Base Se Thoda Zyada Ka Square: 104², 112², 13² | Speed Maths Ch 13

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📘 MATHS⚡ Speed Maths · Level 3 (Squares, Roots aur Division)Ch 13 · Fast Squaring Near BasePost 13.2 / 13.4⏱ 35–40 min🎒 Pehle aana chahiye: Post 13.1 (square near base: deficit, left, right ke boxes), Chapter 7 (Post 7.3: 100 se zyada wale numbers ka base method), Chapter 6 (15², 105² jaise square)
Base Se Thoda Zyada Ka Square: 104², 112², 13² ⬆️

Post 13.1 mein 97² jaise square kiye: number base se kam tha, to left mein subtract karna pada. Aaj reverse: number base se thoda zyada, jaise 104², 112², 13². Trick wahi, bas ek word badlega: “subtract karo” ki jagah “add karo”. Aur chitra mein dekhenge ki yeh kyun hai, Chapter 7 (Post 7.3) ki tarah.

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⏱ Before Timer: Pehle Purane Method Se

Pehle seedha method, ek doosre number pe dekho: 103². 103² matlab 103 × 103. Doosre 103 ko tod lo: 103 = 100 + 3.

  1. 103 × 100 = 10300 (peeche do 0).
  2. 103 × 3: 100 × 3 = 300 aur 3 × 3 = 9, to 300 + 9 = 309.
  3. Add karo: 10300 + 309 = 10609. To 103² = 10609.

Ab isi method se 104² 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.

104² = ?

Mera time: ______ second

Jawab dekho

104² = 104 × 104 = 10816

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

Chahiye: Post 13.1 ka square wala base method, aur kitna zyada (number − base). Sab neeche ke boxes mein dobara. Naye word:

📖 Word jo is post mein aayenge

  • Kitna zyada (zyada): number − base. 104 → 104 − 100 = 4 zyada. 112 → 12 zyada.
  • Chhota path (base 100): 101 se 120 jaise number ka “zyada” = uske last do digit: 107 → 07 = 7.
  • Left part (base se zyada wala number): number + zyada. 104 + 4 = 108. Yeh kitne saikde.
  • Right part: zyada × zyada. 4 × 4 = 16. Base 100 mein 2 boxes.
  • Teen-digit left: 108 | 16 mein left 108 saikde = 10800. Jawab 5 digit ka: 10816.

📏 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.

Place box: Hazaar, Saikda, Dahai, Ikai ke boxes, unme 165 likhaHazaar1000 waleSaikda100 wale1Dahai10 wale6Ikai1 wale5165 = 1 saikda + 6 dahai + 5 ikai = 100 + 60 + 5
NumberKaise padheinKis cheez se bana
98attaanve9 dahai (90) + 8 ikai
165ek sau painsath1 saikda (100) + 6 dahai (60) + 5 ikai
999nau sau ninyaanve9 saikde (900) + 9 dahai (90) + 9 ikai
1011ek hazaar gyaarah1 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.

Place boxes mein round number ke saath sum: 100 + 65 = 165, 100 + 2 = 102, 200 + 6 = 206, 1000 + 11 = 1011H = Hazaar, S = Saikda, D = Dahai, I = Ikai.Chhota number RIGHT se (ikai ke neeche ikai).100 + 65HS11D066I055+= 165100 + 2HS11D00I022+= 102200 + 6HS22D00I066+= 2061000 + 11H11S00D011I011+= 1011
  • 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 baar 47: har line mein 4 dande aur 7 cube, total 6 lineLine 1= 40 + 7 = 47Line 2= 40 + 7 = 47Line 3= 40 + 7 = 47Line 4= 40 + 7 = 47Line 5= 40 + 7 = 47Line 6= 40 + 7 = 474 dande (4 dahai = 40)7 chhote cube (7 ikai = 7)Sab dande: 6 baar 4 = 24 dande= 24 dahai = 240Sab cube: 6 baar 7 = 426 baar 47 = 240 + 42 = 282
Chitra: 6 line, har line mein 4 dande aur 7 cube. Sab dande: 6 × 4 = 24 dahai = 240. Sab cube: 6 × 7 = 42.

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.

4 line mein 16-16 dots: 16 × 4 = 4 × 16Line 1Line 2Line 3Line 4Leti line se count karo: 4 line, har line mein 16 dots4 baar 16 = 16 + 16 + 16 + 16 = 64Khade column se count karo: 16 column, har ek mein 4 dots16 baar 4 = 4 + 4 + … (16 baar) = 64Dots wahi hain, to 16 × 4 = 4 × 16 = 64
Chitra: 4 line × 16 dots = 16 column × 4 dots = 64 dots.

