✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- a(b + c) = ?
- (−2) × (−3) = ?
- 34 × 18 = ?
Answers dekho
- ab + ac
- +6
- 612
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 1 — Distributive Property Kya Hai? dekh lijiye.
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🎬 Kahani: Khet Ka Naya Naksha
Kisaan Ramu ka khet 34 m × 18 m hai (612 varg m). Sarkar ne kaha: “Ek taraf 1 m badha lo.” Ramu ka beta poochhta: “Lambaai badhaayein ya chaudaai? Aur agar dono 1-1 m badhe to?” 🌾
Padosi bola: “Dono 1 m badhe to 2 varg m zyada!” Ramu ko shak hua.
Kitna badhta hai sach mein? Aur agar ek badhe aur doosra ghate to? Aaj chitra se samjhenge, ratenge nahi. 📐
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📖 Concept: Naya Rectangle = Purana + Judi Hui Patti
💡 Is post ki ek badi baat: (a + m)(b + n) ka rectangle chaar tukdon mein bant-ta hai: ab + an + mb + mn. Purana guna ab; baaki teen tukde = badlaav. Har bracket ka har term doosre ke har term se milta hai.
Step 1: Ek number 1 badhe
a(b + 1) = ab + a. 34 × 19 = 612 + 34 = 646.
WHY? Ek khol (column) juda jisme a dot. Lambaai 34 → 34 naye varg m.
Agar a badhe: (a + 1)b = ab + b → 35 × 18 = 612 + 18 = 630.
Step 2: Dono 1-1 badhein
(a + 1)(b + 1) = ab + a + b + 1. 35 × 19 = 612 + 34 + 18 + 1 = 665.
WHY +1? Ek patti a ki, ek b ki, aur kone ka chhota varg 1 × 1. Padosi ne yahi bhoola (aur pattiyaan bhi!).
Check: 35 × 19 = 665 ✅
Step 3: Ek badhe, doosra ghate
(a + 1)(b − 1) = ab + b − a − 1.
34 → 35, 18 → 17: 612 + 18 − 34 − 1 = 595. Ghat gaya! Check: 35 × 17 = 595 ✅
Kab badhega? b − a − 1 > 0, yaani chhote wale ko badhaao: (18 + 1)(34 − 1) = 19 × 33 = 627 > 612.
Step 4: General: Identity 1
(a + m)(b + n) = ab + an + mb + mn. Ghataav ho to m ya n negative lo: sign integer niyam se khud aa jaata.
(a + 2)(b − 3) = ab − 3a + 2b − 6. (a − u)(b − v) = ab − av − ub + uv (minus × minus = plus).
Step 5: Do tareeke se kholo
Tareeka A (ek bracket ko ek term maano): (a + 1)(b + 1) = (a + 1)b + (a + 1)1 = ab + b + a + 1.
Tareeka B (chaar teer / grid): har term ko har term se: a·b, a·1, 1·b, 1·1.
Kab kaunsa? A shuruaat mein samajhne ke liye. B tez aur bade brackets (teen terms) ke liye: grid banao, koi jodi chhootegi nahi.
🪜 Atak Gaye? Aise Socho
- Do brackets? Grid banao: ek ke terms upar, doosre ke side mein.
- Har khaane mein guna (sign ke saath).
- Like terms jodo.
- Badlaav poochha ho: purana ab hata do.
- a = 2, b = 3 jaisi chhoti values se check.
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🔐 Deep Concept: Kone Ka Chhota Varg: Sabse Bhoola Hua Tukda
Jab dono taraf badhao, log aksar sirf do pattiyaan ginte (an + mb) aur kone ka tukda mn bhool jaate. Chitra mein woh saaf dikhta!
Kyun badlaav “ab” pe nirbhar? (a + 1)(b + 1) − ab = a + b + 1: bade numbers mein 1 badhaane ka asar zyada. 100 × 100 → 101 × 101: +201! 2 × 2 → 3 × 3: sirf +5.
