Post 14.1 aur 14.2 mein sirf perfect squares ka root nikaala. Par 50, 200, 10 jaise number kisi poore number ka square nahi hain. Inka root whole number nahi, par estimate laga sakte hain! Aaj chaar step: (1) kin do squares ke beech, (2) sabse paas whole number, (3) half check (7.5² = 56.25), aur (4) ek decimal tak (√50 ≈ 7.1). Calculator ki zaroorat nahi.
⏱ Before Timer: Pehle Purane Method Se
Pehle “try karke” method (doosre number pe, taaki timer wale sawaal ka jawab pehle na dikhe). √20: estimates se point wale (decimal) number try karo aur multiply karo (point ka matlab neeche DECIMAL box mein):
- 4 × 4 = 16: chhota. 5 × 5 = 25: bada.
- 4.5 × 4.5 = 20.25 (45 × 45 = 2025, point do jagah): thoda bada.
- 4.4 × 4.4 = 19.36 (44 × 44 = 1936, point do jagah): thoda chhota.
- 20.25 − 20 = 0.25 aur 20 − 19.36 = 0.64: 20.25 zyada paas. To √20 ≈ 4.5.
Kai decimal multiply, aur pata nahi kahan se shuru karein. Ab timer wala sawaal isi tarah try karke kijiye.
Ab phone ya watch mein stopwatch (time measure karne wali watch) chalu kijiye. Neeche wala sawaal isi “try karke” method se kijiye (upar jaisa: estimate lagao, multiply karke dekho), copy-pencil ke saath. Jaise hi jawab mil jaaye, stopwatch rok dijiye aur apna time likh lijiye.
Mera time: ______ second
Jawab dekho
√50 ≈ 7.1
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 14.1 ka √ aur perfect squares ki list, Post 13.4 ka “agla square”, aur Chapter 6 ke 5 wale square (75² = 5625). Naya: decimal (point) wale number, comparison ke word (chhota, bada, beech), aur ≈. Sab neeche ke boxes mein. Naye word:
📖 Word jo is post mein aayenge
- Estimate (estimate): sahi jawab ke bahut paas ka jawab, jab bilkul sahi nikaalna mushkil ho.
- ≈ (lagbhag equal): √50 ≈ 7.1 padho “root 50 lagbhag 7.1”.
- Decimal point: 7.1 mein beech wala bindu: 7 aur ek tenth part (box dekho).
- Distance: do numbers ka difference, bade mein se chhota subtract karo: 50 aur 49 ki distance 1.
- Sabse paas whole number: woh whole number jo sahi jawab ke sabse paas ho: √50 ke liye 7.
- Metre, square metre: metre length ki ikaai hai (lagbhag ek long step). Square metre area ki ikaai: 1 metre × 1 metre ke square ki jagah.
📏 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.
🔄 Square root = “reverse square” (√ ka matlab)
Square mein hum number dete hain aur poochte hain “iska square kya?” 7 → 7 × 7 = 49. Square root mein reverse poochte hain: “Kis number ka square 49 hai?” Jawab 7, kyunki 7 × 7 = 49. Isliye square root ko reverse square samjho (Hindi mein “varg-mool”: mool matlab jad, yaani woh number jisse square uga).
Sign √ (square root ka sign): square root dikhane ke liye number ke aage √ lagate hain. Yeh ek sahi-ka-tick (✓) jaisa dikhta hai, jiski long roof number ke upar faili hoti hai. √49 padho “49 ka square root” ya “root 49”. √49 = 7 ka matlab: 7 × 7 = 49.
| Square (aage) | Square root (reverse) |
|---|---|
| 3 × 3 = 9, yaani 3² = 9 | √9 = 3 |
| 7 × 7 = 49, yaani 7² = 49 | √49 = 7 |
| 10 × 10 = 100, yaani 10² = 100 | √100 = 10 |
| 12 × 12 = 144, yaani 12² = 144 | √144 = 12 |
Check hamesha aasaan: jo root mila, use usi se multiply karo. Wahi number wapas aaye to root sahi. √144 = 12? 12 × 12 = 144 ✅.
Dhyaan: hum sirf 0, 1, 2, 3 … jaise numbers (0 ya usse bade) ki baat kar rahe hain. √ ka sign hamesha aisa hi root batata hai: √49 = 7.
↔️ Comparison ke word: chhota, bada, beech mein
Is chapter mein numbers ki comparison hum shabdon mein likhenge, koi sign nahi:
- “49, 50 se chhota hai”: 49 kam hai.
- “64, 50 se bada hai”: 64 zyada hai.
- “16, 22 se chhota ya equal hai”: 16 ya to 22 se kam hai, ya bilkul 22.
- “50, 49 aur 64 ke beech hai”: 50, 49 se bada aur 64 se chhota.
🔸 Decimal (point) wale numbers: 7.5 ka matlab
Kabhi jawab do whole numbers ke beech mein hota hai. Use likhne ke liye point (.) lagate hain, jise decimal point kehte hain. Point ke left wala part poore number, aur right wala part “ek ke pieces”.
