✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- x × y = 48 aur x = 6, to y = ?
- Speed double ho to time?
- 5 × 12 aur 10 × 6 barabar hain?
Answers dekho
- y = 8
- Aadha
- Haan, dono 60
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 7 — Inverse Proportion Kya Hai? pehle padh lijiye, phir yahan aaiye.
— 📌 charumam.com
🎬 Kahani: Gupta Ji Ki Diwali Wali Tension
Gupta ji ka ghar paint hona tha. Thekedaar bola, “6 painter, 10 din mein kaam ho jaayega.”
Gupta ji ghabraaye: “Diwali to 4 din baad hai!” 😟
Unki beti Neha ne turant hisaab lagaaya: “Papa, 6 × 10 = 60 painter-din ka kaam hai. 4 din mein chahiye to 60 ÷ 4 = 15 painter bulaao!”
Thekedaar hairaan. 😄 Aaj seekhenge Neha jaisa sochna: inverse proportion ke asli sawaal, step-by-step.
— 📌 charumam.com
📖 Concept: Inverse Proportion Ke Sawaal Kaise Hal Karein
Step 1: Pehle maan ke chalne wali baatein (assumptions)
Har aise sawaal mein kuch baatein chupchaap maani jaati hain:
- Saare painter ek jaisi speed se kaam karte hain.
- Roz utne hi ghante kaam hota hai.
- Zyada log aane se koi dhakka-mukki ya rukawat nahi hoti.
WHY zaroori? Inhi baaton ki wajah se “painter × din” fixed rehta hai. Koi baat toot jaaye, to inverse proportion bhi toot jaata hai.
Step 2: Fixed cheez (k) nikaalo
Painter × din = 6 × 10 = 60 painter-din. Yahi poora kaam hai.
| Painter | 6 | 10 | 12 | 15 | 20 |
|---|---|---|---|---|---|
| Din | 10 | 6 | 5 | 4 | 3 |
WHY “painter-din”? 1 painter-din = ek painter ka ek din ka kaam. Isse kaam ko ek unit mein naap lete hain, jaise doori km mein.
Step 3: Formula lagao: $x_1 y_1 = x_2 y_2$
$6 \times 10 = 15 \times y_2 \Rightarrow y_2 = 4$ din
Ya unitary tareeka: 1 painter ko 6 × 10 = 60 din lagenge. 15 painter ko 60 ÷ 15 = 4 din.
WHY dono tareeke? Dono ek hi baat hain. Jo samajh mein aaye, woh lagao.
Step 4: Beech mein badlaav wale sawaal
Maan lo 6 painter ne kaam shuru kiya. 2 din baad 2 aur painter aa gaye. Kaam kab khatam hoga?
- 2 din mein hua: 6 × 2 = 12 painter-din
- Bacha kaam: 60 − 12 = 48 painter-din
- Ab 8 painter: 48 ÷ 8 = 6 din aur
- Kul = 2 + 6 = 8 din
WHY aise? Jo ho chuka, woh ho chuka. Sirf bache hue kaam pe naya inverse proportion lagao.
Step 5: Jawab ko “sense” se check karo
Painter badhe (6 → 15), to din ghatne chahiye (10 → 4) ✅. Agar din badh gaye, to pakka galti hai!
WHY? Ek second ka check, aur direct/inverse wali sabse badi galti pakad mein.
— 📌 charumam.com
🔐 Deep Concept: Kya 600 Painter 1 Ghante Mein Paint Kar Denge?
Formula kehta hai: 60 painter-din ÷ 600 painter = 0.1 din (lagbhag 1 ghanta, agar din 10 ghante ka ho). Par kya yeh sach mein hoga? 😂
Nahi! Ek ghar mein 600 log khade bhi nahi ho sakte. Seedhiyaan kam padengi, ek-doosre se takraayenge. Assumption “koi rukawat nahi” toot gayi.
Gehri baat: Maths ka model ek daayre (range) mein hi sahi hota hai. 6 se 20 painter tak theek, 600 pe nahi. Isi tarah:
- Ek gaana 3 musician 4 minute mein bajaate hain. 6 musician 2 minute mein? Nahi! Gaana utna hi lamba hai. Yahan inverse proportion hai hi nahi.
