 × 97, yaani 97². Tab trick aur chhoti ho jaati hai: kitna kam, utna aur subtract karo, phir deficit ka square. Square (²) kya hai, base kya hai, right part ke boxes kya hain, sab neeche shuru se dobara samjhaaya hai.
⏱ Before Timer: Pehle Purane Method Se
Pehle school wala method (long multiply), ek doosre number pe dekho: 98². 98² matlab 98 × 98. Doosre 98 ko tod lo: 98 = 90 + 8.
- 98 × 8 (column method, right se): 8 × 8 = 64 → 4 likho, 6 carry. 8 × 9 = 72, carry 6 add karo: 72 + 6 = 78. To 98 × 8 = 784.
- 98 × 90: pehle 98 × 9 = 882 (9 × 8 = 72 → 2 likho, 7 carry; 9 × 9 = 81, 81 + 7 = 88), phir peeche 0 (× 10) → 8820.
- Add karo: 784 + 8820 = 9604. To 98² = 9604.
Teen bade step, kai carry. Ab isi method se 97² khud karo, timer ke saath (jawab abhi mat dekhna).
Ab phone ya watch mein stopwatch (time measure karne wali watch) chalu kijiye. Neeche wala sawaal isi school wale method (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
97² = 97 × 97 = 9409
Abhi trick nahi dekhni! Post ke end mein After Timer mein yahi sawaal trick se karenge aur dono time ki compare karenge. 😊
— 📌 charumam.com
✅ Pehle Yeh Aata Hai
Is post mein chahiye: square ka matlab (²), base aur kitna kam (Chapter 7), 100 se subtract karna (Chapter 3), aur wall ( | ) ke right part ke boxes. Sab neeche ke boxes mein dobara hai. Pehle naye word:
📖 Word jo is post mein aayenge
- Square (²): number × wahi number. 97² = 97 × 97. ² ka matlab “× 2” nahi.
- Base: paas wala round number: 10, 100 ya 1000. 97 ka base 100.
- Deficit (kitna kam): base − number. 97 → 100 − 97 = 3 kam.
- Left part: number − apni deficit. 97 − 3 = 94. Yeh base 100 mein “kitne saikde” count karta hai.
- Right part: deficit ka square: 3² = 3 × 3 = 9. Base 100 mein isko 2 boxes milte hain: 09.
- Wall ( | ): left aur right part ke beech ki lakeer, sirf samajhne ke liye. Asli jawab mein dono saath likhte hain: 94 | 09 → 9409.
📏 Bade numbers padhna aur round number mein add karna
Har digit ki ek jagah (place) hoti hai. Right se count karo: 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 times 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 add karna (place box rule): chhote number ko right se likho, yaani ikai ke neeche ikai, dahai ke neeche dahai. Phir har boxes 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.
✖️ Multiply kya hai? (Multiplication)
Multiply (sign ×) matlab ek hi number ko baar-baar add karna. “6 × 4” ko bolte hain “6 times 4”, matlab 4 ko 6 baar add karo:
6 × 4 = 4 + 4 + 4 + 4 + 4 + 4 = 24
Isi tarah 6 × 7 matlab 7 ko 6 baar add karo. Running total (har baar ka naya sum) 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.)
Pieces mein multiply 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, total utna hi rahega:
6 × 47: 6 baar 40 = 240, 6 baar 7 = 42, total 240 + 42 = 282
Multiply mein carry 1 se bada ho sakta hai: sum mein carry aksar 1 hota hai, par multiply 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 add karo: 24 + 4 = 28. Jawab 282.
Multiply mein jagah convert karo, jawab wahi: 16 × 4 aur 4 × 16 ek hi cheez hain. Neeche dots dekho: 4 line, har line mein 16 dots. Line se count karo to 4 baar 16; khade column se count karo to 16 baar 4. Dots wahi hain, to jawab bhi wahi: 64. Isliye jo aasaan lage, us tarah add karo.
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 add karo: 24 + 4 = 28. Neeche 28 likho (8 dahai mein, 2 saikde mein).
- Jawab: 282.
10 se multiply: 3 × 10 matlab 10 ko 3 baar add karna: 10 + 10 + 10 = 30. Kisi number ko 10 se multiply karo to woh har digit ek jagah left shift jaata hai aur ikai mein 0 aa jaata hai: 3 → 30, 30 → 300.
🟦 Square kya hai? Chhota ² ka matlab (Chapter 6 yaad karo)
Kisi number ko usi number se multiply karna us number ka square (Hindi mein “varg”) kehlata hai. Ise number ke upar-right mein chhota ² likh kar dikhate hain. 7² padho “7 ka square” ya “7 square”, matlab 7 × 7 = 49. Dhyaan: 7² ka matlab 7 × 2 nahi hai (woh 14 hota)!