- 7.5 padho “saat point paanch” ya “saadhe saat”: 7 aur half. Jaise ₹7.50 = 7 rupees 50 money (50 money = half rupaya).
- 0.1 = ek ka 10 mein se 1 part (tenth part). Jaise ek roti ke 10 equal pieces karo, unme se 1. To 7.1 = 7 aur ek tenth part, 7 se thoda sa zyada.
- 0.25 = ek ka chautha part (one-fourth), jaise 25 money = ₹1 ka chautha part.
Kaun bada? Point ke left wala part pehle dekho: 8.1 bada hai 7.9 se. Left same ho to right dekho: 7.5 bada hai 7.1 se (5 pieces, 1 pieces se zyada).
Point wale number ka square kaise? Point hata kar multiply karo, phir jawab mein point waapas lagao, dono numbers ke point ke baad jitne digit the, utne right se gin kar. 7.5 × 7.5: point remove karo → 75 × 75 = 5625 (Post 6.1: 7 × 8 = 56, peeche 25). Dono 7.5 mein point ke baad 1-1 digit, total 2. To 5625 mein right se 2 digit gin kar point: 56.25.
Yeh kyun sahi hai? 7.5 asal mein 75 ka tenth part hai (75 ÷ 10; ÷ matlab equal parts mein divide karna, Chapter 5). Multiply mein do baar tenth part liya, to jawab 5625 ka hundredth part: 5625 ÷ 100 = 56.25. (100 se divide karna = point do jagah left.)
Point wale numbers ka subtract karna: money ki tarah socho, point ke neeche point rakh kar. ₹50.41 − ₹50 = ₹0.41 (sirf 41 money). ₹10 − ₹9.61 = ₹0.39 (39 money), kyunki ₹9.61 + ₹0.39 = ₹10 (61 money aur 39 money milkar 100 money, yaani ₹1).
🔁 Chhota sa warm-up
- 7² aur 8² = ?
- 64 − 50 = ?
- 75² = ?
Answers dekho
- 49 aur 64
- 14
- 5625 (Chapter 6: 7 × 8 = 56, peeche 25)
Pehla jawab hi aaj ka step 1 hai: 50, 49 aur 64 ke beech!
— 📌 charumam.com
🎬 Story: Square Farming
Village mein Kabir ke paas 200 square metre land ka square field banana hai. Baadh (fence) lagane wale Ramesh chacha ne poocha: “Har kinaara kitne metre ka hoga? Wire utni hi laaunga.”
Kabir ne list dekhi: 14² = 196, 15² = 225. “200 to list mein hai hi nahi!” Didi Noor boli: “Koi baat nahi. 200, 196 ke bahut paas hai. To kinaara 14 se thoda sa zyada.”
Phir Noor ne 14.1 ka square nikaala: 198.81 (point remove karke 141² = 140² + 140 + 141 = 19600 + 281 = 19881, point do jagah). Aur 14.2 ka: 201.64 (142² = 19881 + 141 + 142 = 20164, point do jagah). “200 inke beech, 198.81 ke zyada paas. Kinaara lagbhag 14.1 metre!” Ramesh chacha wire le aaye, aur bilkul pura pada. 🎯
— 📌 charumam.com
🪄 Trick Step by Step: Beech, Paas, Half, Ek Decimal
💡 Is post ki ek badi baat: (1) Beech: jo number diya hai, woh kin do perfect squares ke beech hai? (50: 49 = 7² aur 64 = 8²) To root bhi 7 aur 8 ke beech. (2) Paas: dono squares se distance nikaalo (50 − 49 = 1, 64 − 50 = 14); jiske paas, wahi sabse paas whole number (7). (3) Half: saadhe wale number ka square (jaise 7.5²): chhota whole number × agla number (7 × 8 = 56), peeche .25 lagao, to 56.25 (Chapter 6). Jo number diya hai, woh is se chhota ho to root saadhe wale se chhota (50, 56.25 se chhota, to √50, 7 aur 7.5 ke beech); bada ho to saadhe wale se bada. (4) Ek decimal: “agla square” se 7.1², 7.2² … ek-ek karke nikaalo (7.1² = 50.41, 7.2² = 51.84), aur jaise hi square diye hue number se bada ho jaaye, ruk jao. 50 ke liye 7.1² = 50.41 pe hi ruk gaye. Phir jo paas ho woh chuno: √50 ≈ 7.1.
Step 1: Kin do squares ke beech?
√50: list mein 50 se chhota sabse bada square 49 = 7², aur bada sabse chhota 64 = 8². 50, 49 aur 64 ke beech hai, to √50, 7 aur 8 ke beech. (Bada number, bada root.)
Step 2: Distance se sabse paas whole number
50 − 49 = 1, aur 64 − 50 = 14. 50, 49 ke bahut paas, to √50 ka sabse paas whole number 7.