Copy pe column mein (28 + 15 jaisa):

4 4 7 × 6 —– 2 8 2
  1. Ikai: 6 × 7 = 42. Neeche ikai mein 2 likho. 42 ke 4 dahai carry hain: chhota 4 dahai wale digit (4) ke upar likho.
  2. Dahai: 6 × 4 = 24, usme upar likha carry 4 add karo: 24 + 4 = 28. Neeche 28 likho (8 dahai mein, 2 saikde mein).
  3. 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.

SquareMatlabJawab
3²3 × 39
7²7 × 749
10²10 × 10100
35²35 × 351225 (Post 6.1 ki trick)
104²104 × 10410816 (aaj ki trick!)

Chapter 7 se relation: 104² bas 104 × 104 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.

BaseBase mein kitne zeroJawab ka left part kya count karta hai
101 zerokitne dahai (10 wale)
1002 zerokitne saikde (100 wale)
10003 zerokitne 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.
Number line: base 100 se kitna kam, kitna zyadaNumber line: base 100 se kitna kam, kitna zyada9697989910010110210310497: 3 kam103: 3 zyada96: 4 kam104: 4 zyadaNaarangi teer = base se kitna kam (base − number). Hara teer = kitna zyada (number − base).
Chitra: base 100 ke paas wale numbers; naarangi teer “kitna kam”, hara teer “kitna zyada”.

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.

NumberBase 100 seKaise nikala
991 kam100 − 99 = 1
973 kam9 − 9 = 0, 10 − 7 = 3 → 03
964 kam100 − 96 = 4
928 kam100 − 92 = 8
8812 kam9 − 8 = 1, 10 − 8 = 2 → 12
1044 zyada104 − 100 = 4
11212 zyada112 − 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 ✅.
Wall ( | ) ke dono parts place value ke dabbon meinWall ( | ) ke dono parts place value ke dabbon mein9Hazaar3Saikda1Dahai2Ikai93 | 12 → 9312(base 100: right mein 2 digit)9Hazaar5Saikda0Dahai6Ikai95 | 06 → 9506(base 100: right mein 2 digit)
Chitra: wall ke left mein left part, right mein right part; right ko utne hi boxes jitne base mein zero.

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

  1. 107 − 100 = ?
  2. 7² = ?
  3. 114 × 100 + 49 = ?
Answers dekho
  1. 7
  2. 49 (7 × 7)
  3. 11449 (pehle multiply: 11400, phir sum)

Teeno aa gaye? Yahi milkar 107² = 11449 banta hai!

— 📌 charumam.com

🎬 Story: Cricket Ground Ka Square

School ke practice ground pe ek square part hai: har taraf 104 step. PT sir ne poocha: “Agar har chhote square boxes (1 step × 1 step) mein ek chhota cone rakhein, to kitne cone chahiye?” Matlab 104 line, har line mein 104 boxes: 104 × 104.

Aryan bola: “Sir, 104 to 100 se 4 zyada hai. Post 13.1 mein subtract karte the, yahan jodenge: 104 + 4 = 108. Aur 4 × 4 = 16.”

Phir Aryan ne poora jawab bola: “108 aur 16 saath likho: 10816 cone!” Sir ne phone pe calculator chalaaya: 10816. Bilkul sahi! 🏏

Aaj seekhenge ki base se zyada wale number ka square kaise, aur sum kyun karte hain. ⬆️

— 📌 charumam.com

🪄 Trick Step by Step: Kitna Zyada, Utna Aur Add karo, Zyada Ka Square

💡 Is post ki ek badi baat: Number base se kitna zyada hai, woh nikaalo. Left = number + zyada. Right = zyada × zyada, base ke zero jitne boxes mein (kam pade to aage 0, zyada ho to carry). 104²: 108 | 16 → 10816.

Step 1: Base chuno, zero count karo

Sawaal: 104². 104 sabse zyada 100 ke paas, to base = 100, 2 zero, right ko 2 boxes.

Step 2: Kitna zyada?

104 − 100 = 4. To 104, base se 4 zyada. (Chhota path: last do digit 04 = 4.)

Step 3: Left: number + zyada

104 + 4 = 108. Post 13.1 mein deficit ghatayi thi; yahan number base se aage hai, to zyada add karo.

Step 4: Right: zyada ka square

4² = 4 × 4 = 16. Do digit, 2 boxes poore bhar gaye.

Step 5: Saath likho aur check

108 | 16. Left 108 saikde. Pehle multiply, phir sum: 108 × 100 + 16 = 10800 + 16 = 10816 ✅. Estimate: 104² to 100² = 10000 se thoda zyada hona chahiye. 10816 theek!