Name “square” kyun? Square ka ek matlab square bhi hai (jiski all four bhujaayein, yaani kinaare, equal hon). 7 line banao, har line mein 7 dots: dots ka ek square ban jaata hai, aur total dots 7 × 7 = 49. To 7² = 49 dots ka square.
| Square | Matlab | Jawab |
|---|---|---|
| 3² | 3 × 3 | 9 |
| 7² | 7 × 7 | 49 |
| 10² | 10 × 10 | 100 |
| 35² | 35 × 35 | 1225 (Post 6.1 ki trick) |
| 97² | 97 × 97 | 9409 (aaj ki trick!) |
Chapter 7 se relation: 97² bas 97 × 97 hai, yaani base method ka woh sawaal jisme dono numbers ek jaise hain. Isliye dono numbers base se utne hi door hote hain, aur trick aur bhi aasaan ho jaati hai.
🎯 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 “place” 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 part kya count karta 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 part kitne digit ka” isi se tay hoga.
📏 Base 100 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.
100 se kitna kam? Chapter 3 wali trick: Post 3.1 mein seekha tha ki 100 se subtract karne ke liye dahai digit ko 9 se poora karo aur ikai digit ko 10 se poora karo. 100 − 97: dahai 9 → 9 − 9 = 0, ikai 7 → 10 − 7 = 3. Jawab 03, yaani 3. Reverse counting se bhi: 97 se 98, 99, 100: teen step.
| Number | Base 100 se | Kaise nikala |
|---|---|---|
| 99 | 1 kam | 100 − 99 = 1 |
| 97 | 3 kam | 9 − 9 = 0, 10 − 7 = 3 → 03 |
| 96 | 4 kam | 100 − 96 = 4 |
| 92 | 8 kam | 100 − 92 = 8 |
| 88 | 12 kam | 9 − 8 = 1, 10 − 8 = 2 → 12 |
| 104 | 4 zyada | 104 − 100 = 4 |
| 112 | 12 zyada | 112 − 100 = 12 |
Chhota path: 100 se zyada wale (101 se 120 jaise) ka “kitna zyada” bas uske last do digit hain: 107 → 07 = 7, 112 → 12. Kyunki 112 = 100 + 12.
🧱 Wall ( | ), right part ke digit, aur carry
Is trick mein jawab do parts mein banta hai: left part | right part. Beech ki khadi lakeer ( | ) ko “wall” samjho. Yeh sirf hamare samjhne ke liye hai; asli jawab mein wall nahi likhte, dono parts saath-saath likh dete hain.
Base 100 mein 2 zero hain. Isliye left part kitne saikde count karta hai, aur right part ko jawab mein do boxes milte hain. Rule:
Right parts mein digit = base ke zero (2)
- Right chhota pade (kam digit) → aage 0 bharo.
- Right bada pade (zyada digit) → sabse left wala extra part left part mein add karo (carry).
- 93 | 12: 12 mein pehle se do digit hain. 93 saikde + 12 = 9300 + 12 = 9312.
- 95 | 6: 6 sirf ek digit hai. Use 06 likho: 0 dahai mein, 6 ikai mein. 95 saikde + 6 = 9500 + 6 = 9506. Agar 0 bhool gaye to “956” ban jaata, jo 9506 se bahut chhota hai (galat!). Yeh bilkul Post 6.3 wala 09 hai.
- Carry: 77 | 132: 132 mein teen digit, par jagah sirf do. 132 = 1 saikda + 32. Woh 1 saikda left ke saikdon mein jaata hai: 77 + 1 = 78 → 78 | 32 = 7832. Pakka: 7700 + 132 = 7832 ✅.
Pehle multiply, phir sum (zaroori rule): jab ek hi line mein × aur + dono hon, to pehle × (multiply) karo, phir + (sum). 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 rule se padhna.
🔁 Chhota sa warm-up
- 100 − 96 = ?
- 4² = ?
- 92 × 100 + 16 = ?
Answers dekho
- 4
- 16 (4 × 4)
- 9216 (pehle multiply: 9200, phir sum)
Teeno aa gaye? Yahi milkar 96² = 9216 banta hai! Neeche dekho kaise.
— 📌 charumam.com
🎬 Story: Square Aangan Ki Tiles
Zoya ke place ka aangan bilkul square hai: length mein 97 tiles, chaudai mein bhi 97 tiles. Papa ne poocha: “Poore aangan mein kitni tiles lagengi? Shop pe order dena hai.”
Bhai Kabir ne copy nikaali: 97 × 7, phir 97 × 90, phir sum… Zoya ne mann mein socha: “97 to 100 se 3 kam hai. 97 − 3 = 94. Aur 3 × 3 = 9, do boxes mein 09.”