Step 3: Half check: 7.5²
7.5²: point remove karo → 75² = 5625 (Chapter 6: 7 × 8 = 56, peeche 25). Point do jagah (DECIMAL box) → 56.25. 50, 56.25 se chhota hai, to √50, 7 aur 7.5 ke beech.
Step 4: Ek decimal: agla square
7.1²: point remove karo → 71² = 70² + 70 + 71 = 4900 + 70 + 71 = 5041 (Post 13.4). Point do jagah → 50.41. 50.41 pehle hi 50 se bada! To √50, 7.0 aur 7.1 ke beech.
Step 5: Paas wala chuno, check karo
50 − 49 = 1, aur 50.41 − 50 = 0.41. 50, 50.41 ke zyada paas. To √50 ≈ 7.1. Check: 7.1 × 7.1 = 50.41, 50 ke bahut paas ✅
Poora example, step-by-step: √200
Step 1 (beech): 14² = 196 aur 15² = 225. 200, 196 aur 225 ke beech hai, to √200, 14 aur 15 ke beech. Distance: 200 − 196 = 4, aur 225 − 200 = 25. 200, 196 ke zyada paas, to sabse paas whole number 14.
Step 2 (half check): 14.5² nikaalo. Point remove karo: 145² (Chapter 6: 14 × 15 = 210, peeche 25) = 21025. Point do jagah: 14.5² = 210.25. 200, 210.25 se chhota hai, to √200, 14.5 se chhota: 14 aur 14.5 ke beech.
Step 3 (ek decimal): 14.0 se 0.1-0.1 increase karo. Har square “agla square” se (square + number + agla number), point hata kar: 141² = 19600 + 140 + 141 = 19881, to 14.1² = 198.81; 142² = 19881 + 141 + 142 = 20164, to 14.2² = 201.64. To √200, 14.1 aur 14.2 ke beech.
Step 4 (kiske paas): 200 − 198.81 = 1.19, aur 201.64 − 200 = 1.64. 200, 198.81 ke zyada paas. To √200 ≈ 14.1 (≈ matlab “lagbhag”).
Aur ek: √80 (bade sire se reverse chalo)
Step 1 (beech): 8² = 64 aur 9² = 81. 80, 64 aur 81 ke beech hai, to √80, 8 aur 9 ke beech. Distance: 80 − 64 = 16, aur 81 − 80 = 1. 80, 81 ke zyada paas, to sabse paas whole number 9.
Step 2 (half check): 8.5² nikaalo. Point remove karo: 85² (Chapter 6: 8 × 9 = 72, peeche 25) = 7225. Point do jagah: 8.5² = 72.25. 80, 72.25 se bada hai, to √80, 8.5 se bada: 8.5 aur 9 ke beech.
Step 3 (ek decimal): 80 bade sire 9.0² = 81.00 ke zyada paas hai, to 9.0 se 0.1-0.1 subtract karo. Har square “pichla square” se (square − number − pichla number), point hata kar: 89² = 8100 − 90 − 89 = 7921, to 8.9² = 79.21. To √80, 8.9 aur 9.0 ke beech.
Step 4 (kiske paas): 80 − 79.21 = 0.79, aur 81.00 − 80 = 1.00. 80, 79.21 ke zyada paas. To √80 ≈ 8.9 (≈ matlab “lagbhag”).
Dhyaan: jab diya hua number bade square ke zyada paas ho (jaise 80, jo 81 ke paas hai), to bade sire se pichla square (square − number − pichla number) se neeche aao. Kam step lagte hain.
Half wale squares (Chapter 6 se, yaad rakhne layak)
| Number | Point hata kar | Square |
|---|---|---|
| 1.5 | 15² (1 × 2 = 2, peeche 25) = 225 | 1.5² = 2.25 |
| 2.5 | 25² (2 × 3 = 6, peeche 25) = 625 | 2.5² = 6.25 |
| 3.5 | 35² (3 × 4 = 12, peeche 25) = 1225 | 3.5² = 12.25 |
| 4.5 | 45² (4 × 5 = 20, peeche 25) = 2025 | 4.5² = 20.25 |
| 5.5 | 55² (5 × 6 = 30, peeche 25) = 3025 | 5.5² = 30.25 |
| 6.5 | 65² (6 × 7 = 42, peeche 25) = 4225 | 6.5² = 42.25 |
| 7.5 | 75² (7 × 8 = 56, peeche 25) = 5625 | 7.5² = 56.25 |
| 8.5 | 85² (8 × 9 = 72, peeche 25) = 7225 | 8.5² = 72.25 |
| 9.5 | 95² (9 × 10 = 90, peeche 25) = 9025 | 9.5² = 90.25 |
🆘 Atak gaye? Aise socho
- Beech: list mein diye hue number se chhota aur bada square (50: 49 aur 64).
- Distance: diya hua number − chhota square, aur bada square − diya hua number (50 − 49 = 1, 64 − 50 = 14).