104 (4 zyada) × 104 (4 zyada) ————– 108 | 16 → 10816

Right 1 digit ka? Aage 0 (Post 13.1 jaisa)

103² (base 100): 103 − 100 = 3, yaani 3 zyada. Left: wahi zyada add karo: 103 + 3 = 106. Right: 3² = 3 × 3 = 9. Base 100 mein right ko 2 boxes chahiye, to aage 0 bharo: 09. Jawab: 106 | 09 → 10609.

0 bhool gaye to “1069” likhoge, jo 10000 se bhi chhota hai. Galat!

Ek nazar mein: 101 se 110 tak

SquareBase se distanceLeftRight (distance²)Jawab
101²1 zyada101 + 1 = 1021 × 1 = 1102 | 01 → 10201
102²2 zyada102 + 2 = 1042 × 2 = 4104 | 04 → 10404
103²3 zyada103 + 3 = 1063 × 3 = 9106 | 09 → 10609
104²4 zyada104 + 4 = 1084 × 4 = 16108 | 16 → 10816
105²5 zyada105 + 5 = 1105 × 5 = 25110 | 25 → 11025
106²6 zyada106 + 6 = 1126 × 6 = 36112 | 36 → 11236
107²7 zyada107 + 7 = 1147 × 7 = 49114 | 49 → 11449
108²8 zyada108 + 8 = 1168 × 8 = 64116 | 64 → 11664
109²9 zyada109 + 9 = 1189 × 9 = 81118 | 81 → 11881
110²10 zyada110 + 10 = 12010 × 10 = 100120 | 100 → 121 | 00 → 12100

Right 100 ya zyada? Carry

112² (base 100): 112 − 100 = 12, yaani 12 zyada. Left: wahi zyada add karo: 112 + 12 = 124. 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): 124 + 1 = 125. Right mein bacha 44. Jawab: 124 | 144 → 125 | 44 → 12544.

(12 × 12: 12 ko 10 baar = 120, 12 ko 2 baar = 24, total 144.) Place value se pakka: 124 saikde + 144 = 12400 + 144 = 12544 ✅.

Base 10: 11 se 19 tak ke square

Base 10 mein right ko 1 box. 11 se 19 ka “zyada” bas unka ikai digit hai (13 = 10 + 3, to 3 zyada).

SquareBase se distanceLeftRight (distance²)Jawab
11²1 zyada11 + 1 = 121 × 1 = 112 | 1 → 121
12²2 zyada12 + 2 = 142 × 2 = 414 | 4 → 144
13²3 zyada13 + 3 = 163 × 3 = 916 | 9 → 169
14²4 zyada14 + 4 = 184 × 4 = 1618 | 16 → 19 | 6 → 196
15²5 zyada15 + 5 = 205 × 5 = 2520 | 25 → 22 | 5 → 225
16²6 zyada16 + 6 = 226 × 6 = 3622 | 36 → 25 | 6 → 256
17²7 zyada17 + 7 = 247 × 7 = 4924 | 49 → 28 | 9 → 289
18²8 zyada18 + 8 = 268 × 8 = 6426 | 64 → 32 | 4 → 324
19²9 zyada19 + 9 = 289 × 9 = 8128 | 81 → 36 | 1 → 361

(15² wali line: 20 | 25. Right ke 25 mein 2 dahai aur 5: 2 dahai left mein → 22 | 5 → 225. Chapter 6 ki trick bhi yahi deti hai: 1 × 2 = 2, peeche 25 → 225 ✅.)

(14² se 19² tak right 2 digit ka hai, par base 10 mein box sirf 1. To right ka dahai wala digit left mein jaata hai (carry), aur yeh carry 1 se bada bhi ho sakta hai. 16²: 36 = 3 dahai + 6, to 22 + 3 = 25 → 256 (carry 3). 19²: 81 = 8 dahai + 1, to 28 + 8 = 36 → 361 (carry 8).)

🆘 Atak gaye? Aise socho

  • Zyada: number − base. 107 − 100 = 7.
  • Left: base se zyada wala number → add karo: 107 + 7 = 114.
  • Right: zyada × zyada = 49. 2 boxes.
  • Kam digit: 103² → right 9 → 09.
  • Carry: 112² → 124 | 144 → 125 | 44.
  • Estimate: 100 se upar wale ka square 10000 se upar, 5 digit ka.

— 📌 charumam.com

🔍 Yeh Kaam Kyun Karta Hai?

Chitra (Post 7.3 jaisa): 104² matlab 104 × 104 tiles ka square. Ise chaar parts mein break karo:

  1. Bada neela square: 100 × 100 = 10000.
  2. Right mein patli naarangi patti: 4 column, har ek 100 tall: 4 × 100 = 400.
  3. Neeche patli hari patti: 4 line, har ek 100 long: 4 × 100 = 400.
  4. Corners mein chhota gulaabi square: 4 × 4 = 16.