Zoya turant boli: “9409 tiles!” Kabir ka jawab copy pe 2 minute baad aaya: woh bhi 9409. Papa ne hand milaya. 🤝
Zoya: “Chapter 7 ka base method hai, bas jab dono number ek jaise hon, to ek hi deficit do baar aati hai.” Aaj yahi seekhenge. 🟦
— 📌 charumam.com
🪄 Trick Step by Step: Kitna Kam, Utna Aur Subtract karo, Deficit Ka Square
💡 Is post ki ek badi baat: Number base se kitna kam hai, woh nikaalo (yeh “deficit”). Left = number − deficit. Right = deficit × deficit (deficit ka square), utne boxes mein jitne base mein zero (kam pade to aage 0, zyada ho to carry). 97²: 94 | 09 → 9409.
Step 1: Base chuno, zero count karo
Sawaal: 97², yaani 97 × 97. 97 sabse zyada 100 ke paas hai, to base = 100. 100 mein 2 zero, to right part ko 2 boxes milenge.
Step 2: Deficit nikaalo
100 − 97 = 3. To 97, base se 3 kam hai. (Chapter 3 ki trick: dahai 9 → 9 − 9 = 0, ikai 7 → 10 − 7 = 3, yaani 03 = 3.)
Step 3: Left: number − deficit
97 − 3 = 94. Chapter 7 mein left part “ek number − doosre ki deficit” tha (tirchha). Yahan dono number 97 hain, dono ki deficit 3, to bas 97 − 3.
Step 4: Right: deficit ka square
3² = 3 × 3 = 9. Right ko 2 boxes chahiye aur 9 ek hi digit hai, to aage 0: 09.
Step 5: Dono saath likho aur check
94 | 09. Left 94 matlab 94 saikde. Pehle multiply, phir sum: 94 × 100 + 9 = 9400 + 9 = 9409. Asal mein bas dono parts saath likh do: 94 aur 09 → 9409 ✅
Sabse zaroori: right part 1 digit ka aaye to aage 0
98²: 98 → 2 kam. Left: 98 − 2 = 96. Right: 2 × 2 = 4. Base 100 mein right ko 2 boxes, to 04. Jawab 96 | 04 → 9604. Agar 0 bhool gaye to “964” ban jaata, jo bahut chhota hai: 98² to 90 × 90 = 8100 se bada hona chahiye.
Ek nazar mein: 90 se 99 tak ke square
| Square | Base se distance | Left | Right (distance²) | Jawab |
|---|---|---|---|---|
| 99² | 1 kam | 99 − 1 = 98 | 1 × 1 = 1 | 98 | 01 → 9801 |
| 98² | 2 kam | 98 − 2 = 96 | 2 × 2 = 4 | 96 | 04 → 9604 |
| 97² | 3 kam | 97 − 3 = 94 | 3 × 3 = 9 | 94 | 09 → 9409 |
| 96² | 4 kam | 96 − 4 = 92 | 4 × 4 = 16 | 92 | 16 → 9216 |
| 95² | 5 kam | 95 − 5 = 90 | 5 × 5 = 25 | 90 | 25 → 9025 |
| 94² | 6 kam | 94 − 6 = 88 | 6 × 6 = 36 | 88 | 36 → 8836 |
| 93² | 7 kam | 93 − 7 = 86 | 7 × 7 = 49 | 86 | 49 → 8649 |
| 92² | 8 kam | 92 − 8 = 84 | 8 × 8 = 64 | 84 | 64 → 8464 |
| 91² | 9 kam | 91 − 9 = 82 | 9 × 9 = 81 | 82 | 81 → 8281 |
| 90² | 10 kam | 90 − 10 = 80 | 10 × 10 = 100 | 80 | 100 → 81 | 00 → 8100 |
(90² wali line dekho: right 10 × 10 = 100, teen digit. 100 = 1 saikda + 0, to 1 saikda left mein: 80 + 1 = 81, right 00. Jawab 8100 ✅. Yahi “carry” neeche samjhaaya hai.)
Right part 100 ya zyada: carry karo
88² (base 100): 100 − 88 = 12, yaani 12 kam. Left: wahi deficit subtract karo: 88 − 12 = 76. Right: 12² = 12 × 12 = 144. Right ko sirf 2 boxes milte hain, aur 144 mein 3 digit hain. Tod lo: 144 = 100 + 44, yaani 1 saikda (100) aur 44. Left part saikde hi count karta hai, to woh 1 saikda left mein bhejo (carry): 76 + 1 = 77. Right mein bacha 44. Jawab: 76 | 144 → 77 | 44 → 7744.
(12 × 12 kaise? 12 ko 10 baar = 120, 12 ko 2 baar = 24, total 120 + 24 = 144.) Place value se pakka: 76 saikde + 144 = 7600 + 144 = 7744 ✅.
89² (base 100): 100 − 89 = 11, yaani 11 kam. Left: wahi deficit subtract karo: 89 − 11 = 78. Right: 11² = 11 × 11 = 121. Right ko sirf 2 boxes milte hain, aur 121 mein 3 digit hain. Tod lo: 121 = 100 + 21, yaani 1 saikda (100) aur 21. Left part saikde hi count karta hai, to woh 1 saikda left mein bhejo (carry): 78 + 1 = 79. Right mein bacha 21. Jawab: 78 | 121 → 79 | 21 → 7921.