- Half: saadhe wale number ka square = chhota whole number × agla number, peeche .25 (7.5²: 7 × 8 = 56 → 56.25).
- Ek decimal: agla (ya pichla) square, point do jagah.
- ≈ likho, = nahi: √50 ≈ 7.1.
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Beech wala rule kyun? Bada number, bada square: 7, 8 se chhota, to 7² bhi 8² se chhota. Reverse bhi sach: square bada, to root bada. 50, 49 aur 64 ke beech, to root bhi 7 aur 8 ke beech.
Chitra se: 50 dots ka square banao. 7 × 7 = 49 mein poora square ban jaata hai, 1 dot bachta hai. 8 × 8 = 64 ke liye 14 dots kam. To square ki bhuja 7 se thodi si zyada: √50 ≈ 7.1.
Estimate ek decimal tak kyun kaafi hai?
7.1² = 50.41 aur 50 mein sirf 0.41 ka difference, yaani 41 money jitna. Exam mein “lagbhag” ya “ek decimal tak” poochha jaata hai. Zyada sahi chahiye to 7.07, 7.08 jaise do decimal tak bhi isi tarah ja sakte ho (aage ki class mein “long division” wala method bhi hai).
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Method Se Check)
Example 1: √10 lagbhag kitna? (ek decimal tak)
Step 1 (beech): 3² = 9 aur 4² = 16. 10, 9 aur 16 ke beech hai, to √10, 3 aur 4 ke beech. Distance: 10 − 9 = 1, aur 16 − 10 = 6. 10, 9 ke zyada paas, to sabse paas whole number 3.
Step 2 (half check): 3.5² nikaalo. Point remove karo: 35² (Chapter 6: 3 × 4 = 12, peeche 25) = 1225. Point do jagah: 3.5² = 12.25. 10, 12.25 se chhota hai, to √10, 3.5 se chhota: 3 aur 3.5 ke beech.
Step 3 (ek decimal): 3.0 se 0.1-0.1 increase karo. Har square “agla square” se (square + number + agla number), point hata kar: 31² = 900 + 30 + 31 = 961, to 3.1² = 9.61; 32² = 961 + 31 + 32 = 1024, to 3.2² = 10.24. To √10, 3.1 aur 3.2 ke beech.
Step 4 (kiske paas): 10 − 9.61 = 0.39, aur 10.24 − 10 = 0.24. 10, 10.24 ke zyada paas. To √10 ≈ 3.2 (≈ matlab “lagbhag”).
✔️ Normal method se check: 3.2 × 3.2 = 10.24, 10 ke paas ✅ (3.1 × 3.1 = 9.61 thoda door)
Example 2: √20 lagbhag kitna? (ek decimal tak)
Step 1 (beech): 4² = 16 aur 5² = 25. 20, 16 aur 25 ke beech hai, to √20, 4 aur 5 ke beech. Distance: 20 − 16 = 4, aur 25 − 20 = 5. 20, 16 ke zyada paas, to sabse paas whole number 4.
Step 2 (half check): 4.5² nikaalo. Point remove karo: 45² (Chapter 6: 4 × 5 = 20, peeche 25) = 2025. Point do jagah: 4.5² = 20.25. 20, 20.25 se chhota hai, to √20, 4.5 se chhota: 4 aur 4.5 ke beech.
Step 3 (ek decimal): 20 bade sire 4.5² = 20.25 ke zyada paas hai, to 4.5 se 0.1-0.1 subtract karo. Har square “pichla square” se (square − number − pichla number), point hata kar: 44² = 2025 − 45 − 44 = 1936, to 4.4² = 19.36. To √20, 4.4 aur 4.5 ke beech.
Step 4 (kiske paas): 20 − 19.36 = 0.64, aur 20.25 − 20 = 0.25. 20, 20.25 ke zyada paas. To √20 ≈ 4.5 (≈ matlab “lagbhag”).
✔️ Normal method se check: 4.5 × 4.5 = 20.25, 20 ke bahut paas ✅
Example 3: √30 lagbhag kitna? (ek decimal tak)
Step 1 (beech): 5² = 25 aur 6² = 36. 30, 25 aur 36 ke beech hai, to √30, 5 aur 6 ke beech. Distance: 30 − 25 = 5, aur 36 − 30 = 6. 30, 25 ke zyada paas, to sabse paas whole number 5.
Step 2 (half check): 5.5² nikaalo. Point remove karo: 55² (Chapter 6: 5 × 6 = 30, peeche 25) = 3025. Point do jagah: 5.5² = 30.25. 30, 30.25 se chhota hai, to √30, 5.5 se chhota: 5 aur 5.5 ke beech.
Step 3 (ek decimal): 30 bade sire 5.5² = 30.25 ke zyada paas hai, to 5.5 se 0.1-0.1 subtract karo. Har square “pichla square” se (square − number − pichla number), point hata kar: 54² = 3025 − 55 − 54 = 2916, to 5.4² = 29.16. To √30, 5.4 aur 5.5 ke beech.