Pehle teen: 10000 + 400 + 400 = 10800, yaani 108 saikde. Yeh 108 kahan se? 100 + 4 + 4, yaani 104 + 4: number + zyada! Aur bacha corner 4 × 4 = 16 = right part. Total 10800 + 16 = 10816.

Post 13.1 se difference: wahan patti cut karke hatani padi thi (isliye subtract karna). Yahan number base se bada hai, to pattiyan judti hain (isliye add karna).

Corner dono baar add karte kyun hain? Corner tiles ka ek asli piece hai: ek chhota square, jisme sach mein tiles hain (empty jagah nahi). Tiles ginte waqt jo piece maujood hai, use count mein add karna hi padta hai. Post 13.1 mein patti cut karke upar lagaayi, to 94 × 100 ke boxes ke baahar 3 × 3 = 9 tiles ka corner latak gaya. Woh tiles bhi square ki thi, isliye 9400 mein 9 add kiya. Yahan (13.2) bada square aur dono pattiyan milkar 108 saikde ban gaye, par corners ke 4 × 4 = 16 tiles un mein nahi aaye. Yeh extra piece hai jo bacha hua hai, isliye 10800 mein 16 add karo. To left part kabhi decrease hota hai, kabhi increase hota hai (patti hatai ya judi), par right part (corner) hamesha add karte hain.

104 × 104 = 108 × 100 + 4 × 4 = 10816104 × 104: chaudai 100 + 4, oonchai 100 + 4 (measurement se nahi)100 × 100100 × 4100 × 44 × 410041004Neela: 100 × 100 = 10000Naarangi: 100 ko 4 baar = 400Hara: 100 ko 4 baar = 400Teeno milkar: 10800= 108 ke peeche 2 zero (104 + 4 = 108)Gulaabi corner: 4 × 4 = 16104 × 104 = 10800 + 16 = 10816
Chitra: 104 × 104 = 100 × 100 + 4 × 100 + 4 × 100 + 4 × 4. Pehle teen milkar 108 saikde, corner 16. Total 10816.

Tiles se count karke dekho: 13²

13 × 13: 10 × 10 = 100, do pattiyan 3 × 10 = 30 aur 30, corner 3 × 3 = 9. 100 + 30 + 30 = 160, yaani 16 dahai (13 + 3 = 16), phir + 9 → 169.

13 × 13: 16 dahai aur corner 3 × 3 = 9, total 16913 × 13: 13 column, har column mein 13 tile103103Neela: 10 × 10 = 100Naarangi: 3 column × 10 = 30Hara: 3 line × 10 = 30Neela + naarangi + hara = 160= 16 dahai (13 + 3 = 16)Gulaabi corner: 3 × 3 = 913 × 13 = 160 + 9 = 169
Chitra: 13 × 13 = 160 + 9 = 169 (16 dahai aur corner 3 × 3).

— 📌 charumam.com

✏️ Solved Examples (Har Ek Normal Method Se Check)

Example 1: 107² = ?

107² (base 100): 107 − 100 = 7, yaani 7 zyada. Left: wahi zyada add karo: 107 + 7 = 114. Right: 7² = 7 × 7 = 49 (2 boxes, poore bhar gaye). Jawab: 114 | 49 → 11449.

✔️ Normal method se check: 107 × 100 = 10700, 107 × 7 = 749 (700 + 49), phir 10700 + 749 = 11449 ✅

Example 2: 111² = ?

111² (base 100): 111 − 100 = 11, yaani 11 zyada. Left: wahi zyada add karo: 111 + 11 = 122. 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): 122 + 1 = 123. Right mein bacha 21. Jawab: 122 | 121 → 123 | 21 → 12321.

(11 × 11: Chapter 4 ki trick: 1 | 1 + 1 | 1 → 121.)

✔️ Normal method se check: Doosre 111 ko tod lo: 111 = 100 + 11. To 111 × 111 = 111 × 100 + 111 × 11. 111 × 100 = 11100 (peeche do 0). 111 × 11 (Chapter 4 ki trick, Post 4.2): pehla digit 1, phir 1 + 1 = 2, phir 1 + 1 = 2, last digit 1 → 1221. Add karo: 11100 + 1221 = 12321 ✅

Example 3: 16² = ? (base 10, carry)

16² (base 10): 16 − 10 = 6, yaani 6 zyada. Left: wahi zyada add karo: 16 + 6 = 22. Right: 6² = 6 × 6 = 36. Right ko sirf 1 box milta hai, aur 36 mein 2 digit hain. Tod lo: 36 = 30 + 6, yaani 3 dahai (30) aur 6. Left part dahai hi count karta hai, to woh 3 dahai left mein bhejo (carry): 22 + 3 = 25. Right mein bacha 6. Jawab: 22 | 36 → 25 | 6 → 256.