(11 × 11: Chapter 4 ki × 11 trick: 1 | 1 + 1 | 1 → 121.)
Base 10 pe bhi wahi trick (1-digit square)
Base 10 mein 1 zero, to right part ko sirf 1 box, aur left part dahai count karta hai.
| Square | Base se distance | Left | Right (distance²) | Jawab |
|---|---|---|---|---|
| 9² | 1 kam | 9 − 1 = 8 | 1 × 1 = 1 | 8 | 1 → 81 |
| 8² | 2 kam | 8 − 2 = 6 | 2 × 2 = 4 | 6 | 4 → 64 |
| 7² | 3 kam | 7 − 3 = 4 | 3 × 3 = 9 | 4 | 9 → 49 |
| 6² | 4 kam | 6 − 4 = 2 | 4 × 4 = 16 | 2 | 16 → 3 | 6 → 36 |
(6² wali line: right 4 × 4 = 16, par box sirf 1. 16 = 1 dahai + 6. Woh 1 dahai left mein: 2 + 1 = 3. Jawab 3 | 6 → 36 ✅.)
🆘 Atak gaye? Aise socho
- Base kaunsa? 6 se 9 → base 10. 88 se 99 → base 100.
- Deficit: base − number. 100 − 94 = 6.
- Left: number − deficit (subtract karo, add karo mat). 94 − 6 = 88.
- Right: deficit × deficit. 6 × 6 = 36. Base 100 → 2 boxes; 1 digit aaye to aage 0.
- Right zyada bada? Uska extra (saikda) left mein add karo: 76 | 144 → 77 | 44.
- Check: jawab hamesha base × base se thoda kam hoga (kyunki number base se thoda kam hai). 97² → base 100, to 100 × 100 = 10000 se thoda kam: 9409 ✅. 8² → base 10, to 10 × 10 = 100 se thoda kam: 64 ✅.
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Chitra se samjho (Chapter 7 jaisa). 97² matlab tiles ka ek square: 97 column, har column mein 97 tile.
- Right se 3 column ki ek patli patti cut kar lo (3 = deficit). Ab bacha part 94 column chauda hai (97 − 3 = 94). Yahi left part hai.
- Kaati hui patti ko ghuma kar bache parts ke upar laga do. Bache parts ki oonchai 97 thi, patti 3 tall, to ab oonchai 97 + 3 = 100, yaani poora base!
- Ab ek bada box hai: 94 column × 100 = 9400 tile. Par patti 97 long thi aur bacha part sirf 94 chauda, to patti ka 3 × 3 wala corner baahar latak gaya. Woh corner = 3 × 3 = 9 tile = right part (deficit ka square).
- Total tile: 9400 + 9 = 9409. Koi tile nayi nahi bani, koi gayab nahi hui: sirf jagah badli.
Neeche chitra mein yahi dikhaaya hai (chitra measurement ke hisaab se nahi, samjhane ke liye hai).
Chhota example tiles se: 8²
8 × 8 ka square: 2 column ki patti cut karo (8 − 2 = 6 bache), upar lagao (oonchai 8 + 2 = 10). Box 6 × 10 = 60, baahar corner 2 × 2 = 4. Total 64.
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Method Se Check)
Example 1: 96² = ?
96² (base 100): 100 − 96 = 4, yaani 4 kam. Left: wahi deficit subtract karo: 96 − 4 = 92. Right: 4² = 4 × 4 = 16 (2 boxes, poore bhar gaye). Jawab: 92 | 16 → 9216.
✔️ Normal method se check: 96 to 100 se 4 kam hai, to 96 ko 100 baar lo, phir 4 baar subtract karo. 96 × 100: 96 × 10 = 960, phir × 10 = 9600 (har × 10 pe peeche ek 0). 96 × 4 (column): 4 × 6 = 24 → 4 likho, 2 carry; 4 × 9 = 36, 36 + 2 = 38 → 384. Ab 9600 − 384 = 9216 ✅
Example 2: 99² = ?
99² (base 100): 100 − 99 = 1, yaani 1 kam. Left: wahi deficit subtract karo: 99 − 1 = 98. Right: 1² = 1 × 1 = 1. Base 100 mein right ko 2 boxes chahiye, to aage 0 bharo: 01. Jawab: 98 | 01 → 9801.
✔️ Normal method se check: 99 × 99, Chapter 10 ki × 99 trick: 99 × 100 = 9900, phir 99 subtract karo: 9900 − 99 = 9801 ✅
Example 3: 89² = ? (carry wala)
89² (base 100): 100 − 89 = 11, yaani 11 kam. Left: wahi deficit subtract karo: 89 − 11 = 78. Right: 11² = 11 × 11 = 121. Right ko sirf 2 boxes milte hain, aur 121 mein 3 digit hain. Tod lo: 121 = 100 + 21, yaani 1 saikda (100) aur 21. Left part saikde hi count karta hai, to woh 1 saikda left mein bhejo (carry): 78 + 1 = 79. Right mein bacha 21. Jawab: 78 | 121 → 79 | 21 → 7921.
