Distributive Property Kya Hai? | Class 8 Maths Book 1 Ch 6

📘 Class 8 · Ganita Prakash Book 1Ch 6 · We Distribute, Yet Things MultiplyPost 1 / 7⏱ 25–30 min

Distributive Property Kya Hai? 🧱

IS POST MEIN

Pehle Yeh Aata Hai?Kahani Concept Deep Concept Examples Real Life Galti 10 MCQs FAQs Practice HOTS Olympiad Recap

✅ Pehle Yeh Aata Hai?

Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:

5 × 10 = ?
Like terms kya hote?
a × a ko kaise likhte?

Answers dekho

50
Same letter-numbers wale (3x, 5x)
a²

Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Book 1 Ch 2 (exponents) aur Ch 5 dekh lijiye.

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🎬 Kahani: Tiffin Ki Dukaan Ka Hisaab

Sonu ki maa tiffin service chalaati hain. Har tiffin mein 6 roti aur 4 sabzi-katori. Aaj 5 tiffin bane. Sonu ne gina: “6 + 4 = 10 cheezein, 10 × 5 = 50.” 🍱

Maa ne alag gina: “Roti 5 × 6 = 30, katori 5 × 4 = 20, kul 50.” Dono ka jawab same!

Kya yeh sanyog hai? Nahi, yahi hai distributive property: guna ko tukdon mein baant do, phir jodo. Aaj iska “kyun” chitra se samjhenge. 🧠

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📖 Concept: Guna = Area; Area Ko Tukdon Mein Baanto

💡 Is post ki ek badi baat: a × (b + c) ek rectangle ka area hai jiski chaudaai (b + c). Use do tukdon mein kaato: a × b aur a × c. Tukdon ka jod = poora area. Isliye a(b + c) = ab + ac, har number ke liye. Poora chapter isi ek idea pe khada hai.

Step 1: Chitra: dots ya area

5 panktiyon aur 6 + 4 kholon wala rectangle, do hisson mein: 5 × 6 = 30 aur 5 × 4 = 20, kul 50

5 panktiyaan, 6 + 4 = 10 khol. Poora: 5 × 10 = 50. Tukde: 30 + 20 = 50 ✅

WHY kaam karta? Rectangle kaatne se koi dot gayab nahi hota, na naya aata. Bas do baar mein gine.

Step 2: Ghataav pe bhi: a(b − c) = ab − ac

7 × 48 = 7 × (50 − 2) = 350 − 14 = 336.

WHY? 50 khol ka rectangle banao, phir 2 khol ka tukda hata do. Hataaya hua area = 7 × 2.

Step 3: Ulta bhi: (a + b)c = ac + bc

(a + b)c = c(a + b) (guna ka kram badal sakte) = ca + cb = ac + bc.

Rectangle ko khada kaato ya leta, baat ek hi.

Step 4: Bracket mein kai terms

4a(3a − 2b + 5) = 4a × 3a − 4a × 2b + 4a × 5 = 12a² − 8ab + 20a.

Har term ko bahar wale se guna karo, sign ke saath. Phir dekho: koi like terms? Yahan nahi (a², ab, a sab alag).

Step 5: Ulta chalna: common nikaalo (factor)

6x + 9y = 3(2x + 3y). Distributive ko ulta padha: dono terms mein 3 common.

Kab kaunsa? Hisaab aasaan karna ho to kholo (7 × 48). Expression chhota karna ho ya divisibility dikhani ho to common nikaalo (Ch 5 ka 6a + 6b = 6(a + b) yaad?).

🪜 Atak Gaye? Aise Socho

Bracket ke bahar wala number/letter pehchaano.
Andar ke har term pe ek-ek teer banao.
Har teer pe guna karo, sign saath le jaao.
Like terms jodo.
Ek chhoti value (a = 1, b = 2) daal ke dono taraf check karo.

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🔐 Deep Concept: Distributive: Jod Aur Guna Ka Pul

Jod aur guna do alag operation hain. Distributive property batati hai woh aapas mein kaise milte: guna, jod ke upar “baant” jaata hai.

Par ulta nahi! Jod guna pe nahi baant-ta: 2 + (3 × 4) = 14, par (2 + 3) × (2 + 4) = 30 ✗. Aur guna guna pe bhi nahi: 2 × (3 × 4) = 24, par (2 × 3) × (2 × 4) = 48 ✗.

Itihaas: Brahmagupta (628 CE) ne Brāhmasphuṭasiddhānta mein ise khaṇḍa-guṇanam (tukdon mein guna) kaha: “Guna karne wale ko tukdon mein todo jinka jod wahi ho, har tukde se guna karo, aur jodo.” Euclid ne isse geometry mein, Aryabhata ne algebra mein istemaal kiya.