(6 × 6 = 36: 6 ko 6 baar add karo: 6, 12, 18, 24, 30, 36.)

✔️ Normal method se check: 16 × 16 = 160 + 96 (16 ko 10 baar, phir 16 ko 6 baar: 16, 32, 48, 64, 80, 96) = 256 ✅

Example 4: 120² = ? (zyada 20)

120² (base 100): 120 − 100 = 20, yaani 20 zyada. Left: wahi zyada add karo: 120 + 20 = 140. Right: 20² = 20 × 20 = 400. Right ko sirf 2 boxes milte hain, aur 400 mein 3 digit hain. Tod lo: 400 = 400 + 0, yaani 4 saikde (400) aur 0. Left part saikde hi count karta hai, to woh 4 saikde left mein bhejo (carry): 140 + 4 = 144. Right mein bacha 0, 2 boxes mein 00. Jawab: 140 | 400 → 144 | 00 → 14400.

(20 × 20: 2 × 2 = 4, phir dono 20 ke peeche ek-ek 0, to do 0: 400.)

✔️ Normal method se check: 12 × 12 = 144, aur 120 × 120 mein do 0 aur: 14400 ✅

— 📌 charumam.com

💡 CharuMam Ki Smart Trick

Post 13.1 wali “Base Wala Square” trick yahan reverse form mein: “jitna zyada hai, utna aur zyada karo, aur zyada ka square lagao”. 104 mein 4 zyada, to 4 aur add karo (108), aur 4 ka square (16) lagao.

— 📌 charumam.com

🧭 Kab Use Karein, Kab Nahi

SituationTrick use karein?Kyun
101 se 110 (104²)✅ HaanZyada chhota, right aasaan.
11 se 19 (13²)✅ HaanBase 10, right 1 box; 14 se 19 tak carry (14²: right 16 → 1 dahai left mein).
111 se 115 (112²)✅ Haan, carry ke saathRight 121 se 225, carry 1 ya 2.
105², 115² (5 pe khatam)✅ Dono105²: 5 zyada → 110 | 25 → 11025 (Chapter 6: 10 × 11 = 110, peeche 25). 115²: 15 zyada → left 115 + 15 = 130, right 15 × 15 = 225. 225 = 200 + 25, yaani 2 saikde aur 25: 2 saikde left mein (carry 2): 130 + 2 = 132 → 132 | 25 → 13225 (Chapter 6: 11 × 12 = 132, peeche 25).
140², 160² (bahut zyada)❌ NahiZyada 40, 60: right bahut bada. 14 × 14 jaisa socho aur peeche 00.
996², 1007² (1000 ke paas)➡️ Post 13.3Base 1000, right 3 boxes.

— 📌 charumam.com

❌→✅ Common Mistakes

Mistake 1: Zyada add karne ki jagah subtract kar diya

Sawaal: 104²

❌ Galat: 104 − 4 = 100 → 100 | 16 → 10016.

✅ Sahi: base se zyada wale number mein add karo: 104 + 4 = 108 → 10816.

Mistake 2: 0 bharna bhool gaye

Sawaal: 103²

❌ Galat: 106 | 9 → 1069.

✅ Sahi: 2 boxes: 09 → 10609. (100 se upar ka square 10000 se upar, 5 digit.)

Mistake 3: Carry nahi bheja

Sawaal: 112²

❌ Galat: 124 | 144 → 124144.

✅ Sahi: 144 = 100 + 44 → 1 saikda left mein: 125 | 44 → 12544.

Mistake 4: Square ki jagah double

Sawaal: 104²

❌ Galat: right 4 × 2 = 8 → 108 | 08 → 10808.

✅ Sahi: right 4 × 4 = 16 → 10816.

🧠 Rule of thumb: Base se zyada wala number: number + zyada | zyada², right ko base ke zero jitne boxes.

— 📌 charumam.com

🏃 Speed Drill: 15 Sawaal, Timer Ke Saath

Timer lagao. 15 square, sab base se zyada. Target: 5 minute.

  1. 101² = ?
  2. 102² = ?
  3. 103² = ?
  4. 104² = ?
  5. 105² = ?
  6. 106² = ?
  7. 107² = ?
  8. 108² = ?
  9. 109² = ?
  10. 11² = ?
  11. 12² = ?
  12. 13² = ?
  13. 14² = ?
  14. 111² = ?
  15. 113² = ?