✔️ Normal method se check: Column (Before Timer jaisa), 89 = 80 + 9. 89 × 9: 9 × 9 = 81 → 1 likho, 8 carry; 9 × 8 = 72, 72 + 8 = 80 → 801. 89 × 80: pehle 89 × 8: 8 × 9 = 72 → 2 likho, 7 carry; 8 × 8 = 64, 64 + 7 = 71 → 712, phir peeche 0 (× 10) → 7120. Add karo: 7120 + 801 = 7921 ✅
Example 4: 7² = ? (base 10)
7² (base 10): 10 − 7 = 3, yaani 3 kam. Left: wahi deficit subtract karo: 7 − 3 = 4. Right: 3² = 3 × 3 = 9 (1 boxes, poore bhar gaye). Jawab: 4 | 9 → 49.
✔️ Normal method se check: 7 ko 7 baar add karo (running total): 7, 14, 21, 28, 35, 42, 49 ✅
— 📌 charumam.com
💡 CharuMam Ki Smart Trick
Yeh CharuMam ki “Base Wala Square” trick hai. Ek line mein yaad rakho: “jitna kam hai, utna aur kam karo, aur deficit ka square lagao”. Bilkul aaj ki trick: 97 mein 3 kam, to 3 aur kam karo (94), aur 3 ka square (09) lagao. Yeh trick Chapter 7 ke base method ka hi ek chhota form hai, square ke liye.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| 90 se 99 ka square (97²) | ✅ Haan | Deficit chhoti (1 se 10), right ka square aasaan. |
| 6 se 9 ka square (8²) | ✅ Haan | Base 10, right 1 box. |
| 85 se 89 (88²) | ✅ Haan, carry ke saath | Deficit 11 se 15, right 3 digit, carry 1 ya 2. |
| 95² (5 pe khatam) | ✅ Dono chalenge | Base: 90 | 25. Chapter 6: 9 × 10 = 90, peeche 25. Dono 9025. |
| 67², 54² (base se door) | ❌ Nahi | Deficit 33 ya 46: right ka square hi mushkil. Seedha multiply behtar. |
| 104², 112² (base se zyada) | ➡️ Post 13.2 | Wahan deficit nahi, “zyada”: left mein add karna hoga. |
— 📌 charumam.com
❌→✅ Common Mistakes
Mistake 1: Right mein 0 bharna bhool gaye
Sawaal: 98²
❌ Galat: 96 | 4 → 964.
✅ Sahi: base 100 mein right ko 2 boxes: 4 → 04. Jawab 9604. Estimate check: 98² to 8100 (90²) se bada, 964 bahut chhota.
Mistake 2: Left mein deficit add kar di
Sawaal: 97²
❌ Galat: 97 + 3 = 100 → 100 | 09 → 10009.
✅ Sahi: number base se kam hai, to deficit subtract karo: 97 − 3 = 94 → 9409. (97² to 100² = 10000 se chhota hona hi chahiye.)
Mistake 3: Deficit ka square nahi, double kar diya
Sawaal: 97²
❌ Galat: right 3 × 2 = 6 → 94 | 06 → 9406.
✅ Sahi: right = deficit × deficit = 3 × 3 = 9 → 09 → 9409. (² ka matlab “usi number se multiply”, “2 se multiply” nahi.)
Mistake 4: Carry left mein nahi bheja
Sawaal: 88²
❌ Galat: 76 | 144 → seedha likh diya 76144 (5 digit!).
✅ Sahi: right ko sirf 2 boxes. 144 = 1 saikda + 44 → 76 + 1 = 77 → 77 | 44 → 7744.
🧠 Rule of thumb: Base se kam wala number: number − deficit | deficit², right ko utne boxes jitne base mein zero.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 sawaal, sab square. Target: 5 minute. Pehle har number ki deficit likh lo, phir trick.
- 99² = ?
- 98² = ?
- 97² = ?
- 96² = ?
- 95² = ?
- 94² = ?
- 93² = ?
- 92² = ?
- 91² = ?
- 9² = ?
- 8² = ?
- 7² = ?
- 6² = ?
- 89² = ?
- 88² = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- 9801
- 9604
- 9409
- 9216
- 9025
- 8836
- 8649
- 8464
- 8281
- 81
- 64
- 49
- 36
- 7921
- 7744
Kaise kiya (kuch):
93² (base 100): 100 − 93 = 7, yaani 7 kam. Left: wahi deficit subtract karo: 93 − 7 = 86. Right: 7² = 7 × 7 = 49 (2 boxes, poore bhar gaye). Jawab: 86 | 49 → 8649.
