✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 30 × 4 aur 60 × 2 mein kya common hai?
- Direct proportion mein kya same rehta hai?
- 120 km, 40 km/h ki speed se kitne ghante?
Answers dekho
- Dono = 120
- Ratio (y ÷ x)
- 120 ÷ 40 = 3 ghante
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 1 — Proportion Ka Recap mein direct proportion dobara padh lijiye, phir yahan aaiye.
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🎬 Kahani: Nani Ke Ghar Ka Safar
Riya ki Nani ka gaon 120 km door hai. Ghar mein bahas chal rahi thi.
Bhaiya bole, “Main cycle se jaaunga, 20 km/h pe 6 ghante!” Papa hanse, “Car 60 km/h pe chalaayenge, sirf 2 ghante.”
Riya ne dimaag lagaaya: “Speed 3 guna hui, to time 3 guna kam ho gaya!” 🤔
Ab tak humne dekha tha, ek badhe to doosra bhi badhe (direct proportion). Yahan ek badha to doosra ghata, aur woh bhi utne hi guna! Is rishte ka naam hai inverse proportion. 😉
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📖 Concept: Ek Badhe, Doosra Utne Hi Guna Ghate
Step 1: Table banao aur dekho
Doori fixed: 120 km.
| Speed (km/h) | 20 | 30 | 40 | 60 | 120 |
|---|---|---|---|---|---|
| Time (ghante) | 6 | 4 | 3 | 2 | 1 |
Speed 20 → 40 (double) to time 6 → 3 (aadha). Speed 20 → 60 (3 guna) to time 6 → 2 (tihaai).
WHY aisa? Doori utni hi tay karni hai. Tez chaloge to kam samay lagega, aur jitne guna tez, utne guna kam samay.
Step 2: Guna karke dekho: product same!
20 × 6 = 120, 30 × 4 = 120, 40 × 3 = 120, 60 × 2 = 120, 120 × 1 = 120.
Har baar speed × time = 120 (yaani doori). Yahi inverse proportion ki asli pehchaan hai.
WHY guna? Direct mein ratio fixed tha (y ÷ x). Inverse mein guna fixed rehta hai, kyunki ek ka badhna doosre ke ghatne se “cancel” ho jaata hai.
Step 3: Definition aur formula
Do quantities x aur y inverse proportion mein hain, agar unka guna hamesha same rahe:
$x \times y = k$ (k = constant)
Isliye kisi bhi do jodi ke liye:
$x_1 \times y_1 = x_2 \times y_2$
Ise aise bhi likhte hain: $x \propto \dfrac{1}{y}$ (“x, y ke ulte ke proportional hai”).
WHY yeh formula kaam ka? Teen cheezein pata hon to chauthi turant nikal aati hai. Jaise 20 km/h pe 6 ghante; 30 km/h pe? 20 × 6 = 30 × y → y = 4 ghante.
Step 4: Har “ek badhe, doosra ghate” inverse nahi hota!
Riya ke paas 12 laddoo hain. Woh jitne khud khaaye, bhai ko utne kam milenge:
| Riya | 2 | 4 | 6 |
|---|---|---|---|
| Bhai | 10 | 8 | 6 |
| Guna | 20 | 32 | 36 |
Ek badha, doosra ghata, par guna same nahi. Yahan jod same hai (12). Yeh inverse proportion nahi hai.
WHY dhyaan dena? Pehchaan “ghatna” nahi, “utne hi guna ghatna” hai. Hamesha guna check karo.
Step 5: Direct vs Inverse, aamne-saamne
| Direct | Inverse | |
|---|---|---|
| x double ho to y | Double | Aadha |
| Kya same rehta hai | y ÷ x | x × y |
| Formula | $\dfrac{y_1}{x_1} = \dfrac{y_2}{x_2}$ | $x_1 y_1 = x_2 y_2$ |
| Example | Kilo aur daam | Speed aur time (fixed doori) |
WHY yeh table? Sawaal mein sabse pehle yahi tay karna hai ki rishta kaunsa hai. Galat rishta, galat jawab.
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🔐 Deep Concept: Yaad Hai Map Wala Raaz?