- Ek baby 9 mahine mein paida hota hai… aage aap samajh gaye! 😄
Isliye har sawaal mein pehle poochho: “Kya sach mein kaam baanta ja sakta hai?”
sirf charumam.com readers ke liye 🔐
✏️ Solved Examples
Example 1 (Easy): 4 ek jaise nal ek tanki 6 ghante mein bharte hain. 3 nal kitne ghante mein bharenge?
Nal × ghante = 4 × 6 = 24. 3 nal → 24 ÷ 3 = 8 ghante.
WHY sahi hai? Nal kam hue, to time badha ✅
📌 charumam.com
Example 2 (Medium): Ek hostel mein 120 students ka raashan 20 din chalega. 30 naye students aa gaye. Ab raashan kitne din chalega?
Students × din = 120 × 20 = 2,400. Ab students = 150.
Din = 2,400 ÷ 150 = 16 din.
WHY sahi hai? Zyada log, kam din ✅. Assumption: har student roz utna hi khaata hai.
📌 charumam.com
Example 3 (NCERT style): Ek school mein roz 8 period, har ek 45 minute ka hota hai. Agar school ka samay utna hi rakhte hue 9 period kar diye jaayein, to har period kitne minute ka hoga?
Kul samay = 8 × 45 = 360 minute (fixed).
Naya period = 360 ÷ 9 = 40 minute.
WHY sahi hai? Period zyada, har period chhota ✅. Assumption: break ka samay alag hai aur badla nahi.
📌 charumam.com
Neha ka mantra: “Fixed cheez nikaalo, phir divide karo. Beech mein badlaav ho, to pehle bacha hua kaam!” 🎉
— 📌 charumam.com
🏠 Real Life Mein Yeh Sawaal
- 🎉 Shaadi ki taiyaari: 5 halwai 800 logon ka khaana 8 ghante mein. Jaldi chahiye? Halwai badhao.
- 🚌 School trip: Bus 50 km/h pe 3 ghante. Highway pe 60 km/h? 2.5 ghante.
- 🌾 Khet ki katai: Zyada mazdoor ya zyada machine, kam din.
- 🏕️ Camp ka raashan: Mehmaan badhe, raashan jaldi khatam.
- 🪑 Function ki kursiyaan: 360 kursiyaan, har line mein zyada kursi, to kam lines.
Har jagah wahi sawaal: kya fixed hai, aur kya assumptions sach hain?
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Kaam, doori, raashan jaisa poora total fixed hai | ✅ Pehle k = x × y nikaalo | Yahi poora kaam hai |
| Kaam beech mein badla (log aaye/gaye) | ✅ Pehle hua kaam ghatao, phir bache pe lagao | Ho chuka kaam dobara nahi hota |
| Ghante/din bhi badle | ✅ Mazdoor × din × ghante = fixed | Teeno milke kaam banate hain |
| Kaam baanta nahi ja sakta (gaana, baby) | ❌ Inverse mat lagao | Zyada log se samay kam nahi hota |
| Jawab mein dono saath badh gaye | ❌ Dobara check | Inverse mein ek badhe to doosra ghate |
— 📌 charumam.com
❌→✅ Worked Mistake
Sawaal: 6 painter 10 din mein kaam karte hain. 15 painter kitne din mein?
❌ Galat: “10/6 = y/15 → y = 10 × 15 ÷ 6 = 25 din.”
Galti kya hai? Direct proportion laga diya! Zyada painter aur zyada din? Sense check fail ❌
✅ Sahi:
- Inverse hai: 6 × 10 = 15 × y
- y = 60 ÷ 15 = 4 din ✅
Yaad rakho: Sawaal shuru karne se pehle khud se poochho: “Ek badhega to doosra badhega ya ghatega?”
sirf charumam.com pe milenge!
📝 10 MCQs
Q1. 5 mazdoor ek kaam 12 din mein karte hain. 10 mazdoor kitne din mein?
