✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 6 notebook ₹180 ki, to 10 kitne ki?
- 5 mazdoor 12 din, to 10 mazdoor kitne din?
- 1/3 + 1/2 = ?
Answers dekho
- ₹300 (direct)
- 6 din (inverse)
- 5/6
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 1 — Proportion Ka Recap aur Post 8 — Inverse Proportion Ke Sawaal dobara dekh lijiye, phir yahan aaiye.
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🎬 Kahani: Quiz Show: “Direct Ya Inverse?”
School ke maths quiz mein ek naya round tha: “Direct ya Inverse?” Host sawaal padhta, buzzer dabaake jawab dena tha.
Pehla sawaal: “Zyada log, raashan kitne din?” Arjun ne turant buzzer dabaaya: “Inverse!” ✅
Doosra: “Do nal ek saath, ek 3 ghante, ek 2 ghante. Saath mein kitna?” Arjun bola, “3 + 2 = 5 ghante!” ❌ Poori class hans padi. 😅
Do nal milke zyada time kaise lenge? Aaj seekhenge pehchaanna, milke kaam wale sawaal, aur woh tricky cases jahan bade-bade fans jaate hain. 😉
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📖 Concept: Pehchaan Ke 5 Hathiyaar
Step 1: Double test
Ek quantity double karo, doosri ka kya hua?
- Double → Direct
- Aadhi → Inverse
- Kuch aur (4 guna, ya sirf thoda badla) → Dono nahi
WHY? Yeh dimaag mein 2 second ka experiment hai. Kaafi sawaal yahin solve ho jaate hain.
Step 2: Table test: ratio ya guna?
| Check | Same aaya | Matlab |
|---|---|---|
| y ÷ x | Haan | Direct |
| x × y | Haan | Inverse |
| Dono nahi | — | Koi proportion nahi |
WHY table? Jab sawaal mein numbers ki list ho, to andaaze ki zaroorat hi nahi. Seedha hisaab lagao.
Step 3: Shabdon pe nahi, “fixed cheez” pe dhyaan
“Zyada” aur “kam” shabd dhokha dete hain. Poochho: kya fixed hai?
- Daam per kilo fixed → kilo aur kul daam direct
- Kul kaam fixed → mazdoor aur din inverse
- Speed fixed → doori aur time direct; doori fixed → speed aur time inverse
WHY? Ek hi speed-time-doori mein dono rishte chhupe hain. Kya fixed hai, wahi tay karta hai.
Step 4: Milke kaam: time nahi, “speed” jodo
Nal A tanki 3 ghante mein bharta hai, nal B 2 ghante mein. Dono saath?
- A ek ghante mein: 1/3 tanki
- B ek ghante mein: 1/2 tanki
- Saath mein ek ghante mein: 1/3 + 1/2 = 5/6 tanki
- Poori tanki: 1 ÷ 5/6 = 6/5 ghante = 1 ghanta 12 minute
$\dfrac{1}{T} = \dfrac{1}{a} + \dfrac{1}{b}$
WHY time nahi jodte? Time aur kaam-ki-speed inverse hain. Saath kaam karne pe speed judti hai, time nahi. Aur jawab hamesha sabse tez wale se bhi kam aayega!
Step 5: “Dono nahi” wale cases bhi pehchaano
- Square ki side aur area: side double → area 4 guna. Direct nahi!
- Umar aur lambaai: koi fixed rishta nahi.
- Jod fixed (12 laddoo baantna): ek badhe, doosra ghate, par inverse nahi.
- Taxi ka kiraya (₹50 fixed + ₹15 per km): doori double to kiraya double nahi.
WHY? Sabse badi galti yahi hai ki har sawaal pe zabardasti proportion laga dena.
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🔐 Deep Concept: Ek Saath Direct Aur Inverse: Compound Proportion
Kabhi ek sawaal mein do rishte saath hote hain. Socho: 6 mazdoor 10 din mein 60 m deewar banaate hain. 9 mazdoor 90 m deewar kitne din mein banaayenge?