Post 2 — Map Ka Scale (RF) Kya Hai? mein humne dekha tha: map doori × (1 cm ki km) hamesha same aata tha, kyunki zameen ki doori fixed thi. Woh bhi inverse proportion tha! Map jitna “chhota scale”, shehar utne paas dikhte hain.
Inverse ka graph kaisa dikhta hai? Speed-time ke points (20, 6), (30, 4), (40, 3), (60, 2), (120, 1) graph pe lagao. Yeh seedhi line nahi, ek jhuki hui curve banti hai jo dono axes ki taraf jaati hai par unhe kabhi chhoo nahi paati.
Kyun nahi chhooti? Speed 0 ho to time kitna? 0 × kuch bhi = 0, 120 kabhi nahi banega. Matlab ruk gaye to kabhi nahi pahunchoge! Isliye inverse proportion mein koi value 0 nahi ho sakti.
Ek aur gehri baat: Agar x aur y inverse hain, to x aur $\dfrac{1}{y}$ direct hain. Speed 20, 40, 60 ke liye 1/time = 1/6, 1/3, 1/2. Speed double → 1/time bhi double ✅. “Inverse” = “ulte ke saath direct”.
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✏️ Solved Examples
Example 1 (Easy): Kya yeh table inverse proportion mein hai? x: 2, 4, 8 aur y: 40, 20, 10.
Guna: 2 × 40 = 80, 4 × 20 = 80, 8 × 10 = 80. Sab same.
Haan, inverse proportion, k = 80.
WHY sahi hai? x double (2 → 4) to y aadha (40 → 20) ✅
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Example 2 (Medium): x aur y inverse proportion mein hain. x = 3 pe y = 20 hai. (a) x = 4 pe y? (b) y = 30 pe x?
k = 3 × 20 = 60.
(a) 4 × y = 60 → y = 15. (b) x × 30 = 60 → x = 2.
WHY sahi hai? x badha (3 → 4) to y ghata (20 → 15). y badha (20 → 30) to x ghata (3 → 2) ✅
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Example 3 (NCERT style): Mona ke paas toffee ke liye ₹600 ka budget hai. ₹5 wali toffee → 120 toffee, ₹10 wali → 60, ₹15 wali → 40. Agar toffee ₹12 ki ho, to kitni milegi? Kya daam aur sankhya inverse hain?
Guna: 5 × 120 = 10 × 60 = 15 × 40 = 600. Haan, inverse (k = budget = ₹600).
₹12 pe: 12 × n = 600 → n = 50 toffee.
WHY sahi hai? 12 × 50 = 600 ✅. Mehangi toffee, kam toffee.
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Riya ka formula: “Kuch fixed hai (doori, budget, kaam)? To dono ka guna wahi fixed cheez hai!” 🎉
— 📌 charumam.com
🏠 Real Life Mein Inverse Proportion
- 🚗 Safar: Fixed doori pe speed aur time.
- 👷 Kaam: Ek deewar banaani hai. Zyada mazdoor, kam din (agle post mein detail).
- 🍚 Raashan: Ghar mein mehmaan aa gaye. Utna hi raashan ab kam din chalega.
- 🍕 Pizza party: 8 slice, 4 dost → 2-2 slice. 8 dost → 1-1 slice. Log × slice = 8.
- 🚰 Tanki: Ek jaise nal. Zyada nal kholo, tanki jaldi bharegi.
Har jagah ek cheez fixed hai: doori, kaam, raashan, pizza. Wahi hai k.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Guna same aa raha hai (x × y = k) | ✅ Inverse proportion | Yahi definition hai |
| Ratio same aa raha hai (y ÷ x = k) | ❌ Inverse nahi, direct | Dono saath badh rahe hain |
| Jod same hai (jaise laddoo baantna) | ❌ Na direct, na inverse | Ghatna hai, par “utne guna” nahi |
| x double → y aadha | ✅ Inverse ka shortcut test | Guna same rehta hai |
| Koi value 0 aa rahi hai | ❌ Inverse ho hi nahi sakta | 0 × kuch = 0, k nahi banega |
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❌→✅ Worked Mistake
Sawaal: Phone ki battery: 1 ghante baad 90%, 2 ghante baad 80%, 3 ghante baad 70%. 4 ghante baad kitni?