Solution dekho
Sahi jawab: (b)
- (a) galat: Direct laga diya, zyada mazdoor zyada din nahi.
- (b) sahi: 5 × 12 = 60, 60 ÷ 10 = 6 ✅
- (c) galat: 12 + 5, jodne ki galti.
- (d) galat: 12 − 5, ghataane ki galti.
Q2. 4 nal tanki 6 ghante mein bharte hain. 8 nal?
Solution dekho
Sahi jawab: (b)
- (a) galat: Direct maan liya.
- (b) sahi: 24 ÷ 8 = 3 ✅
- (c) galat: Nal double hue, to time aadha, tihaai nahi.
- (d) galat: 6 + 4, jod diya.
Q3. 50 logon ka raashan 30 din chalta hai. 75 logon ke liye?
Solution dekho
Sahi jawab: (b)
- (a) galat: Direct, zyada log zyada din? Nahi.
- (b) sahi: 1,500 ÷ 75 = 20 ✅
- (c) galat: 30 − 5, galat ghataav.
- (d) galat: 30 − 25, galat ghataav.
Q4. 9 mazdoor 8 din kaam karein, to kul kitne mazdoor-din?
Solution dekho
Sahi jawab: (b)
- (a) galat: Jod diya, guna karna tha.
- (b) sahi: 9 × 8 = 72 ✅
- (c) galat: 9 − 8, farak liya.
- (d) galat: Galat guna.
Q5. Ek kaam 6 mazdoor 12 din mein karte hain. 9 din mein karna ho to kitne mazdoor?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 72 ÷ 9 = 8 ✅
- (b) galat: 6 × 9 ÷ 12, direct maan liya.
- (c) galat: Din aur mazdoor barabar kyun?
- (d) galat: 6 + 9, jod diya.
Q6. Mazdoor-din wale sawaal mein sabse zaroori assumption kya hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: Tab mazdoor × din fixed nahi rahega.
- (b) sahi: Ek jaisi speed se hi guna fixed rehta hai ✅
- (c) galat: Kaam fixed hona chahiye.
- (d) galat: Yeh zaroori nahi, speed same honi chahiye.
Q7. 360 kursiyaan, har line mein 24 to 15 lines. Har line mein 30 kursi ho to kitni lines?
Solution dekho
Sahi jawab: (b)
- (a) galat: Direct maan liya.
- (b) sahi: 360 ÷ 30 = 12 ✅
- (c) galat: Galat divide.
- (d) galat: 15 + 6, jod diya.
Q8. Cycle 15 km/h pe 4 ghante leti hai. 20 km/h pe?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 60 ÷ 20 = 3 ✅
- (b) galat: Direct maan liya.
- (c) galat: 4 + 5, jod diya.
- (d) galat: 4 − 3, galat.
Q9. 300 pages ki kitaab, roz 25 page padho to 12 din. 10 din mein khatam karni ho to roz kitne page?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 300 ÷ 10 = 30 ✅
- (b) galat: Kam din mein kam page? Nahi.
- (c) galat: Galat hisaab.
- (d) galat: 25 + 2, jodne ki galti.
Q10. 10 mazdoor kaam 15 din mein karte. 5 din baad 5 mazdoor chale gaye. Bacha kaam kitne din mein?
Solution dekho
Sahi jawab: (b)
- (a) galat: Mazdoor ghate, to din badhne chahiye.
- (b) sahi: Kul 150, hua 50, bacha 100. 100 ÷ 5 = 20 din ✅
- (c) galat: Bacha kaam alag hai.
- (d) galat: Poore 150 ko 5 se baant diya, hua kaam ghataana bhool gaye.
— 📌 charumam.com
❓ FAQs (Bachchon Ke Asli Sawaal)
1. Kaise pata karein ki sawaal inverse ka hai?
Khud se poochho: “Agar ek quantity double karoon, to doosri aadhi hogi?” Haan → inverse. Mazdoor-din, speed-time, nal-ghante, log-raashan: aksar inverse.