6² (base 10): 10 − 6 = 4, yaani 4 kam. Left: wahi deficit subtract karo: 6 − 4 = 2. Right: 4² = 4 × 4 = 16. Right ko sirf 1 box milta hai, aur 16 mein 2 digit hain. Tod lo: 16 = 10 + 6, yaani 1 dahai (10) aur 6. Left part dahai hi count karta hai, to woh 1 dahai left mein bhejo (carry): 2 + 1 = 3. Right mein bacha 6. Jawab: 2 | 16 → 3 | 6 → 36.
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⏰ One-Minute Challenge
Sirf 1 minute (60 second). Timer lagao, aur jitne ho sakein utne sawaal mann mein (bina copy ke) karo. Sirf jawab likhna hai.
- 9² = ?
- 8² = ?
- 99² = ?
- 98² = ?
- 97² = ?
- 95² = ?
- 93² = ?
- 92² = ?
- 90² = ?
- 87² = ?
| Rating | Kitne sahi, kitne time mein |
|---|---|
| 🌱 Beginner | 60 second mein 5 se 7 sahi. Badhiya start! |
| 🚀 Pro | 60 second mein 8 ya 9 sahi |
| 🏆 Champion | 10 mein 10 sahi, 60 second ke andar |
Answers dekho
- 81
- 64
- 9801
- 9604
- 9409
- 9025
- 8649
- 8464
- 8100
- 7569
Dhyaan:
87² (base 100): 100 − 87 = 13, yaani 13 kam. Left: wahi deficit subtract karo: 87 − 13 = 74. Right: 13² = 13 × 13 = 169. Right ko sirf 2 boxes milte hain, aur 169 mein 3 digit hain. Tod lo: 169 = 100 + 69, yaani 1 saikda (100) aur 69. Left part saikde hi count karta hai, to woh 1 saikda left mein bhejo (carry): 74 + 1 = 75. Right mein bacha 69. Jawab: 74 | 169 → 75 | 69 → 7569.
90² (base 100): 100 − 90 = 10, yaani 10 kam. Left: wahi deficit subtract karo: 90 − 10 = 80. Right: 10² = 10 × 10 = 100. Right ko sirf 2 boxes milte hain, aur 100 mein 3 digit hain. Tod lo: 100 = 100 + 0, yaani 1 saikda (100) aur 0. Left part saikde hi count karta hai, to woh 1 saikda left mein bhejo (carry): 80 + 1 = 81. Right mein bacha 0, 2 boxes mein 00. Jawab: 80 | 100 → 81 | 00 → 8100.
5 se kam sahi? Koi baat nahi! Trick wala section ek baar aur padho aur kal phir try karo. Har day thoda better, bas yahi chahiye. 💪
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📝 10 MCQs (Har Option Ka Reason)
MCQ matlab ek sawaal ke chaar jawab (options) diye hote hain, unme se ek sahi chunna hai.
Q1. 97² = ?
Jawab aur reason dekho
Sahi jawab: (a) 9409
- (a) sahi ✅: 3 kam: 97 − 3 = 94, 3 × 3 = 9 → 09 → 9409.
- (b) galat ❌: 9406: right mein 3 × 2 = 6 kiya (square nahi, double).
- (c) galat ❌: 10009: left mein deficit add kar di (97 + 3 = 100).
- (d) galat ❌: 949: right mein 0 bharna bhool gaye (94 | 9).
Q2. 98² = ?
Jawab aur reason dekho
Sahi jawab: (d) 9604
- (a) galat ❌: 964: 0 bharna bhool gaye (96 | 4).
- (b) galat ❌: 9602: right mein deficit ka square ki jagah deficit hi likh di (02).
- (c) galat ❌: 10004: left mein deficit add kar di (98 + 2 = 100).
- (d) sahi ✅: 2 kam: 98 − 2 = 96, 2 × 2 = 4 → 04 → 9604.
Q3. Base 100 wale square mein right part kitne boxes ka hota hai?
Jawab aur reason dekho
Sahi jawab: (c) 2
- (a) galat ❌: 3 boxes base 1000 mein (Post 13.3).
- (b) galat ❌: Nahi: kam aayein to aage 0, zyada aayein to carry.
- (c) sahi ✅: 100 mein 2 zero, to right ko 2 boxes.
- (d) galat ❌: 1 box base 10 mein hota hai.
Q4. 88² = ?
Jawab aur reason dekho
Sahi jawab: (b) 7744
- (a) galat ❌: 7844: carry 1 ki jagah 2 bhej diya (76 + 2 = 78 → 78 | 44). 144 mein sirf 1 saikda hai (144 = 100 + 44), to carry 1.
- (b) sahi ✅: 12 kam: 88 − 12 = 76, 12 × 12 = 144 → 1 saikda left mein: 77 | 44 → 7744.