❌ Galat: “Time badha, battery ghati, to inverse! 1 × 90 = 4 × y → y = 22.5%.”
Galti kya hai? Guna check hi nahi kiya: 1 × 90 = 90, 2 × 80 = 160, 3 × 70 = 210. Guna same nahi, to inverse nahi! ❌
✅ Sahi:
- Har ghante battery 10% kam ho rahi hai (farak same hai).
- 4 ghante baad = 70 − 10 = 60%.
Yaad rakho: “Ek badhe, doosra ghate” sirf shak hai, saboot nahi. Saboot = guna same.
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📝 10 MCQs
Q1. Inverse proportion mein kya same rehta hai?
Solution dekho
Sahi jawab: (d)
- (a) galat: Jod same rehna laddoo baantne jaisa hai, inverse nahi.
- (b) galat: Farak same rehna battery jaisa hai.
- (c) galat: Ratio same rehna direct proportion hai.
- (d) sahi: x × y = k, yahi inverse hai ✅
Q2. Fixed doori pe speed double ho jaaye, to time?
Solution dekho
Sahi jawab: (b)
- (a) galat: Tez chaloge to zyada time kaise lagega?
- (b) sahi: Speed × time fixed, to time aadha ✅
- (c) galat: Speed badli to time bhi badlega.
- (d) galat: Yeh ulti disha mein bhi galat factor hai.
Q3. x × y = 60 hai. x = 4 pe y = 15. x = 6 pe y?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 6 × 10 = 60 ✅
- (b) galat: 15 + 2, jodne ki galti.
- (c) galat: 15 − 6, ghataane ka koi matlab nahi.
- (d) galat: 15 × 6, direct maan liya.
Q4. Kaunsi table inverse proportion mein hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: Ratio same (5), yeh direct hai.
- (b) sahi: Guna 60, 60, 60 ✅
- (c) galat: Jod same (10), inverse nahi.
- (d) galat: y = x × x, koi proportion nahi.
Q5. Kaunsa jodi inverse proportion mein hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: Zyada aam, zyada daam, direct.
- (b) sahi: Tez = jaldi, guna = doori ✅
- (c) galat: Zyada time, zyada doori, direct.
- (d) galat: Koi fixed rishta hi nahi.
Q6. 120 km, 40 km/h ki speed se kitna time?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 120 ÷ 40 = 3 ✅
- (b) galat: 120 − 40, ghata diya.
- (c) galat: 120 ÷ 30 kar diya.
- (d) galat: 120 + 40, jod diya.
Q7. x aur y inverse hain. x = 8 pe y = 9. x = 12 pe y?
Solution dekho
Sahi jawab: (b)
- (a) galat: 9 × 12 ÷ 8, direct maan liya.
- (b) sahi: 8 × 9 = 72, 72 ÷ 12 = 6 ✅
- (c) galat: 9 − 4, farak ghata diya.
- (d) galat: 8 × 12, galat guna.
Q8. Riya aur bhai ke beech 12 laddoo baante. Kya Riya ke laddoo aur bhai ke laddoo inverse proportion mein hain?
Solution dekho
Sahi jawab: (b)
- (a) galat: Sirf ghatna kaafi nahi.
- (b) sahi: 2 × 10 = 20, 4 × 8 = 32. Guna alag, jod same ✅
- (c) galat: 12 to jod hai, guna nahi.
- (d) galat: Direct mein dono saath badhte, yahan nahi.
Q9. x aur y inverse hain. x ko 3 guna kar diya, to y?
Solution dekho
Sahi jawab: (b)
- (a) galat: Yeh direct hota.
- (b) sahi: Guna same rakhne ke liye y ÷ 3 ✅
- (c) galat: x badla to y badlega.
- (d) galat: 3 ghataana farak hai, guna nahi.
Q10. Ek pizza ke 8 slice. 4 dost ho to 2-2 slice. 8 dost ho to?
Solution dekho
Sahi jawab: (b)
- (a) galat: Zyada dost, zyada slice? Nahi.
- (b) sahi: 4 × 2 = 8 × 1 = 8 ✅
- (c) galat: Dost badle to hissa badlega.
- (d) galat: Pizza hi 8 slice ka hai!
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Inverse proportion ka seedha matlab kya hai?