2. Mazdoor-din ya painter-din kya hota hai?
Kaam naapne ki unit. 1 mazdoor-din = ek mazdoor ka ek din ka kaam. 6 mazdoor 10 din = 60 mazdoor-din kaam.
3. Beech mein mazdoor aayein ya jaayein to?
Pehle jitna kaam ho chuka, use ghatao. Bache hue kaam ko naye mazdooron se divide karo. Aakhir mein kul din jodo (Step 4).
4. Agar roz ke ghante bhi badlein to?
Tab teen cheezein: mazdoor × din × ghante = fixed kaam. Jaise 8 painter, roz 6 ghante, 10 din = 480 painter-ghante.
5. Kya jawab fraction mein aa sakta hai?
Din ya ghante mein haan (jaise 2.5 ghante). Par mazdoor ya kursiyaan aadhe nahi hote. Aisa aaye to sawaal ka matlab socho: aksar upar wala poora number lena padta hai (kam se kam itne chahiye).
6. Assumptions likhna zaroori hai kya?
Exam mein kabhi-kabhi poochha jaata hai. Achhi aadat hai ek line likhna: “Maana ki sab mazdoor ek jaisi speed se kaam karte hain.” Isse samajh bhi saaf dikhti hai.
— 📌 charumam.com
🏋️ Practice Zone (12 Sawaal)
Q1. Easy 3 mazdoor ek kaam 8 din mein karte hain. 6 mazdoor kitne din mein?
Solution dekho
24 ÷ 6 = 4 din.
Q2. Easy 240 mithaai, har dabbe mein 12 to 20 dabbe. Har dabbe mein 16 ho to kitne dabbe?
Solution dekho
240 ÷ 16 = 15 dabbe.
Q3. Easy Car 40 km/h pe 6 ghante leti hai. 60 km/h pe?
Solution dekho
240 ÷ 60 = 4 ghante.
Q4. Easy 2 pump tanki 9 ghante mein khaali karte hain. 3 pump?
Solution dekho
18 ÷ 3 = 6 ghante.
Q5. Medium Ek camp mein 1,200 logon ka raashan 28 din ka hai. 400 aur log aa gaye. Ab kitne din?
Solution dekho
1,200 × 28 = 33,600. ÷ 1,600 = 21 din.
Q6. Medium 25 mazdoor ek kaam 36 din mein karte hain. 30 din mein khatam karna ho to kitne aur mazdoor chahiye?
Solution dekho
25 × 36 = 900. 900 ÷ 30 = 30 mazdoor. Aur chahiye = 5.
Q7. Medium Roz 7 period, har ek 45 minute. Utne hi samay mein 9 period ho to har period?
Solution dekho
7 × 45 = 315 minute. 315 ÷ 9 = 35 minute.
Q8. Medium ₹1,800 ka inaam barabar baanta jaata hai. 6 jeetne wale ho to har ek ko ₹300. 9 ho to?
Solution dekho
1,800 ÷ 9 = ₹200.
Q9. NCERT style 12 mazdoor ek kaam 20 din mein karte. 8 din baad 4 mazdoor aur aa gaye. Kaam kul kitne din mein khatam hoga?
Solution dekho
Kul = 240. 8 din mein = 96. Bacha = 144. 16 mazdoor → 144 ÷ 16 = 9 din. Kul = 8 + 9 = 17 din.
Q10. NCERT style 6 nal ek tanki 5 ghante mein bharte hain. 2 ghante baad 3 nal band kar diye. Tanki kul kitne ghante mein bharegi?
Solution dekho
Kul = 30 nal-ghante. 2 ghante mein = 12. Bacha = 18. 3 nal → 6 ghante. Kul = 2 + 6 = 8 ghante.
Q11. NCERT style 200 logon ka raashan 30 din ka. 10 din baad 40 log chale gaye. Bacha raashan kitne din chalega?
Solution dekho
Bacha raashan = 200 × 20 = 4,000 vyakti-din. 160 log → 4,000 ÷ 160 = 25 din.