Ek badhe, ek ghate: 20 × 20 = 400, 21 × 19 = 399, 22 × 18 = 396. Jod same rakhne pe guna sabse bada jab dono barabar. Isliye same ghere (perimeter) mein square ka area sabse zyada!
Unifying idea: har bracket-guna = rectangle ka area tukdon mein. Signs, negative numbers, kai terms, sab ek grid mein. 🎯
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✏️ Solved Examples
Example 1 (Easy): a × b mein a ko 3 badhao, b ko 2. Badlaav?
(a + 3)(b + 2) − ab = 2a + 3b + 6.
Check: a = 4, b = 5: 7 × 7 − 20 = 29; 8 + 15 + 6 = 29 ✅
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Example 2 (Medium): (x − 4)(y + 5) kholo.
xy + 5x − 4y − 20.
Check: x = 6, y = 1: 2 × 6 = 12; 6 + 30 − 4 − 20 = 12 ✅
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Example 3 (NCERT style): Ek number 3 badhe, doosra 2 ghate, guna wahi: teen udaharan.
(a + 3)(b − 2) = ab − 2a + 3b − 6 → 3b − 2a = 6. a = 6, b = 6: 9 × 4 = 36 = 6 × 6 ✅; a = 12, b = 10: 15 × 8 = 120 ✅; a = 3, b = 4: 6 × 2 = 12 ✅
Check: teeno mein 3b − 2a = 6 ✅
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Ramu ka faisla: chaudaai (18) 1 badhaao → +34; lambaai (34) badhaao → +18. Ramu ne chhoti side ko badhaaya: 34 varg m mile! Dono badhte to +53, padosi ke “2” se kahin zyada. 🌾
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🏠 Guna Ke Badlaav Roz Ki Zindagi Mein
- 🏠 Kamra badhaana: 12 × 10 ft se 13 × 11: 12 + 10 + 1 = 23 varg ft zyada.
- 📺 TV screen: lambaai-chaudaai 1-1 inch badhe to area kaafi zyada badhta.
- 🎟️ Ticket aur bheed: 200 log × ₹50; ₹5 badhaaya, 10 log kam: (200 − 10)(50 + 5) = 10450 vs 10000.
- 🌾 Khet ki baad: same baad (perimeter) mein square khet sabse zyada fasal.
- 🧮 Mental check: 21 × 19 = 20² − 1 = 399.
Ticket wala hisaab dukaandaar roz karte: daam badhaao to grahak ghatte; guna (kamaai) kab badhta, yahi identity batati.
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Ek number +1 | ✅ + doosra number | a(b + 1) = ab + a |
| Dono +1 | ✅ + a + b + 1 | Kona mat bhoolo |
| Ek +, ek − | ✅ Identity 1, n negative | Sign niyam |
| Teen terms ka bracket | ✅ Grid | Koi jodi na chhoote |
| Badlaav poochha | ✅ Purana ab hatao | Baaki tukde |
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❌→✅ Worked Mistake
Sawaal: (a + 2)(b + 5) kholo.
❌ Galat: ab + 10. Sirf pehle-pehle aur aakhri-aakhri ko guna kiya!
✅ Sahi: chaar tukde: ab + 5a + 2b + 10. Check: a = 1, b = 1: 3 × 6 = 18; 1 + 5 + 2 + 10 = 18 ✅ (galat wala 11 deta)
Doosri common galti: (a − 3)(b − 2) mein aakhri term −6 likhna. Minus × minus = +6.
🧠 Rule of thumb: Do brackets = chaar tukde. Ginti karo: 2 × 2 = 4 terms (like terms jodne se pehle).
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📝 10 MCQs
Q1. a(b + 1) − ab = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Ek khol juda, usme a dot ✅
- (b) galat: b tab badhta jab a badhe.
- (c) galat: 1 sirf kone ka tukda hai.
- (d) galat: Dono tab jab dono badhein.
Q2. (a + 1)(b + 1) − ab = ?
Solution dekho
Sahi jawab: (b)
- (a) galat: Kone ka 1 × 1 bhool gaye.