Do cheezein jinka guna hamesha same rahe. Ek kisi factor se badhe, to doosri utne hi factor se ghate. Jaise speed double, time aadha.
2. Kya har “ek badhe, doosra ghate” inverse hai?
Nahi! Laddoo baantna (jod same) ya battery ghatna (farak same) inverse nahi hai. Hamesha x × y nikaal ke check karo.
3. k ka matlab kya hota hai?
k woh fixed cheez hai jo dono ka guna hai: doori (speed × time), poora kaam (mazdoor × din), budget (daam × sankhya). k ka asli matlab samjho, sawaal aasaan ho jaayega.
4. x ∝ 1/y ka kya matlab hai?
“x, y ke ulte (1/y) ke proportional hai.” Yeh inverse proportion ka symbol hai. Iska matlab x × y = k.
5. Inverse proportion ka graph kaisa hota hai?
Seedhi line nahi, ek curve jo neeche jhukti jaati hai. Yeh axes ke paas jaati hai par unhe kabhi chhoo nahi paati (Deep Concept dekho).
6. Kya inverse mein koi value zero ho sakti hai?
Nahi. Agar x = 0 ho to x × y = 0, jabki k fixed non-zero number hai. Speed 0 matlab kabhi nahi pahunchoge!
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy Kya yeh inverse proportion hai? x: 1, 2, 4 aur y: 24, 12, 6.
Solution dekho
Guna: 24, 24, 24. Haan, k = 24.
Q2. Easy x × y = 36. x = 9 ho to y?
Solution dekho
y = 36 ÷ 9 = 4.
Q3. Easy 60 km ki doori, 15 km/h ki speed se. Kitna time?
Solution dekho
60 ÷ 15 = 4 ghante.
Q4. Easy Fixed doori pe speed 4 guna ho jaaye, to time ka kya hoga?
Solution dekho
Time 4 guna kam (÷ 4) ho jaayega.
Q5. Medium Kya yeh inverse proportion hai? x: 5, 10, 25 aur y: 50, 25, 10.
Solution dekho
Guna: 250, 250, 250. Haan, k = 250.
Q6. Medium x aur y inverse hain, x = 2 pe y = 18. Table bharo: x = 3 pe y = ?, y = 6 pe x = ?, x = 12 pe y = ?
Solution dekho
k = 36. y = 12, x = 6, y = 3.
Q7. Medium Kya yeh inverse proportion hai? x: 2, 3, 4 aur y: 9, 8, 7.
Solution dekho
Guna: 18, 24, 28. Nahi. (Yahan jod 11 same hai.)
Q8. Medium Pen ke liye ₹480 ka budget. ₹12 wala pen kitne aayenge? ₹16 wala?
Solution dekho
480 ÷ 12 = 40 pen; 480 ÷ 16 = 30 pen. Daam × sankhya = 480 ✅
Q9. NCERT style Ek rectangle ka area 48 sq cm fixed hai. Lambaai 8 cm, 12 cm aur 16 cm ho to chaudaai? Kya lambaai aur chaudaai inverse hain?
Solution dekho
Chaudaai = 6 cm, 4 cm, 3 cm. Lambaai × chaudaai = 48 hamesha, to haan, inverse.
Q10. NCERT style Ek bori chawal 30 logon ke liye 8 din chalti hai. 40 log ho to kitne din?
Solution dekho
Log × din = 30 × 8 = 240. 40 log → 240 ÷ 40 = 6 din.
Q11. NCERT style x aur y inverse hain. x = 6 pe y = 10. (a) x = 15 pe y? (b) y = 12 pe x?
Solution dekho
k = 60. (a) y = 4. (b) x = 5.
Q12. NCERT style Direct, inverse ya koi nahi? (a) Fixed speed se padhne pe pages aur time (b) Fixed inaam ki rakam mein jeetne wale aur har ek ka hissa (c) Square ki side aur perimeter (d) Umar aur vajan
Solution dekho
(a) Direct (b) Inverse (c) Direct (perimeter = 4 × side) (d) Koi nahi.