Mera time: ______ · Sahi: ____ / 15

Answers dekho (pehle khud karo!)
  1. 10201
  2. 10404
  3. 10609
  4. 10816
  5. 11025
  6. 11236
  7. 11449
  8. 11664
  9. 11881
  10. 121
  11. 144
  12. 169
  13. 196
  14. 12321
  15. 12769

Kaise kiya (kuch):

113² (base 100): 113 − 100 = 13, yaani 13 zyada. Left: wahi zyada add karo: 113 + 13 = 126. 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): 126 + 1 = 127. Right mein bacha 69. Jawab: 126 | 169 → 127 | 69 → 12769.

(13 × 13 = 169, upar tiles wale chitra mein.)

14² (base 10): 14 − 10 = 4, yaani 4 zyada. Left: wahi zyada add karo: 14 + 4 = 18. 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): 18 + 1 = 19. Right mein bacha 6. Jawab: 18 | 16 → 19 | 6 → 196.

— 📌 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.

  1. 101² = ?
  2. 103² = ?
  3. 105² = ?
  4. 108² = ?
  5. 110² = ?
  6. 11² = ?
  7. 12² = ?
  8. 15² = ?
  9. 104² = ?
  10. 102² = ?
RatingKitne sahi, kitne time mein
🌱 Beginner60 second mein 5 se 7 sahi. Badhiya start!
🚀 Pro60 second mein 8 ya 9 sahi
🏆 Champion10 mein 10 sahi, 60 second ke andar
Answers dekho
  1. 10201
  2. 10609
  3. 11025
  4. 11664
  5. 12100
  6. 121
  7. 144
  8. 225
  9. 10816
  10. 10404

Dhyaan:

110² (base 100): 110 − 100 = 10, yaani 10 zyada. Left: wahi zyada add karo: 110 + 10 = 120. 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): 120 + 1 = 121. Right mein bacha 0, 2 boxes mein 00. Jawab: 120 | 100 → 121 | 00 → 12100.

15² (base 10): 15 − 10 = 5, yaani 5 zyada. Left: wahi zyada add karo: 15 + 5 = 20. Right: 5² = 5 × 5 = 25. Right ko sirf 1 box milta hai, aur 25 mein 2 digit hain. Tod lo: 25 = 20 + 5, yaani 2 dahai (20) aur 5. Left part dahai hi count karta hai, to woh 2 dahai left mein bhejo (carry): 20 + 2 = 22. Right mein bacha 5. Jawab: 20 | 25 → 22 | 5 → 225.

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. 💪

— 📌 charumam.com

📝 10 MCQs (Har Option Ka Reason)

MCQ matlab ek sawaal ke chaar jawab (options) diye hote hain, unme se ek sahi chunna hai.

Q1. 104² = ?

(a) 10416
(b) 10816
(c) 10016
(d) 10808
Jawab aur reason dekho

Sahi jawab: (b) 10816

  • (a) galat ❌: 10416: left mein zyada add karna bhool gaye (104 | 16).
  • (b) sahi ✅: 4 zyada: 104 + 4 = 108, 4 × 4 = 16 → 10816.
  • (c) galat ❌: 10016: zyada add karne ki jagah subtract kar diya (104 − 4 = 100).
  • (d) galat ❌: 10808: right mein 4 × 2 = 8 kiya (double, square nahi).

Q2. 103² = ?

(a) 10609
(b) 1069
(c) 10606
(d) 10009
Jawab aur reason dekho

Sahi jawab: (a) 10609

  • (a) sahi ✅: 3 zyada: 106 | 09 → 10609.
  • (b) galat ❌: 1069: 0 bharna bhool gaye.
  • (c) galat ❌: 10606: right mein 3 × 2 = 6 (double).
  • (d) galat ❌: 10009: left mein add karna bhool gaye (100 | 09).

Q3. 112² = ?

(a) 124144
(b) 12444
(c) 10144
(d) 12544
Jawab aur reason dekho

Sahi jawab: (d) 12544

  • (a) galat ❌: 124144: carry nahi bheja.
  • (b) galat ❌: 12444: 144 ka 1 saikda left mein add karna bhool gaye.
  • (c) galat ❌: 10144: left mein zyada add karne ki jagah subtract kar diya (112 − 12 = 100), phir 100 | 144 → carry → 101 | 44.
  • (d) sahi ✅: 12 zyada: 124 | 144 → 125 | 44 → 12544.

Q4. Base se zyada wale number ke square mein left part kaise?

(a) Base − zyada
(b) Zyada × zyada
(c) Number + zyada
(d) Number − zyada
Jawab aur reason dekho

Sahi jawab: (c) Number + zyada

  • (a) galat ❌: 100 − 4 = 96: yeh kisi kaam ka nahi yahan.
  • (b) galat ❌: Yeh right part hai.
  • (c) sahi ✅: Haan: 104 + 4 = 108. Pattiyan judti hain.
  • (d) galat ❌: Subtract karna tab, jab number base se kam ho (Post 13.1).