Unifying idea: mushkil guna = aasaan tukde. Aage ke saare identities, (a + b)², fast multiplication, sab isi ek pul pe chalte hain. 🎯

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✏️ Solved Examples

Example 1 (Easy): 9 × 104 mann mein.

9 × (100 + 4) = 900 + 36 = 936.

Check: 936 ÷ 9 = 104 ✅

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Example 2 (Medium): −3p(2p − 5q + 1) kholo.

−3p × 2p = −6p², −3p × (−5q) = +15pq, −3p × 1 = −3p → −6p² + 15pq − 3p.

Check: p = 1, q = 1: baayan −3(2 − 5 + 1) = 6; daayan −6 + 15 − 3 = 6 ✅

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Example 3 (NCERT style): 17 × 6 + 17 × 4 sabse aasaan tareeke se.

Ulta distributive: 17 × (6 + 4) = 17 × 10 = 170.

Check: 102 + 68 = 170 ✅

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Sonu aur maa: dono sahi the. Sonu ne poora rectangle gina (5 × 10), maa ne tukde (30 + 20). Agle din 12 tiffin: maa ne turant kaha “72 roti, 48 katori, 120 cheezein”! 🍱

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🏠 Distributive Property Roz Ki Zindagi Mein

🛒 Shopping: 4 kilo seb ₹98 kilo: 4 × 100 − 4 × 2 = ₹392.
🎒 School kit: 30 bachche, har ek ko 2 copy + 1 pen: 30 × 2 + 30 × 1 = 90 cheezein.
🏠 Ghar ka farsh: L-shape kamra = do rectangles ke area ka jod.
🍬 Mithai ke dabbe: 8 dabbe, har ek mein 12 laddoo + 6 barfi = 8 × 18 = 144.
💰 Salary: 25 din × (₹400 + ₹50 bonus) = 25 × 400 + 25 × 50.

Dukaandaar roz yeh karte: “₹99 × 6? 600 − 6 = ₹594!” Distributive = mental maths ka superpower. 💪

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🧭 Kab Use Karein, Kab Nahi

Situation	Kya karein	Kyun
Hisaab aasaan karna	✅ Kholo (7 × 48 = 7 × 50 − 7 × 2)	Paas ka gol number
Expression chhota karna	✅ Common nikaalo	Ulta distributive
Kai terms bracket mein	✅ Har term ko guna	Sign dhyaan
Jod guna pe?	❌ Nahi baant-ta	2 + 3 × 4
Guna guna pe?	❌ Nahi	2(3 × 4) ≠ 6 × 8

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❌→✅ Worked Mistake

Sawaal: 5(2x + 3) kholo.

❌ Galat: 10x + 3. Sirf pehle term ko guna kiya!

✅ Sahi: 5 × 2x + 5 × 3 = 10x + 15. Check: x = 1: 5 × 5 = 25; 10 + 15 = 25 ✅

Doosri common galti: −2(a − 3) = −2a − 6. Sign! −2 × (−3) = +6 → −2a + 6.

🧠 Rule of thumb: Bahar wala andar ke HAR term se milta hai, sign ke saath.

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📝 10 MCQs

Q1. 6(a + 7) = ?

(a) 6a + 7

(b) 6a + 42

(c) a + 42

(d) 13a

Solution dekho

Sahi jawab: (b)

(a) galat: 7 ko bhi 6 se guna karna tha.
(b) sahi: 6 × a + 6 × 7 = 6a + 42 ✅
(c) galat: a ko bhi 6 se guna karna tha.
(d) galat: 6 aur 7 jodna galat hai.

Q2. 8 × 97 aasaan tareeke se?

(a) 800 − 24

(b) 800 − 3

(c) 720 + 7

(d) 8 × 100 + 3

Solution dekho

Sahi jawab: (a)

(a) sahi: 8 × (100 − 3) = 800 − 24 ✅
(b) galat: 3 ko bhi 8 se guna karna tha.
(c) galat: Yeh tukde 97 nahi banaate.
(d) galat: 97 = 100 − 3, plus nahi.

Q3. −4(x − 2) = ?

(a) −4x − 8

(b) −4x + 8

(c) 4x − 8

(d) −4x − 2

Solution dekho

Sahi jawab: (b)

(a) galat: −4 × −2 = +8, minus nahi.
(b) sahi: −4 × −2 = +8, to −4x + 8 ✅
(c) galat: x ka sign ulta ho gaya.
(d) galat: 2 ko bhi −4 se guna karna tha.