- (b) sahi: Sahi jawab: Do pattiyaan + kona ✅
- (c) galat: Pattiyaan bhi to judti hain.
- (d) galat: ab to purana hissa hai.
Q3. 35 × 19 agar 34 × 18 = 612?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 612 + 34 + 18 + 1 = 665 ✅
- (b) galat: Sirf 2 jodna galat hai.
- (c) galat: 646 to sirf 34 × 19 hai.
- (d) galat: 630 to sirf 35 × 18 hai.
Q4. (x + 3)(y − 2) = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Chaar tukde, sign sahi ✅
- (b) galat: Beech ke tukde chhoot gaye.
- (c) galat: Beech ke sign ulte ho gaye.
- (d) galat: 3 × (−2) = −6 hota, +6 nahi.
Q5. (a − 2)(b − 5) ka aakhri term?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Minus × minus = plus 10 ✅
- (b) galat: Do minus ka guna plus hota.
- (c) galat: 2 aur 5 ka guna chahiye, jod nahi.
- (d) galat: Jod nahi, guna 10 chahiye.
Q6. 20 × 20, 21 × 19, 22 × 18: sabse bada?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Barabar hone pe guna max ✅
- (b) galat: 399 hai, 400 se kam, isliye yeh galat.
- (c) galat: 396 hai, aur bhi kam, isliye yeh galat.
- (d) galat: 400, 399, 396 alag hain.
Q7. Kab (a + 1)(b − 1) > ab?
Solution dekho
Sahi jawab: (a)
- (a) sahi: b − a − 1 > 0 chahiye ✅
- (b) galat: Tab to guna ghat-ta hai.
- (c) galat: 34, 18 mein ghata tha.
- (d) galat: 19 × 33 mein badha tha.
Q8. Do 2-term brackets ke guna mein kitne tukde?
Solution dekho
Sahi jawab: (c)
- (a) galat: Har term har term se milta.
- (b) galat: Teen nahi, kona bhi gino.
- (c) sahi: Sahi jawab: 2 × 2 = 4 tukde ✅
- (d) galat: 6 tab jab 2 aur 3 terms.
Q9. (2a + 1)(a + 3) = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 2a² + 6a + a + 3 ✅
- (b) galat: Beech ke tukde bhool gaye.
- (c) galat: a × 1 wala tukda chhoota.
- (d) galat: Guna ki jagah jod kar diya.
Q10. 100 × 100 se 101 × 101: badlaav?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 100 + 100 + 1 = 201 ✅
- (b) galat: Pattiyaan bhool gaye, isliye yeh galat.
- (c) galat: Ek hi patti gini, isliye yeh galat.
- (d) galat: Kone ka 1 chhoot gaya.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Identity 1 kya hai?
(a + m)(b + n) = ab + an + mb + mn: do brackets ke guna ka general niyam.
2. Negative numbers pe chalta?
Haan. Integers pe distributive property sach, isliye identity bhi.
3. Ghataav ke liye alag formula?
Zaroorat nahi: m ya n negative lo. Signs khud sahi aate.
4. Grid kaise banaate?
Ek bracket ke terms upar, doosre ke baayein. Har khaane mein dono ka guna.
5. Teen terms × do terms?
3 × 2 = 6 tukde, phir like terms jodo.
6. Aage kya?
Ise mental maths mein: 11, 101 se tez guna: Post 3 — Fast Multiplication Ki Tricks.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy a × b mein b ko 1 badhao: badlaav?
Solution dekho
+a. Check: 5 × 7 → 5 × 8: +5 ✅
Q2. Easy (x + 2)(y + 3) kholo.
Solution dekho
xy + 3x + 2y + 6. Check: x = y = 1 → 12 ✅
Q3. Easy 26 × 15 = 390. 27 × 16?
Solution dekho
390 + 26 + 15 + 1 = 432. Check ✅
Q4. Easy (p − 1)(q + 4) kholo.
Solution dekho
pq + 4p − q − 4. Check: p = 2, q = 1 → 5 ✅
Q5. Medium (3a − 2)(a + 5) kholo.