- Deewar badi → din zyada (direct)
- Mazdoor zyada → din kam (inverse)
Tareeka: Kaam ko mazdoor-din mein naapo. 60 m = 6 × 10 = 60 mazdoor-din, to 1 m = 1 mazdoor-din.
90 m = 90 mazdoor-din. 9 mazdoor → 90 ÷ 9 = 10 din.
$\dfrac{M_1 \times D_1}{W_1} = \dfrac{M_2 \times D_2}{W_2}$
Check: $\dfrac{6 \times 10}{60} = \dfrac{9 \times 10}{90} = 1$ ✅
Gehri baat: Deewar 1.5 guna hui (din × 1.5), mazdoor bhi 1.5 guna hue (din ÷ 1.5). Dono asar cancel, din wahi 10! Yahi proportional reasoning ki khoobsurti hai.
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✏️ Solved Examples
Example 1 (Easy): Direct, inverse ya dono nahi? (a) Petrol aur doori (fixed mileage) (b) Bachche aur har ek ko mili toffee (fixed toffee) (c) Square ki side aur area (d) Pages aur printing ka time (fixed speed)
(a) Direct (b) Inverse (c) Dono nahi (side × 2 → area × 4) (d) Direct
WHY? Har baar poocha: kya fixed hai, aur double test kya kehta hai.
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Example 2 (Medium): Nal A tanki 4 ghante mein bharta hai, nal B 6 ghante mein. Dono saath kitne samay mein?
1 ghante mein: 1/4 + 1/6 = 3/12 + 2/12 = 5/12 tanki.
Poori tanki: 12/5 ghante = 2 ghante 24 minute.
WHY sahi hai? Jawab 4 ghante (tez wala) se bhi kam ✅
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Example 3 (NCERT style): Table mein khaali jagah bharo. (i) x: 3, 6, 9 aur y: 12, 24, ? (ii) x: 3, 6, 9 aur y: 12, 6, ?
(i) 12 ÷ 3 = 24 ÷ 6 = 4 → direct, y = 9 × 4 = 36.
(ii) 3 × 12 = 6 × 6 = 36 → inverse, y = 36 ÷ 9 = 4.
WHY sahi hai? Same x, par alag rishta, to alag jawab. Pehle rishta, phir hisaab.
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Arjun ka badla: Agle round mein usne pehle socha “kya fixed hai?”, phir buzzer dabaaya. Saare sahi! 🏆
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🏠 Real Life Mein Pehchaan
- ⛽ Petrol pump: Litre aur daam direct. Bike ka mileage fixed, to petrol aur doori bhi direct.
- 🍽️ Ghar ki rasoi: Mehmaan zyada, sabzi zyada (direct); par fixed khaana, to har ek ka hissa kam (inverse).
- 🚿 Do baaltiyaan, do nal: Saath kholo to jaldi bharti hain, time ki jagah speed judti hai.
- 🚕 Taxi ka meter: Fixed charge + per km, isliye pure direct nahi.
- 📱 Data pack: Fixed GB mein zyada video, kam din chalega (inverse).
Rozmarra ki zindagi mein yeh teen sawaal poochho: kya fixed hai? Double karne pe kya hoga? Kahin fixed charge to nahi?
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Ek double → doosra double, ratio same | ✅ Direct: $\dfrac{y_1}{x_1} = \dfrac{y_2}{x_2}$ | Saath badhte hain |
| Ek double → doosra aadha, guna same | ✅ Inverse: $x_1 y_1 = x_2 y_2$ | Fixed total baanta gaya |
| Do log/nal saath kaam karein | ✅ 1/a + 1/b jodo | Speed judti hai, time nahi |
| Ek sawaal mein dono asar | ✅ Kaam ko mazdoor-din mein naapo | Compound proportion |
| Fixed charge, square/area, umar | ❌ Proportion mat lagao | Na ratio same, na guna |
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❌→✅ Worked Mistake
Sawaal: Nal A tanki 3 ghante mein, nal B 2 ghante mein bharta hai. Dono saath kitne samay mein?