- (c) galat ❌: 76144: carry left mein nahi bheja, seedha likh diya.
- (d) galat ❌: 7644: 144 ka sirf 44 likha, 1 saikda add karna bhool gaye.
Q5. 8² = ? (base 10)
Jawab aur reason dekho
Sahi jawab: (a) 64
- (a) sahi ✅: 2 kam: 8 − 2 = 6, 2 × 2 = 4 → 64.
- (b) galat ❌: 104: left mein deficit add kar di (8 + 2 = 10).
- (c) galat ❌: 62: right mein deficit ka square ki jagah deficit hi likh di (2).
- (d) galat ❌: 84: left mein deficit ghatai hi nahi (8 | 4).
Q6. 6² = ? (base 10)
Jawab aur reason dekho
Sahi jawab: (d) 36
- (a) galat ❌: 216: carry nahi bheja (2 | 16 seedha likha).
- (b) galat ❌: 26: 16 ka sirf 6 likha, 1 dahai add karna bhool gaye.
- (c) galat ❌: 46: carry 2 maan liya (2 + 2 = 4); 16 mein sirf 1 dahai hai.
- (d) sahi ✅: 4 kam: 6 − 4 = 2, 4 × 4 = 16 → 1 dahai left mein: 3 | 6 → 36.
Q7. 97² mein left part 94 kahan se aaya?
Jawab aur reason dekho
Sahi jawab: (c) 97 − 3
- (a) galat ❌: 100 − 3 = 97, yeh to number hi hai.
- (b) galat ❌: 3 × 3 = 9 right part hai, left nahi.
- (c) sahi ✅: Number − apni deficit: 97 − 3 = 94.
- (d) galat ❌: 97 + 3 = 100; yeh tab hota jab number base se zyada hota.
Q8. 99² = ?
Jawab aur reason dekho
Sahi jawab: (b) 9801
- (a) galat ❌: 9901: left mein deficit ghatai hi nahi (99 | 01).
- (b) sahi ✅: 1 kam: 99 − 1 = 98, 1 × 1 = 1 → 01 → 9801.
- (c) galat ❌: 981: 0 bharna bhool gaye (98 | 1).
- (d) galat ❌: 9800: right part 1 × 1 = 1 likhna bhool gaye (98 | 00).
Q9. Kis square ke liye “base 100, base se kam” wali trick sabse acchi hai?
Jawab aur reason dekho
Sahi jawab: (a) 96²
- (a) sahi ✅: 96 sirf 4 kam: 92 | 16 → 9216.
- (b) galat ❌: 67, 100 se 33 kam: 33 × 33 khud mushkil.
- (c) galat ❌: 54, 100 se 46 kam: bahut door.
- (d) galat ❌: 45 ke liye Chapter 6 ki trick (4 × 5 = 20, peeche 25) behtar.
Q10. 94² ka right part kya hai?
Jawab aur reason dekho
Sahi jawab: (d) 36
- (a) galat ❌: 12: 6 ka double (6 × 2) kar diya, square nahi.
- (b) galat ❌: 06: deficit hi likh di, uska square nahi.
- (c) galat ❌: 16: 94 ki ikai 4 ka square kar diya; square deficit ka lena hai.
- (d) sahi ✅: 94 → 6 kam, 6 × 6 = 36 (2 boxes, poore). Jawab 88 | 36 → 8836.
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🧠 Challenge: HOTS Aur Olympiad Style
HOTS matlab Higher Order Thinking Skills, yaani aise sawaal jinme sirf trick nahi, thoda mind bhi lagana padta hai. Olympiad ek maths competition (pratiyogita) hota hai.
Q1. 99² = 9801. Ab 98², 97², 96² bhi dekho: left part har baar kitna ghat raha hai, aur right part kaisa increase raha hai?
Solution dekho
99² → 98 | 01, 98² → 96 | 04, 97² → 94 | 09, 96² → 92 | 16. Left har baar 2 kam (kyunki deficit 1 increase hui, aur left = number − deficit, dono taraf se 1-1 kam). Right: 1, 4, 9, 16, yaani 1², 2², 3², 4².
Check: 96 × 96 = 9216 (Example 1) ✅
Q2. Kaun bada hai, aur kitna: 97² ya 96 × 98?
Solution dekho
97² = 9409. 96 × 98 (Chapter 7: 4 kam, 2 kam): left 96 − 2 = 94, right 4 × 2 = 8 → 08 → 9408. To 97² sirf 1 bada hai!
Check: 9409 − 9408 = 1 ✅. (Ek number 1 increase karo, doosra 1 subtract karo, to multiply hamesha 1 kam: corner 3 × 3 ki jagah 4 × 2.)
Q3. Ek square (square) field ki har bhuja (kinaara) 95 metre hai. Uska kshetrafal (area) kitna? (Area = length × chaudai)
Solution dekho
Square mein length = chaudai = 95, to area = 95 × 95 = 95². Base 100: 95 → 5 kam, left 95 − 5 = 90, right 5 × 5 = 25 → 9025 square metre.