Q4. 2a(3a + b) = ?

(a) 6a + 2ab

(b) 6a² + 2ab

(c) 5a² + 2ab

(d) 6a² + b

Solution dekho

Sahi jawab: (b)

(a) galat: 2a × 3a = 6a², sirf 6a nahi.
(b) sahi: Sahi jawab: 2a × 3a + 2a × b ✅
(c) galat: 2 × 3 = 6, jod nahi, isliye yeh galat.
(d) galat: b ko bhi 2a se guna karna tha.

Q5. 12x + 18 = ?

(a) 6(2x + 3)

(b) 6(2x + 18)

(c) 3(4x + 18)

(d) 12(x + 18)

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi jawab: Common 6 nikaala ✅
(b) galat: 18 ÷ 6 = 3 hona chahiye.
(c) galat: 18 ÷ 3 = 6 hona chahiye.
(d) galat: 18 ÷ 12 = 18 nahi hota.

Q6. 2 + (3 × 4) = (2 + 3)(2 + 4)?

(a) Haan

(b) Nahi

(c) Kabhi-kabhi

(d) Sirf integers

Solution dekho

Sahi jawab: (b)

(a) galat: 14 aur 30 alag hain, isliye yeh galat.
(b) sahi: Jod guna pe nahi baant-ta ✅
(c) galat: Yahan to bilkul nahi, isliye yeh galat.
(d) galat: Integers mein bhi nahi.

Q7. Khaṇḍa-guṇanam kisne likha?

(a) Brahmagupta

(b) Euclid

(c) Newton

(d) Pascal

Solution dekho

Sahi jawab: (a)

(a) sahi: Brāhmasphuṭasiddhānta (628 CE) ✅
(b) galat: Euclid ne geometry mein use kiya.
(c) galat: Newton bahut baad mein aaye.
(d) galat: Pascal ka yeh kaam nahi tha.

Q8. 23 × 7 + 23 × 3?

(a) 230

(b) 161

(c) 69

(d) 253

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi jawab: 23 × (7 + 3) = 230 ✅
(b) galat: 161 sirf 23 × 7 hai, isliye yeh galat.
(c) galat: 69 sirf 23 × 3 hai, isliye yeh galat.
(d) galat: 253 = 23 × 11 hota, 23 × 10 nahi.

Q9. 3(2x − y + 4) mein y ka term?

(a) −3y

(b) −y

(c) 3y

(d) −3

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi jawab: 3 × (−y) = −3y ✅
(b) galat: y ko bhi 3 se guna karna tha.
(c) galat: Sign minus hai, plus nahi.
(d) galat: Letter y bhi chahiye, isliye yeh galat.

Q10. Rectangle 5 × (6 + 4) ke tukde?

(a) 30 + 20

(b) 11 + 9

(c) 30 + 4

(d) 5 + 10

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi jawab: 5 × 6 aur 5 × 4 ✅
(b) galat: Yeh jod hain, guna nahi.
(c) galat: 4 ko bhi 5 se guna karna tha.
(d) galat: Yeh area nahi, lambaai hai.

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❓ FAQs (Bachchon Ke Asli Sawaal)

1. Distributive property ka naam kyun?

Bahar ka guna andar ke har term pe “baant” (distribute) diya jaata hai.

2. Kya negative numbers pe bhi?

Haan: x(y + z) = xy + xz sab integers (aur fractions) pe sach.

3. Bhaag pe?

(a + b) ÷ c = a ÷ c + b ÷ c sach. Par c ÷ (a + b) ≠ c ÷ a + c ÷ b.

4. Identity kya hai?

Aisi barabari jo letters ki har value pe sach ho, jaise a(b + c) = ab + ac.

5. Chitra kyun zaroori?

Chitra se “kyun” dikhta: area tukdon mein baant-ta hai. Formula bhool jao, chitra yaad rahega.

6. Aage kya?

Do brackets ka guna (a + m)(b + n): Post 2 — Guna Mein Badlaav: Increments.

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🏋️ Practice Zone (12 Sawaal)

Q1. Easy 7(x + 5) kholo.

Solution dekho

7x + 35. Check: x = 1 → 42 ✅

Q2. Easy 6 × 103 mann mein.

Solution dekho

600 + 18 = 618. Check ✅

Q3. Easy 15a + 20 mein common nikaalo.

Solution dekho

5(3a + 4). Check: a = 1 → 35 ✅

Q4. Easy 9 × 49?

Solution dekho

450 − 9 = 441. Check: 21² = 441 ✅

Q5. Medium −5(2y − 3) kholo.