Q5. 13² = ? (base 10)

(a) 1609
(b) 169
(c) 109
(d) 166
Jawab aur reason dekho

Sahi jawab: (b) 169

  • (a) galat ❌: 1609: base 10 mein bhi right ko 2 boxes de diye (16 | 09); base 10 mein sirf 1 box.
  • (b) sahi ✅: 3 zyada: 13 + 3 = 16, 3 × 3 = 9 → 169.
  • (c) galat ❌: 109: left mein subtract kar diya (13 − 3 = 10).
  • (d) galat ❌: 166: right mein 3 × 2 = 6 kiya (double, square nahi).

Q6. 15² = ? (base 10)

(a) 225
(b) 2025
(c) 205
(d) 125
Jawab aur reason dekho

Sahi jawab: (a) 225

  • (a) sahi ✅: 5 zyada: 20 | 25 → 2 dahai left mein → 22 | 5 → 225.
  • (b) galat ❌: 2025: carry nahi bheja (20 | 25 seedha).
  • (c) galat ❌: 205: 25 ka sirf 5 likha, 2 dahai add karna bhool gaye.
  • (d) galat ❌: 125: left mein zyada add karne ki jagah subtract kar diya (15 − 5 = 10), phir 10 | 25 → carry → 12 | 5.

Q7. 108² = ?

(a) 11616
(b) 10064
(c) 10864
(d) 11664
Jawab aur reason dekho

Sahi jawab: (d) 11664

  • (a) galat ❌: 11616: right mein 8 × 2 = 16 (double).
  • (b) galat ❌: 10064: left mein zyada add karne ki jagah subtract kar diya (108 − 8 = 100 → 100 | 64).
  • (c) galat ❌: 10864: left mein add karna bhool gaye (108 | 64).
  • (d) sahi ✅: 8 zyada: 116 | 64 → 11664.

Q8. Reverse: kisi number (100 se thoda zyada) ka square 11236 hai. Number?

(a) 104
(b) 94
(c) 106
(d) 112
Jawab aur reason dekho

Sahi jawab: (c) 106

  • (a) galat ❌: 104² = 10816.
  • (b) galat ❌: 94: distance 6 sahi, par 100 − 6 kar diya (number base se kam maan liya). Left 112, 100 se bada hai, isliye number bhi 100 se zyada hai.
  • (c) sahi ✅: Right 36 = 6 × 6, to zyada 6, number 106. Check left: 106 + 6 = 112 ✅.
  • (d) galat ❌: 112 to left part hai, number nahi.

Q9. Kis square ke liye base 100 (base se zyada) wali trick sabse acchi?

(a) 99²
(b) 106²
(c) 160²
(d) 75²
Jawab aur reason dekho

Sahi jawab: (b) 106²

  • (a) galat ❌: 99 base se kam hai: Post 13.1 wala (98 | 01).
  • (b) sahi ✅: 106 sirf 6 zyada: 112 | 36 → 11236.
  • (c) galat ❌: 160, 100 se 60 zyada: right 3600, bahut bada.
  • (d) galat ❌: 75 ke liye Chapter 6 (7 × 8 = 56, peeche 25).

Q10. 105² = ? (do methods se same)

(a) 11025
(b) 10525
(c) 110250
(d) 10025
Jawab aur reason dekho

Sahi jawab: (a) 11025

  • (a) sahi ✅: Base: 110 | 25. Chapter 6: 10 × 11 = 110, peeche 25 → 11025.
  • (b) galat ❌: 10525: left mein 5 add karna bhool gaye (105 | 25).
  • (c) galat ❌: 110250: right 25 ke peeche 0 laga diya (110 | 250); right sirf 2 boxes ka.
  • (d) galat ❌: 10025: left mein zyada add karne ki jagah subtract kar diya (105 − 5 = 100 → 100 | 25).

— 📌 charumam.com

🧠 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. 101² = 10201. 1001² kya hoga? (Hint: base 1000, right 3 boxes)

Solution dekho

1001 → 1 zyada. Left 1001 + 1 = 1002. Right 1 × 1 = 1 → 3 boxes: 001. Jawab 1002 | 001 → 1002001. Pattern: 11² = 121, 101² = 10201, 1001² = 1002001.

Check: 1001 × 1001 = 1001000 + 1001 = 1002001 ✅

Q2. Ek square hall mein har taraf 105 tiles. Total tiles? Do methods se karo.

Solution dekho

Base 100: 5 zyada → 110 | 25 → 11025. Chapter 6 (Post 6.2): 10 × 11 = 110, peeche 25 → 11025. Dono same!