❌ Galat 1: “3 + 2 = 5 ghante.” Do nal milke ek se bhi dheere? ❌
❌ Galat 2: “Average: (3 + 2) ÷ 2 = 2.5 ghante.” Phir bhi akele B (2 ghante) se zyada! ❌
Galti kya hai? Time jod diya. Time aur speed inverse hain, isliye time nahi jud sakta.
✅ Sahi:
- 1 ghante mein: 1/3 + 1/2 = 5/6 tanki
- Poori tanki: 6/5 ghante = 1 ghanta 12 minute ✅
Yaad rakho: Milke kaam ka jawab sabse tez wale se bhi kam hona chahiye.
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📝 10 MCQs
Q1. Fixed daam pe kilo aur kul daam ka rishta?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Kilo double, daam double ✅
- (b) galat: Zyada kilo, kam daam? Nahi.
- (c) galat: Ratio same hai, to direct hai.
- (d) galat: Rishta badalta nahi.
Q2. Fixed doori pe speed aur time?
Solution dekho
Sahi jawab: (b)
- (a) galat: Tez chaloge to zyada time? Nahi.
- (b) sahi: Speed × time = doori ✅
- (c) galat: Guna same hai, to inverse hai.
- (d) galat: Speed aur time alag cheezein hain.
Q3. Square ki side aur area?
Solution dekho
Sahi jawab: (c)
- (a) galat: Side double, area 4 guna. Direct nahi.
- (b) galat: Area to badhta hai, ghatta nahi.
- (c) sahi: Na ratio same, na guna ✅
- (d) galat: Side 3 pe area 9, barabar nahi.
Q4. Nal A 6 ghante, nal B 3 ghante. Saath mein?
Solution dekho
Sahi jawab: (c)
- (a) galat: Time jod diya.
- (b) galat: Average le liya, B se bhi dheere.
- (c) sahi: 1/6 + 1/3 = 1/2, to 2 ghante ✅
- (d) galat: B akela hi 3 leta hai, saath mein kam lagega.
Q5. Table (2, 10), (4, 20), (6, 30) kaunsa rishta?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Ratio 5 same ✅
- (b) galat: Guna 20, 80, 180, same nahi.
- (c) galat: Ratio same hai.
- (d) galat: Jod 12, 24, 36, same nahi.
Q6. Table (2, 30), (3, 20), (6, 10) kaunsa rishta?
Solution dekho
Sahi jawab: (b)
- (a) galat: Ratio 15, 6.67, 1.67, same nahi.
- (b) sahi: Guna 60, 60, 60 ✅
- (c) galat: Guna same hai.
- (d) galat: Farak bhi same nahi.
Q7. Fixed speed wale printer ke liye pages aur time?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Zyada page, zyada time ✅
- (b) galat: Zyada page, kam time? Nahi.
- (c) galat: Ratio same rehta hai.
- (d) galat: Speed fixed di hai, to pata hai.
Q8. Fixed raashan ke liye parivaar ke log aur din?
Solution dekho
Sahi jawab: (b)
- (a) galat: Zyada log, zyada din? Nahi.
- (b) sahi: Log × din = raashan ✅
- (c) galat: Guna same hai.
- (d) galat: Log aur din alag hain.
Q9. 4 mazdoor 3 din mein 12 m deewar banaate hain. 4 mazdoor 24 m kitne din mein?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Deewar double, mazdoor same → din double ✅
- (b) galat: Inverse maan liya, galat.
- (c) galat: Kaam badha to din badhenge.
- (d) galat: Galat guna.
Q10. Riya ki umar aur lambaai?
Solution dekho
Sahi jawab: (c)
- (a) galat: 10 saal mein 140 cm, to 20 mein 280 cm? Nahi!
- (b) galat: Badi hone pe lambaai ghatti nahi.
- (c) sahi: Koi fixed rishta nahi ✅
- (d) galat: Umar aur lambaai alag units.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Sabse jaldi kaise pehchaanein, direct ya inverse?
Double test lagao: ek ko double karo. Doosra double → direct. Aadha → inverse. Kuch aur → koi proportion nahi.