Solution dekho

−10y + 15. Check: y = 1 → 5 ✅

Q6. Medium 3x(x − 4y + 2) kholo.

Solution dekho

3x² − 12xy + 6x. Check: x = 1, y = 1 → −3 ✅

Q7. Medium 46 × 25 + 54 × 25?

Solution dekho

25 × 100 = 2500. Check ✅

Q8. Medium 2(3a + 1) + 4(a − 2) saral karo.

Solution dekho

6a + 2 + 4a − 8 = 10a − 6. Check: a = 1 → 4 ✅

Q9. NCERT style (2/3)(9a − 6b + 12) kholo.

Solution dekho

6a − 4b + 8. Check: a = b = 1 → 10 ✅

Q10. NCERT style 999 × 7 mann mein.

Solution dekho

7000 − 7 = 6993. Check ✅

Q11. NCERT style a(b − c) + b(c − a) + c(a − b) = ?

Solution dekho

Sab kat jaata → 0. Check: a = 1, b = 2, c = 3 → −1 + 4 − 3 = 0 ✅

Q12. NCERT style x² + 5x mein common nikaalo.

Solution dekho

x(x + 5). Check: x = 2 → 14 ✅

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🧠 HOTS (Sochne Wale Sawaal)

Q1. Kya (a + b) ÷ c = a ÷ c + b ÷ c? Aur c ÷ (a + b)?

Solution dekho

Pehla haan, doosra nahi: 12 ÷ (2 + 4) = 2, par 6 + 3 = 9. Check ✅

Q2. L-shape: 8 × 10 ka rectangle jisme 3 × 4 ka kona kata. Area do tareeke se?

Solution dekho

80 − 12 = 68; ya tukde 8 × 6 + 5 × 4 = 48 + 20 = 68. Check ✅

Q3. n(n + 1) − n(n − 1) = ?

Solution dekho

n² + n − n² + n = 2n. Check: n = 3 → 12 − 6 = 6 ✅

Q4. Kya distributive se dikhao ki 3 lagaataar numbers ka jod 3 ka multiple?

Solution dekho

(n − 1) + n + (n + 1) = 3n = 3 × n. Check: 4 + 5 + 6 = 15 ✅

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🏆 Olympiad Corner

Q1. 37 × 27 + 37 × 73?

Solution dekho

3700.

Q2. 125 × 32 mann mein?

Solution dekho

125 × 8 × 4 = 4000.

Q3. a = 2, b = 3: 4a(b − a) + 2b(a + b)?

Solution dekho

8 + 30 = 38.

Q4. 9999 × 12?

Solution dekho

119988.

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💗 CharuMam Motivation

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🟢 Master Recap

💡 Chaabi: a × (b + c) ek rectangle ka area hai jiski chaudaai (b + c). Use do tukdon mein kaato: a × b aur a × c. Tukdon ka jod = poora area. Isliye a(b + c) = ab + ac, har number ke liye. Poora chapter isi ek idea pe khada hai.
a(b + c) = ab + ac (area ka chitra).
a(b − c) = ab − ac.
Bracket ke har term ko guna, sign ke saath.
Ulta: common nikaalo.
Jod guna pe nahi baant-ta.
Brahmagupta: khaṇḍa-guṇanam.

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👉 Aage Kya?

Ek number badhe to guna kitna badhta? Agla post: Post 2 — Guna Mein Badlaav: Increments.

← Pichla chapter: Ch 5 — Number Play 📚 Chapter 6 ke saare posts Agla: Post 2 — Guna Mein Badlaav: Increments →

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✅ Pehle Yeh Aata Hai?

🎬 Kahani: Tiffin Ki Dukaan Ka Hisaab

📖 Concept: Guna = Area; Area Ko Tukdon Mein Baanto

Step 1: Chitra: dots ya area

Step 2: Ghataav pe bhi: a(b − c) = ab − ac

Step 3: Ulta bhi: (a + b)c = ac + bc

Step 4: Bracket mein kai terms

Step 5: Ulta chalna: common nikaalo (factor)

🪜 Atak Gaye? Aise Socho

🔐 Deep Concept: Distributive: Jod Aur Guna Ka Pul

✏️ Solved Examples

🏠 Distributive Property Roz Ki Zindagi Mein

🧭 Kab Use Karein, Kab Nahi

❌→✅ Worked Mistake

📝 10 MCQs

❓ FAQs (Bachchon Ke Asli Sawaal)

🏋️ Practice Zone (12 Sawaal)

🧠 HOTS (Sochne Wale Sawaal)

🏆 Olympiad Corner

💗 CharuMam Motivation

🟢 Master Recap

👉 Aage Kya?

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