Step 4 (kiske paas): 30 − 29.16 = 0.84, aur 30.25 − 30 = 0.25. 30, 30.25 ke zyada paas. To √30 ≈ 5.5 (≈ matlab “lagbhag”).
✔️ Normal method se check: 5.5 × 5.5 = 30.25 ✅ (5.4 × 5.4 = 29.16)
Example 4: Ek square plot ka area 90 square metre. Har kinaara lagbhag?
Kinaara × kinaara = 90, to kinaara = √90.
Step 1 (beech): 9² = 81 aur 10² = 100. 90, 81 aur 100 ke beech hai, to √90, 9 aur 10 ke beech. Distance: 90 − 81 = 9, aur 100 − 90 = 10. 90, 81 ke zyada paas, to sabse paas whole number 9.
Step 2 (half check): 9.5² nikaalo. Point remove karo: 95² (Chapter 6: 9 × 10 = 90, peeche 25) = 9025. Point do jagah: 9.5² = 90.25. 90, 90.25 se chhota hai, to √90, 9.5 se chhota: 9 aur 9.5 ke beech.
Step 3 (ek decimal): 90 bade sire 9.5² = 90.25 ke zyada paas hai, to 9.5 se 0.1-0.1 subtract karo. Har square “pichla square” se (square − number − pichla number), point hata kar: 94² = 9025 − 95 − 94 = 8836, to 9.4² = 88.36. To √90, 9.4 aur 9.5 ke beech.
Step 4 (kiske paas): 90 − 88.36 = 1.64, aur 90.25 − 90 = 0.25. 90, 90.25 ke zyada paas. To √90 ≈ 9.5 (≈ matlab “lagbhag”).
✔️ Normal method se check: 9.5 × 9.5 = 90.25, 90 ke bahut paas ✅
— 📌 charumam.com
💡 CharuMam Ki Smart Trick
Yeh estimate CharuMam ke “Pehle Dekho” (dekh kar) rule se shuru hota hai: pehle poora part (7) pakdo, phir bacha hua (50 − 49 = 1) dekho. Aur half check Chapter 6 ki “Ek Zyada” (ek zyada se multiply) trick wala hi hai: 75² ke liye 7 × 8. Aage ek aur “Pair Wala” method bina estimate ke bhi root deta hai.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| “Lagbhag” ya “kin do numbers ke beech” | ✅ Step 1–2 | Sirf list chahiye. |
| “Ek decimal tak” | ✅ Step 1–4 | Half check, phir agla square. |
| Perfect square (2209) | ➡️ Post 14.2 | Wahan bilkul sahi root milta hai. |
| Sawaal mein options diye hain | ✅ Step 1–2 kaafi | √50 ke options 6.1, 7.1, 8.1 → beech se hi 7.1. |
| Bada number (√5000) | ✅ Step 1–4 (thoda long) | 70² = 4900 aur 71² = 5041: √5000, 70 aur 71 ke beech. Half: 70.5² = 4970.25 (70 × 71 = 4970, peeche .25), 5000 us se bada. Agla square: 70.6² = 4984.36, 70.7² = 4998.49, 70.8² = 5012.64. 5000, 4998.49 ke zyada paas, to √5000 ≈ 70.7. Aur bade numbers ke dhokhe Post 14.4 mein. |
| Do decimal tak | ⚠️ Long | Aage long division. |
— 📌 charumam.com
❌→✅ Common Mistakes
Mistake 1: Root ko half samajh liya
Sawaal: √50
❌ Galat: 25.
✅ Sahi: 25 × 25 = 625, bahut bada. 50, 49 aur 64 ke beech, to √50 ≈ 7.1.
Mistake 2: Distance galat taraf
Sawaal: √80 ka sabse paas whole number
❌ Galat: 8 (kyunki 8² = 64 “chhota wala”).
✅ Sahi: 80 − 64 = 16, aur 81 − 80 = 1. 81 ke paas, to 9.
Mistake 3: Point galat jagah
Sawaal: 7.5²
❌ Galat: 562.5 (point ek jagah).
✅ Sahi: 7.5 aur 7.5 mein point ke baad total 2 digit, to 5625 → 56.25.
Mistake 4: “=” likh diya
Sawaal: √50 ka estimate
❌ Galat: √50 = 7.1
✅ Sahi: √50 ≈ 7.1, kyunki 7.1 × 7.1 = 50.41, bilkul 50 nahi.
🧠 Rule of thumb: Beech (do squares), paas (distance), half (saadhe wala square, jaise 7.5² = 56.25), ek decimal (agla square, jaise 7.1² = 50.41). Aur ≈ likho.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 sawaal: pehle 8 mein “kin do whole numbers ke beech”, baaki 7 mein “sabse paas whole number”. Target: 5 minute.
- √50: kin do whole numbers ke beech?
- √20: kin do whole numbers ke beech?
- √70: kin do whole numbers ke beech?
- √95: kin do whole numbers ke beech?
- √130: kin do whole numbers ke beech?