Check: 105 = 100 + 5, to 105 × 105 = 105 × 100 + 105 × 5 = 10500 + 525 = 11025 ✅ (105 × 5 = 100 × 5 + 5 × 5 = 500 + 25 = 525)

Q3. 19² base 10 se karo. Carry kitna aaya?

Solution dekho

19 → 9 zyada. Left 19 + 9 = 28. Right 9 × 9 = 81. Base 10 mein 1 box: 81 = 80 + 1, yaani 8 dahai aur 1. 8 dahai left mein: 28 + 8 = 36. Jawab 36 | 1 → 361. Carry 8 tha!

Check: 19 × 19 = 190 + 171 = 361 ✅ (19 ko 10 baar = 190; 19 ko 9 baar = 190 − 19 = 171)

Q4. Kaunsa bada: 104² ya 103 × 105? Kitna?

Solution dekho

104² = 10816. 103 × 105 (Chapter 7): left 103 + 5 = 108, right 3 × 5 = 15 → 10815. To 104² 1 bada.

Check: 10816 − 10815 = 1 ✅ (Post 13.1 wala pattern, upar ki taraf bhi!)

— 📌 charumam.com

🎯 Exam Mein Kahan Kaam Aayega?

ExamKahan kaam aayega
School exam (Class 6–8)Squares ki table (11² se 20²), square ka area.
Maths OlympiadPattern (11², 101², 1001²), reverse sawaal (□² = 11236).
NTSE / MATNumber series: 121, 144, 169, … agla?
SSC / Bank (aage)Simplification mein 104², 112² 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.

— 📌 charumam.com

👩‍🏫 CharuMam Ka Tip

Mantra: “kam ho to subtract karo, zyada ho to add karo, corner hamesha square.” 11² se 19² ki list ek baar is trick se khud banao: yahi list Chapter 14 (square root) mein kaam aayegi! ⬆️

— 📌 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:

104² = ?
Trick se kaise karein (dekho)

104, base 100 se 4 zyada (104 − 100 = 4). Left: 104 + 4 = 108. Right: 4 × 4 = 16 (do boxes, poore). Jawab 108 | 16 → 10816 ✅

Before timeAfter timeKitna tez?
____ second____ secondBefore − 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. 🚀

— 📌 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 mistake hone pe samjhayega:

“Mujhe 12 square ke sawaal do: 4 base 10 ke (11 se 19), 6 base 100 se zyada wale (101 se 110), aur 2 carry wale (111 se 115). Ek-ek karke poochho. Mistake pe samjhao: zyada nikaalo, left = number + zyada, right = zyada × zyada, 2 boxes, kam ho to 0, zyada ho to carry.”

— 📌 charumam.com

❓ FAQs (Students Ke Asli Sawaal)

1. Base se kam wale mein subtract karte hain, base se zyada wale mein add karte kyun?

Chitra dekho: base se kam wale mein patti cut karke hatate hain (subtract karna), base se zyada wale mein pattiyan judti hain (add karna).

2. Right hamesha add karte hain?

Haan, square mein corner hamesha add hota hai: deficit × deficit ya zyada × zyada.

3. 125² is trick se?

Ho sakta hai. 125 → 25 zyada. Left: 125 + 25 = 150. Right: 25 × 25 = 625 (Chapter 6: 2 × 3 = 6, peeche 25). Right ko sirf 2 boxes, aur 625 mein 3 digit: 625 = 600 + 25, yaani 6 saikde aur 25. Woh 6 saikde left mein bhejo (carry 6): 150 + 6 = 156 → 150 | 625 → 156 | 25 → 15625. Par Chapter 6 (12 × 13 = 156, peeche 25) zyada aasaan.

4. 5 digit ka jawab kaise padhein?

10816 = das hazaar aath sau solah.

5. Agle post mein kya hai?

Post 13.3: base 1000 (996², 1007²) aur kaunsi trick kab.

— 📌 charumam.com

🟢 Recap: Aaj Kya Seekha

  • Zyada = number − base.
  • Left = number + zyada (base se zyada wale mein add karo).
  • Right = zyada × zyada, base ke zero jitne boxes.
  • Kam digit → 0 (103² = 10609). Zyada → carry (112² = 12544).
  • Base 10: 11² se 19² (15² = 225).
  • Chitra: do pattiyan + corner.

— charumam.com 🎯

💛 Post Achha Laga? Like, Share, Subscribe, Follow!

Agar aaj kuch naya samajh aaya, to post ko like kijiye, apne doston aur class ke group mein share kijiye, aur daily nayi trick ke liye humein subscribe aur follow kijiye! 🙏

— 📌 charumam.com

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