Solution dekho
3a² + 15a − 2a − 10 = 3a² + 13a − 10. Check: a = 1 → 6 ✅
Q6. Medium (a − 3)(b − 4) − ab?
Solution dekho
−4a − 3b + 12. Check: a = 5, b = 6 → 2 × 2 − 30 = −26; −20 − 18 + 12 = −26 ✅
Q7. Medium 40 × 30 se 41 × 29: badha ya ghata?
Solution dekho
b − a − 1 = −11 → ghata, 11. Check: 1189 vs 1200 ✅
Q8. Medium (y + 7)(y − 7) kholo.
Solution dekho
y² − 7y + 7y − 49 = y² − 49. Check: y = 8 → 15 ✅
Q9. NCERT style (a + b − 2)(a + 1) kholo.
Solution dekho
a² + a + ab + b − 2a − 2 = a² + ab − a + b − 2. Check: a = b = 1 → 0 ✅
Q10. NCERT style Ek 4 badhe, doosra 3 ghate, guna same: ek udaharan.
Solution dekho
(a + 4)(b − 3) = ab ⇔ 4b − 3a = 12 → a = 4, b = 6: 8 × 3 = 24 = 4 × 6 ✅. Check ✅
Q11. NCERT style −(2y + 3)(y − 4) kholo.
Solution dekho
−(2y² − 5y − 12) = −2y² + 5y + 12. Check: y = 1 → 15 ✅
Q12. NCERT style Calendar ke 2 × 2 khaane a, a + 1, a + 7, a + 8: tirchhe guna ka farak?
Solution dekho
a(a + 8) aur (a + 1)(a + 7) = a² + 8a + 7 → farak hamesha 7. Check: 3 × 11 = 33, 4 × 10 = 40 ✅
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. 15 × 45 aur 16 × 44: bina guna bada kaun?
Solution dekho
(15 + 1)(45 − 1) = 675 + 45 − 15 − 1 = 704 → 16 × 44 bada. Check: 704 > 675 ✅
Q2. (a + b)(c + d)(e + f) mein kitne tukde?
Solution dekho
2 × 2 × 2 = 8. Check: teeno 1 + 1 → 8 = 1 × 8 ✅
Q3. Ek rectangle 20 × 12: perimeter same rakh ke sabse bada area?
Solution dekho
16 × 16 = 256 (240 se zyada). Check: perimeter 64 ✅
Q4. (k + 1)(k + 2) − (k + 3) hamesha?
Solution dekho
k² + 3k + 2 − k − 3 = k² + 2k − 1 → hamesha 2 nahi (k = 1 → 2, k = 2 → 7). Check ✅
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🏆 Olympiad Corner
Q1. 49 × 51?
Solution dekho
50² − 1 = 2499.
Q2. a + b = 10, ab = 21: (a + 1)(b + 1)?
Solution dekho
21 + 10 + 1 = 32.
Q3. (x − 1)(x + 1)(x² + 1)?
Solution dekho
x⁴ − 1.
Q4. 101 × 99 − 100²?
Solution dekho
−1.
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💗 CharuMam Motivation
Ramu ke bete ne chitra se dekha: kone ka chhota varg! Har baar jab “kitna badhega” poocho, rectangle bana lo: jawab saamne. 📐
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🟢 Master Recap
- 💡 Chaabi: (a + m)(b + n) ka rectangle chaar tukdon mein bant-ta hai: ab + an + mb + mn. Purana guna ab; baaki teen tukde = badlaav. Har bracket ka har term doosre ke har term se milta hai.
- a(b + 1) = ab + a.
- (a + 1)(b + 1) = ab + a + b + 1.
- (a + 1)(b − 1) = ab + b − a − 1.
- (a + m)(b + n) = ab + an + mb + mn.
- Ghataav: negative m/n lo.
- Jod same, guna max jab barabar.
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👉 Aage Kya?
Is identity se 11, 101, 99 ke saath bijli jaisa guna: Post 3 — Fast Multiplication Ki Tricks.
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