- √150: kin do whole numbers ke beech?
- √250: kin do whole numbers ke beech?
- √300: kin do whole numbers ke beech?
- √26: sabse paas whole number?
- √60: sabse paas whole number?
- √80: sabse paas whole number?
- √110: sabse paas whole number?
- √140: sabse paas whole number?
- √170: sabse paas whole number?
- √350: sabse paas whole number?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- 7 aur 8
- 4 aur 5
- 8 aur 9
- 9 aur 10
- 11 aur 12
- 12 aur 13
- 15 aur 16
- 17 aur 18
- 5
- 8
- 9
- 10
- 12
- 13
- 19
Kaise kiya (kuch):
12² = 144 aur 13² = 169. 150, 144 aur 169 ke beech hai, to √150, 12 aur 13 ke beech. Distance: 150 − 144 = 6, aur 169 − 150 = 19. 150, 144 ke zyada paas, to sabse paas whole number 12.
18² = 324 aur 19² = 361. 350, 324 aur 361 ke beech hai, to √350, 18 aur 19 ke beech. Distance: 350 − 324 = 26, aur 361 − 350 = 11. 350, 361 ke zyada paas, to sabse paas whole number 19.
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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.
- 1.5² = ?
- 2.5² = ?
- 3.5² = ?
- 4.5² = ?
- 5.5² = ?
- 6.5² = ?
- 7.5² = ?
- 8.5² = ?
- 9.5² = ?
- 10.5² = ?
| 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
- 2.25
- 6.25
- 12.25
- 20.25
- 30.25
- 42.25
- 56.25
- 72.25
- 90.25
- 110.25
Dhyaan: saadhe wale number ka square = chhota whole number × agla number, peeche .25. 6.5²: 6 × 7 = 42 → 42.25. 10.5²: 10 × 11 = 110 → 110.25.
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. √50 kin do numbers ke beech?
Jawab aur reason dekho
Sahi jawab: (c) 7 aur 8
- (a) galat ❌: Mistake: bade square 64 = 8² ko chhota end maan liya, to ek number aage chale gaye (8 aur 9).
- (b) galat ❌: Mistake: root ko half samjha (50 ÷ 2 = 25).
- (c) sahi ✅: 50, 49 aur 64 ke beech: to √50, 7 aur 8 ke beech.
- (d) galat ❌: Mistake: chhote square 49 = 7² ko bada end maan liya, to ek number neeche chale gaye (6 aur 7).
Q2. √80 ka sabse paas whole number?
Jawab aur reason dekho
Sahi jawab: (b) 9
- (a) galat ❌: 10: agla square 81 chhod ke 100 = 10² le liya.
- (b) sahi ✅: 81 − 80 = 1, bahut paas.
- (c) galat ❌: 8: distance galat taraf: chhote square 64 ko paas maana, jabki 80 − 64 = 16.
- (d) galat ❌: 40: root ko half samjha (80 ÷ 2).
Q3. 7.5² = ?
Jawab aur reason dekho
Sahi jawab: (a) 56.25
- (a) sahi ✅: 7 × 8 = 56, peeche .25.
- (b) galat ❌: 49.25: 7 × 8 ki jagah 7 × 7 = 49 kiya.
- (c) galat ❌: 562.5: 5625 mein point do ki jagah ek hi jagah lagaya.
- (d) galat ❌: 5.625: 5625 mein point do ki jagah teen jagah lagaya.
Q4. √20 ≈ ?
Jawab aur reason dekho
Sahi jawab: (d) 4.5
- (a) galat ❌: 4: step 2 (sabse paas whole number) pe hi ruk gaye, ek decimal nahi nikaala.
- (b) galat ❌: 5: distance galat taraf: 25 ko paas maana (25 − 20 = 5), jabki 20 − 16 = 4.
- (c) galat ❌: 10: root ko half samjha (20 ÷ 2).
- (d) sahi ✅: 4.5² = 20.25, 20 ke bahut paas.
Q5. 7.1² = ?
Jawab aur reason dekho
Sahi jawab: (c) 50.41
- (a) galat ❌: 5.041: 5041 mein point do ki jagah teen jagah lagaya.
- (b) galat ❌: 504.1: 5041 mein point do ki jagah ek hi jagah lagaya.
- (c) sahi ✅: 71² = 5041, point do jagah: 50.41.
- (d) galat ❌: 49.1: 7² = 49 ke peeche .1 chipka diya; poora multiply nahi kiya.
Q6. √200 ≈ ?
Jawab aur reason dekho
Sahi jawab: (b) 14.1
- (a) galat ❌: 15: distance galat taraf: 225 ko paas maana (225 − 200 = 25), jabki 200 − 196 = 4.
- (b) sahi ✅: 14.1² = 198.81, 200 ke paas.
- (c) galat ❌: 14: step 2 (sabse paas whole number) pe hi ruk gaye, ek decimal nahi nikaala.
- (d) galat ❌: 100: root ko half samjha (200 ÷ 2).