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. x aur y inverse hain. Dikhao ki x aur 1/y direct proportion mein hain. (x: 2, 4, 8 aur y: 40, 20, 10 lekar)
Solution dekho
1/y = 1/40, 1/20, 1/10. Ratio x ÷ (1/y) = x × y = 80, 80, 80. Ratio same, to x aur 1/y direct ✅
Q2. x aur y inverse hain. x 25% badh gaya. y kitne % ghatega? (Kya 25%?)
Solution dekho
x × 1.25 → y ÷ 1.25 = y × 0.8. Matlab y 20% ghatega, 25% nahi!
Check: x = 4, y = 10 (k = 40). x = 5 → y = 8. 10 se 8 = 20% kam ✅
Q3. 120 km ka safar 40 km/h pe 3 ghante leta hai. Riya chahti hai 1 ghanta pehle pahunche. Speed kitni badhaani hogi?
Solution dekho
Naya time = 2 ghante. Speed = 120 ÷ 2 = 60 km/h. Badhaani hogi 20 km/h.
Dhyaan do: time 3 se 2 (33% kam), speed 40 se 60 (50% zyada). Inverse mein % barabar nahi hote!
Q4. Mohit kehta hai: “x: 1, 2, 3 aur y: 6, 3, 2 inverse hai, aur x: 1, 2, 3 aur y: 6, 4, 2 bhi, kyunki dono mein y ghat raha hai.” Kaun sahi hai?
Solution dekho
Pehli table: guna 6, 6, 6 → inverse ✅. Doosri: guna 6, 8, 6 → inverse nahi (yahan y har baar 2 kam ho raha hai). Mohit aadha sahi hai.
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🏆 Olympiad Corner
Q1. x, y ke saath inverse proportion mein hai, aur y, z ke saath inverse proportion mein hai. To x aur z ka rishta?
Solution dekho
x × y = k₁ → x = k₁/y. y × z = k₂ → y = k₂/z. To x = k₁z/k₂, yaani x aur z direct proportion mein hain. (Ulta ka ulta = seedha!)
Q2. x aur y inverse hain. Jab x, 8 se 12 hota hai to y, 3 ghat jaata hai. k nikaalo.
Solution dekho
8 × y = 12 × (y − 3) → 8y = 12y − 36 → y = 9. k = 8 × 9 = 72. Check: 12 × 6 = 72 ✅
Q3. Ek car 120 km jaati hai. Agar speed 10 km/h zyada hoti, to 1 ghanta kam lagta. Asli speed?
Solution dekho
Speed v aur v + 10 ke time mein 1 ghante ka farak: 120/v − 120/(v + 10) = 1.
Try karo: v = 30 → 4 ghante; v = 40 → 3 ghante. Farak 1 ✅. Asli speed = 30 km/h.
Q4. Rectangle ka area fixed hai. Lambaai 20% badhaai. Chaudaai kitne % ghategi?
Solution dekho
Lambaai × 1.2 → chaudaai ÷ 1.2 = × 5/6. Ghatega 1/6 = 16⅔%.
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💗 CharuMam Motivation
Socho Riya ki baat: tez chalo to safar chhota lagta hai. Padhaai mein bhi yahi hai. Jitna focus badhaoge, utna kam samay mein samajh aayega. 🚀
Par yaad rakho, speed tabhi kaam aati hai jab raasta sahi ho. Roz thodi mehnat, galti pe sabr, aur concept pe pakad. Manzil zaroor milegi! 🌟
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🟢 Master Recap
- Inverse proportion: x × y = k (guna same).
- Ek kisi factor se badhe, doosra utne hi factor se ghate.
- Formula: $x_1 y_1 = x_2 y_2$; symbol $x \propto \dfrac{1}{y}$.
- k = fixed cheez: doori, kaam, budget, raashan.
- “Ek badhe, doosra ghate” = sirf shak. Saboot = guna check.
- Jod same ya farak same → inverse nahi.
- Direct: ratio same. Inverse: guna same.
- Inverse mein koi value 0 nahi; graph ek curve hai.
— charumam.com 🎯
👉 Aage Kya?
Ab concept pakka! Agle post mein karenge asli inverse proportion ke sawaal: mazdoor aur din, nal aur tanki, raashan aur mehmaan, aur seekhenge ki sawaal ke peeche kaun si maan ke chalne wali baatein (assumptions) hoti hain! 👷
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