2. Milke kaam wale sawaal mein time kyun nahi jodte?
Kyunki saath kaam karne se kaam jaldi hota hai. Time jodne se zyada aayega, jo ulta hai. Isliye har ek ka 1 ghante ka kaam (1/a) jodte hain.
3. Agar ek nal bhare aur doosra khaali kare to?
Bharne wale ka kaam jodo, khaali karne wale ka ghatao: 1/a − 1/b. Jaise A 10 ghante mein bhare, B 15 ghante mein khaali kare: 1/10 − 1/15 = 1/30, to 30 ghante.
4. Compound proportion kya hai?
Jab ek sawaal mein do rishte hon, jaise deewar ki lambaai (direct) aur mazdoor (inverse). Kaam ko mazdoor-din mein naapo, ya $\dfrac{M_1 D_1}{W_1} = \dfrac{M_2 D_2}{W_2}$ lagao.
5. Taxi ka kiraya direct kyun nahi?
Kyunki ₹50 fixed charge doori pe nirbhar nahi. 2 km = 50 + 30 = ₹80, 4 km = 50 + 60 = ₹110. Doori double, kiraya double nahi.
6. Is chapter ke baad kya padhna chahiye?
Book 2 ka agla chapter hai Exploring Some Geometric Themes (jald aa raha hai). Tab tak Class 8 Maths TOC se baaki chapters dekhiye, aur Book 2 Ch 1 — Fractions in Disguise ke percentage posts revise kar lijiye.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy Notebook ki sankhya aur kul daam (fixed daam): direct ya inverse?
Solution dekho
Direct.
Q2. Easy Fixed kaam ke liye mazdoor aur din: direct ya inverse?
Solution dekho
Inverse.
Q3. Easy x: 1, 2, 3 aur y: 7, 14, 21. Rishta aur k?
Solution dekho
Direct, y ÷ x = 7.
Q4. Easy x: 1, 2, 4 aur y: 16, 8, 4. Rishta aur k?
Solution dekho
Inverse, x × y = 16.
Q5. Medium 15 kg chawal ₹900 ka. 25 kg kitne ka?
Solution dekho
Direct. 1 kg = ₹60. 25 kg = ₹1,500.
Q6. Medium 15 logon ka raashan 12 din chalta hai. 20 log ho to?
Solution dekho
Inverse. 180 ÷ 20 = 9 din.
Q7. Medium Pipe A tanki 5 ghante mein bharta hai, pipe B 20 ghante mein. Dono saath?
Solution dekho
1/5 + 1/20 = 5/20 = 1/4. 4 ghante.
Q8. Medium x: 4, 8, 12 aur y: 9, ?, 3. Rishta pehchaano aur bharo.
Solution dekho
4 × 9 = 12 × 3 = 36 → inverse. y = 36 ÷ 8 = 4.5.
Q9. NCERT style Nal A tanki 10 ghante mein bharta hai, nal B poori bhari tanki 15 ghante mein khaali karta hai. Dono khule hon to khaali tanki kitne ghante mein bharegi?
Solution dekho
1/10 − 1/15 = 3/30 − 2/30 = 1/30. 30 ghante.
Q10. NCERT style 8 mazdoor 6 din mein 48 m sadak banaate hain. 12 mazdoor 72 m sadak kitne din mein?
Solution dekho
48 m = 48 mazdoor-din, to 1 m = 1 mazdoor-din. 72 m = 72 mazdoor-din. ÷ 12 = 6 din.
Q11. NCERT style Riya ek kaam 4 din mein, Neha 12 din mein karti hai. Dono saath kitne din mein?
Solution dekho
1/4 + 1/12 = 3/12 + 1/12 = 4/12 = 1/3. 3 din.
Q12. NCERT style Direct, inverse ya dono nahi? (a) Fixed area ke carpet ki lambaai aur chaudaai (b) Square ki side aur perimeter (c) Din ka samay aur taapmaan (d) Fixed mileage pe petrol aur doori
Solution dekho
(a) Inverse (b) Direct (c) Dono nahi (d) Direct
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. Nal A 3 ghante, B 6 ghante, C 2 ghante mein tanki bharte hain. Teeno saath?