Q7. Kaunsa sabse sahi likha hai?
Jawab aur reason dekho
Sahi jawab: (a) √50 ≈ 7.1
- (a) sahi ✅: 7.1² = 50.41: lagbhag, to ≈.
- (b) galat ❌: Mistake: ≈ ki jagah = likha; 7.1² bilkul 50 nahi.
- (c) galat ❌: Mistake: sirf chhote square 49 = 7² ka root likh diya; 7² = 49, 50 nahi.
- (d) galat ❌: Mistake: root ko half samjha (50 ÷ 2).
Q8. √30 ke liye kaunsa sach hai?
Jawab aur reason dekho
Sahi jawab: (d) √30, 5 aur 5.5 ke beech
- (a) galat ❌: Mistake: half check reverse padha: 30, 30.25 se chhota hai, to √30, 5.5 se chhota.
- (b) galat ❌: Mistake: 36 = 6² ko chhota end maan liya; 36, 30 se bada hai.
- (c) galat ❌: Mistake: root ko half samjha (30 ÷ 2).
- (d) sahi ✅: 30, 25 aur 30.25 ke beech (5.5² = 30.25).
Q9. Kaunsa number perfect square NAHI, par uska root 9 aur 10 ke beech hai?
Jawab aur reason dekho
Sahi jawab: (c) 90
- (a) galat ❌: Mistake: sire wala number chun liya: 100 = 10².
- (b) galat ❌: Mistake: root ki jagah number ka half le liya: sire wale 100 ka half 50. Par 50, 49 = 7² aur 64 = 8² ke beech hai, to √50, 7 aur 8 ke beech; 9 aur 10 ke nahi.
- (c) sahi ✅: 90, 81 aur 100 ke beech.
- (d) galat ❌: Mistake: sire wala number chun liya: 81 = 9², perfect square.
Q10. √10 ≈ ?
Jawab aur reason dekho
Sahi jawab: (b) 3.2
- (a) galat ❌: 3: step 2 (sabse paas whole number) pe hi ruk gaye, ek decimal nahi nikaala.
- (b) sahi ✅: 3.2² = 10.24 (0.24 door), 3.1² = 9.61 (0.39 door).
- (c) galat ❌: 3.1: kiske paas galat chuna: 9.61 ko paas maana, jabki 10 − 9.61 = 0.39 aur 10.24 − 10 = 0.24.
- (d) galat ❌: 5: root ko half samjha (10 ÷ 2).
— 📌 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. √300 ek decimal tak?
Solution dekho
Step 1 (beech): 17² = 289 aur 18² = 324. 300, 289 aur 324 ke beech hai, to √300, 17 aur 18 ke beech. Distance: 300 − 289 = 11, aur 324 − 300 = 24. 300, 289 ke zyada paas, to sabse paas whole number 17.
Step 2 (half check): 17.5² nikaalo. Point remove karo: 175² (Chapter 6: 17 × 18 = 306, peeche 25) = 30625. Point do jagah: 17.5² = 306.25. 300, 306.25 se chhota hai, to √300, 17.5 se chhota: 17 aur 17.5 ke beech.
Step 3 (ek decimal): 300 bade sire 17.5² = 306.25 ke zyada paas hai, to 17.5 se 0.1-0.1 subtract karo. Har square “pichla square” se (square − number − pichla number), point hata kar: 174² = 30625 − 175 − 174 = 30276, to 17.4² = 302.76; 173² = 30276 − 174 − 173 = 29929, to 17.3² = 299.29. To √300, 17.3 aur 17.4 ke beech.
Step 4 (kiske paas): 300 − 299.29 = 0.71, aur 302.76 − 300 = 2.76. 300, 299.29 ke zyada paas. To √300 ≈ 17.3 (≈ matlab “lagbhag”).
Check: 17.3 × 17.3 = 299.29, 300 ke bahut paas ✅
Q2. √2 ek decimal tak? (Ek square ka area 2 hai to bhuja?)
Solution dekho
Step 1 (beech): 1² = 1 aur 2² = 4. 2, 1 aur 4 ke beech hai, to √2, 1 aur 2 ke beech. Distance: 2 − 1 = 1, aur 4 − 2 = 2. 2, 1 ke zyada paas, to sabse paas whole number 1.
Step 2 (half check): 1.5² nikaalo. Point remove karo: 15² (Chapter 6: 1 × 2 = 2, peeche 25) = 225. Point do jagah: 1.5² = 2.25. 2, 2.25 se chhota hai, to √2, 1.5 se chhota: 1 aur 1.5 ke beech.
Step 3 (ek decimal): 2 bade sire 1.5² = 2.25 ke zyada paas hai, to 1.5 se 0.1-0.1 subtract karo. Har square “pichla square” se (square − number − pichla number), point hata kar: 14² = 225 − 15 − 14 = 196, to 1.4² = 1.96. To √2, 1.4 aur 1.5 ke beech.