(Metre length measure karne ki ikai hai, lagbhag ek bade step jitni. Square metre area ki ikai hai: 1 metre long aur 1 metre chaudi square jagah = 1 square metre. To field mein aisi 9025 square jagah aayengi.)
Check (Chapter 6): 9 × 10 = 90, peeche 25 → 9025 ✅
Q4. Reverse sawaal: ek number 100 se thoda kam hai. Uske square ke pehle do digit 92 aur last do digit 16 hain (9216). Number kya hai?
Solution dekho
Right part 16 = deficit × deficit, to deficit 4 (4 × 4 = 16). Number = 100 − 4 = 96. Left check: 96 − 4 = 92 ✅.
Check: 96² = 9216 (Example 1) ✅
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🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | Squares, square ka area (bhuja × bhuja), squares ki table. |
| Maths Olympiad | Pattern (99², 98², 97²), reverse sawaal, “kaun bada” wale sawaal. |
| NTSE / MAT | Number series mein squares (81, 64, 49 …) pehchanna. |
| SSC / Bank (aage) | Simplification mein 97², 88² jaise square turant. |
Shabdon ka matlab: Olympiad = school ke students ki maths pratiyogita (competition). NTSE = Class 10 ke students ka ek national scholarship (vazifa) exam. MAT = Mental Ability Test, yaani sochne-samajhne ki pariksha (NTSE ka ek part). SSC / Bank exam = badon ke government naukri ke exam. Simplification = long calculation ko chhota karke jawab nikaalna. Number series = numbers ki ek line jo kisi rule se aage increase hoti hai, aur agla number batana hota hai. Jaise 5, 10, 15, 20, … mein har baar 5 jud raha hai, to agla 25. Class 6–8 = school ki chhathi se aathvi kaksha.
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👩🏫 CharuMam Ka Tip
Mantra: “kitna kam, utna aur kam, phir deficit ka square.” Pehle 3 days sirf 91² se 99² tak daily ek baar bolo, phir 88², 89² (carry wale). Agle post mein number base se zyada hoga: 104², 112²! 🟦
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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)
97, base 100 se 3 kam (100 − 97 = 3). Left: 97 − 3 = 94. Right: 3 × 3 = 9 → do boxes → 09. Jawab 94 | 09 → 9409 ✅
| 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 days ki practice chahiye. Phir yeh purane method 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 mistake hone pe samjhayega:
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❓ FAQs (Students Ke Asli Sawaal)
1. 97² aur 97 × 2 mein kya difference hai?
97² = 97 × 97 = 9409. 97 × 2 = 194. ² ka matlab “usi number se multiply”.
2. Right mein 0 kyun bharte hain?
Base 100 mein left part saikde count karta hai, aur right ko 2 boxes (dahai aur ikai) milte hain. 97² mein right 3 × 3 = 9 hai, to 09 likho: 94 | 09 → 9409, yaani 94 saikde (9400) + 9. Agar 0 na bharein to 94 | 9 → 949 ban jaata hai: yahan 94 ek jagah right shift ho gaya, saikde ki jagah se dahai ki jagah pe (94 saikde = 9400 ki jagah sirf 94 dahai = 940). Place value galat ho gayi, isliye 949 ≠ 9409. 0 bas 94 ko uski sahi jagah pe rakhta hai.
3. Carry kab aata hai?
Jab deficit ka square base jitna ya usse bada ho: base 100 mein deficit 10 ya zyada (10 × 10 = 100).
4. 85² bhi is trick se?
Haan: 85, 100 se 15 kam. Left: 85 − 15 = 70. Right: 15 × 15 = 225 (15 ko 10 baar = 150, 15 ko 5 baar = 75, total 150 + 75 = 225). Right ko sirf 2 boxes, aur 225 mein 3 digit: 225 = 200 + 25, yaani 2 saikde aur 25. Woh 2 saikde left mein bhejo (carry 2): 70 + 2 = 72. Jawab 72 | 25 → 7225. Par Chapter 6 ki trick (8 × 9 = 72, peeche 25) zyada aasaan.
5. Agle post mein kya hai?
Post 13.2: base se zyada wale numbers ka square: 104², 112², 13².
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🟢 Recap: Aaj Kya Seekha
- Square (²) = number × wahi number.
- Deficit = base − number. Left = number − deficit.
- Right = deficit × deficit, base ke zero jitne boxes.
- Kam digit → aage 0 (98² = 9604). Zyada digit → carry (88² = 7744).
- Base 10: right 1 box (6² = 36).
- Chitra: patti cut karke upar lagao, baahar ka corner = deficit².
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👉 Aage Kya?
Ab base se zyada wale numbers ki baari: Post 13.2 mein 104², 112² aur 13². Wahan deficit nahi, “zyada” hoga, aur left mein add karna hoga!
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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