Solution dekho
1/3 + 1/6 + 1/2 = 2/6 + 1/6 + 3/6 = 1. 1 ghanta!
Q2. Arjun kehta hai, “Do log milke kaam karein to time un dono ka average hoga.” Ek example se galat saabit karo.
Solution dekho
A = 2 ghante, B = 2 ghante. Average = 2 ghante. Par saath: 1/2 + 1/2 = 1, to 1 ghanta. Do barabar log milke aadhe samay mein karte hain, average mein nahi.
Q3. Ek car 60 km/h pe 2 ghante chalti hai. Wahi car 45 km/h pe 180 km kitne samay mein jaayegi? Kaunse rishte lage?
Solution dekho
180 ÷ 45 = 4 ghante. Fixed speed pe doori aur time direct; fixed doori pe speed aur time inverse. Doori × speed dono badle, isliye seedha time = doori ÷ speed.
Q4. Square ki side double karne pe area 4 guna hota hai. To area kis cheez ke saath direct proportion mein hai?
Solution dekho
Area = side × side. Area ÷ (side × side) = 1 hamesha. To area, side × side (side ka square) ke saath direct proportion mein hai, side ke saath nahi.
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🏆 Olympiad Corner
Q1. A aur B milke ek kaam 12 din mein karte hain. A akela 20 din mein karta hai. B akela kitne din mein?
Solution dekho
1/12 − 1/20 = 5/60 − 3/60 = 2/60 = 1/30. 30 din.
Q2. Do pipe tanki 12 aur 15 minute mein bharte hain, ek drain use 20 minute mein khaali karta hai. Teeno khule hon to?
Solution dekho
1/12 + 1/15 − 1/20 = 5/60 + 4/60 − 3/60 = 6/60 = 1/10. 10 minute.
Q3. 15 mazdoor roz 8 ghante kaam karke 20 din mein 600 m nahar khodte hain. 20 mazdoor roz 6 ghante, 30 din mein kitni nahar khodenge?
Solution dekho
600 ÷ (15 × 8 × 20) = 600 ÷ 2,400 = 0.25 m per mazdoor-ghanta. Naya = 20 × 6 × 30 × 0.25 = 900 m.
Q4. x, y ke saath direct hai, aur y, z ke saath inverse. x = 6 pe y = 3, aur y = 3 pe z = 8. x = 12 ho to z?
Solution dekho
Direct: y ÷ x = 1/2, to x = 12 pe y = 6. Inverse: y × z = 24, to z = 24 ÷ 6 = 4.
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💗 CharuMam Motivation
Badhaai ho! 🎉 Aapne poora chapter Proportional Reasoning-2 khatam kar liya: ratio, map scale, teen terms wale ratio, baantna, pie chart, aur inverse proportion. Yeh chhoti baat nahi hai!
Arjun ki tarah pehli baar galti hogi, sab hanse bhi to koi baat nahi. Agle round mein wahi jeetta hai jo galti se seekhta hai. Bas focus, roz ki mehnat, aur sabr. Aap final boss jeet chuke ho! 🏆
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🟢 Master Recap
- Double test: double → direct, aadha → inverse, kuch aur → dono nahi.
- Table test: ratio same → direct, guna same → inverse.
- Shabdon pe nahi, “kya fixed hai?” pe dhyaan do.
- Milke kaam: 1/T = 1/a + 1/b. Jawab sabse tez se bhi kam.
- Khaali karne wala ho to uska 1/b ghatao.
- Compound: kaam ko mazdoor-din (ya mazdoor-ghante) mein naapo.
- Square-area, fixed charge, umar → proportion nahi.
- Har jawab ka sense check karo.
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👉 Aage Kya?
Is chapter ke saare 9 posts poore! 🎊 Agla chapter hai Exploring Some Geometric Themes, jo jald aa raha hai. Tab tak chapter hub se koi bhi post revise kar lijiye.
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