Step 4 (kiske paas): 2 − 1.96 = 0.04, aur 2.25 − 2 = 0.25. 2, 1.96 ke zyada paas. To √2 ≈ 1.4 (≈ matlab “lagbhag”).
Check: 1.4 × 1.4 = 1.96 ✅
Q3. Kitne whole numbers aise hain jinka square root 10 aur 11 ke beech ho (bilkul 10 ya 11 nahi)?
Solution dekho
10² = 100, 11² = 121. Root 10 aur 11 ke beech, to number 100 aur 121 ke beech: 100 se bada aur 121 se chhota, yaani 101 se 120. Count: 120 − 101 + 1 = 20. (Post 13.4: 100 aur 121 ka difference 21; beech mein 21 − 1 = 20 number.)
Check: 10 + 11 = 21, aur 21 − 1 = 20 ✅
Q4. √1000 ek decimal tak?
Solution dekho
Step 1 (beech): 31² = 961 aur 32² = 1024. 1000, 961 aur 1024 ke beech hai, to √1000, 31 aur 32 ke beech. Distance: 1000 − 961 = 39, aur 1024 − 1000 = 24. 1000, 1024 ke zyada paas, to sabse paas whole number 32.
Step 2 (half check): 31.5² nikaalo. Point remove karo: 315² (Chapter 6: 31 × 32 = 992, peeche 25) = 99225. Point do jagah: 31.5² = 992.25. 1000, 992.25 se bada hai, to √1000, 31.5 se bada: 31.5 aur 32 ke beech.
Step 3 (ek decimal): 31.5 se 0.1-0.1 increase karo. Har square “agla square” se (square + number + agla number), point hata kar: 316² = 99225 + 315 + 316 = 99856, to 31.6² = 998.56; 317² = 99856 + 316 + 317 = 100489, to 31.7² = 1004.89. To √1000, 31.6 aur 31.7 ke beech.
Step 4 (kiske paas): 1000 − 998.56 = 1.44, aur 1004.89 − 1000 = 4.89. 1000, 998.56 ke zyada paas. To √1000 ≈ 31.6 (≈ matlab “lagbhag”).
Check: 31.6 × 31.6 = 998.56 ✅
— 📌 charumam.com
🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 8) | “Lagbhag square root”, “kin do whole numbers ke beech”, square ka kinaara. |
| Maths Olympiad | Ek range mein kitne numbers ka root, √2 jaise estimates. |
| NTSE / MAT | Options mein se sabse paas wala chuno. |
| SSC / Bank (aage) | Approximation sawaal: √50 ≈ 7.1 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.
— 📌 charumam.com
👩🏫 CharuMam Ka Tip
Mantra: “beech, paas, half, ek decimal.” Half wale squares (1.5² se 9.5²) Chapter 6 se 2 second mein bante hain. Options wale sawaal mein sirf step 1–2 kaafi. Agle post mein Chapter 14 ke tricky sawaal! 🎯
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⏱ After Timer: Ab Trick Se!
Wahi sawaal jo shuru mein kiya tha. Stopwatch phir chalu kijiye, aur is baar aaj ki trick se kijiye:
Trick se kaise karein (dekho)
7² = 49 aur 8² = 64, to √50 7 aur 8 ke beech, 7 ke bahut paas (50 − 49 = 1). 7.1² = 50.41 (71² = 4900 + 70 + 71 = 5041, point do jagah). To √50 ≈ 7.1 ✅
| 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. √50 ka bilkul sahi jawab kya hai?
7.0710… point ke baad digit kabhi khatam nahi hote. Isliye estimate (≈ 7.1) likhte hain.
2. Half check kyun?
Ek hi multiply se pata chal jaata hai ki 7.0–7.5 ya 7.5–8.0 mein dhoondhna hai. Kaam half.
3. Agla square decimal pe kaise?
Point hata kar: 7.1 → 71. 71² = 70² + 70 + 71 = 5041. Point do jagah: 50.41.
4. Har baar 4 step?
Nahi. “Kin do ke beech” ya “sabse paas poora” ho to step 1–2 kaafi.
5. Agle post mein kya hai?
Post 14.4: Chapter 14 ke tricky sawaal: perfect square identify karo, √4000 ka dhokha, area ke sawaal.
— 📌 charumam.com
🟢 Recap: Aaj Kya Seekha
- Beech: 50, 49 aur 64 ke beech, to √50, 7 aur 8 ke beech.
- Paas: 50 − 49 = 1, 64 − 50 = 14, to 7.
- Half: 7.5² = 56.25 (7 × 8 = 56).
- Ek decimal: 7.1² = 50.41, to √50 ≈ 7.1.
- ≈ likho, = nahi.
- Upar ke paas ho to pichla square se neeche aao (√80 ≈ 8.9).
— charumam.com 🎯
👉 Aage Kya?
Ab Chapter 14 ka last post: Post 14.4 mein tricky sawaal: perfect square hai ya nahi, √4000 ka dhokha, area aur “kitna add karein” wale sawaal!
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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