Q12. NCERT style “Ek gaana 3 musician 4 minute mein bajaate hain, to 6 musician 2 minute mein bajaayenge.” Sahi ya galat? Kyun?
Solution dekho
Galat. Gaane ki lambaai fixed hai, kaam baanta nahi ja sakta. Yahan inverse proportion lagta hi nahi. 6 musician bhi 4 minute hi lenge.
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. 8 painter roz 6 ghante kaam karke ghar 10 din mein paint karte hain. 10 painter roz 8 ghante kaam karein, to kitne din?
Solution dekho
Kul = 8 × 6 × 10 = 480 painter-ghante. Roz = 10 × 8 = 80. Din = 480 ÷ 80 = 6 din.
Q2. Ek hall mein 450 kursiyaan lines mein hain. Har line mein 5 kursi zyada kar dein, to 3 lines kam ho jaati hain. Shuru mein kitni lines aur har line mein kitni kursi?
Solution dekho
Lines × kursi = 450. Try: 18 × 25 = 450, aur (18 − 3) × (25 + 5) = 15 × 30 = 450 ✅
Shuru mein 18 lines, har line mein 25 kursi.
Q3. Ek kaam 10 mazdoor 12 din mein karte hain. Shuru se hi kitne aur mazdoor lagaayein ki kaam 4 din pehle khatam ho?
Solution dekho
Kul = 120. Naya samay = 8 din. Mazdoor = 120 ÷ 8 = 15. Aur = 5.
Q4. Gupta ji kehte hain: “60 painter bulaa leta hoon, 1 din mein kaam ho jaayega.” Formula se to sahi hai. Kya asal mein hoga? Do kaaran do.
Solution dekho
Formula: 60 ÷ 60 = 1 din. Par asal mein shaayad nahi: (1) ek ghar mein itne log ek saath kaam nahi kar paayenge, ek-doosre ke raaste mein aayenge; (2) paint ki pehli parat sookhne ka samay fixed hai, use jaldi nahi kar sakte. Assumptions toot jaati hain.
sirf charumam.com pe milenge!
🏆 Olympiad Corner
Q1. 20 mazdoor roz 8 ghante kaam karke 15 din mein kaam karte hain. 24 mazdoor 10 din mein karein, to roz kitne ghante?
Solution dekho
Kul = 20 × 8 × 15 = 2,400. 24 × 10 × h = 2,400 → h = 10 ghante.
Q2. 60 logon ka raashan 20 din ka hai. 5 din baad kuch aur log aa gaye aur raashan sirf 10 din aur chala. Kitne log aaye?
Solution dekho
Bacha = 60 × 15 = 900. 900 ÷ 10 = 90 log. Aaye = 90 − 60 = 30.
Q3. Ek safar 5 ghante mein hota hai. Speed 12 km/h badhaayein to 4 ghante lagte hain. Doori kitni hai?
Solution dekho
5v = 4(v + 12) → v = 48 km/h. Doori = 48 × 5 = 240 km. Check: 60 × 4 = 240 ✅
Q4. Ek bade gear mein 60 daant hain, chhote mein 20. Bada gear 1 minute mein 10 chakkar lagaata hai. Chhota kitne?
Solution dekho
Daant × chakkar = fixed (jitne daant aage badhe). 60 × 10 = 20 × n → n = 30 chakkar. Chhota gear tez ghoomta hai!
sirf charumam.com readers ke liye 🔐
💗 CharuMam Motivation
Neha ne ek chhote se hisaab se Papa ki tension door kar di. Yahi maths ki taakat hai: ghabraahat ki jagah tareeka. 💪
Lambe sawaal se mat daro. Use chhote kadmon mein todo, jaise kaam ko painter-din mein toda. Roz thoda focus, thodi mehnat, aur bahut saara sabr. Aapki “Diwali” bhi time pe taiyaar hogi! 🪔
— 📌 charumam.com
🟢 Master Recap
- Pehle assumptions: sab ek jaisi speed, koi rukawat nahi.
- k nikaalo: mazdoor × din, nal × ghante, log × din.
- Formula $x_1 y_1 = x_2 y_2$, ya unitary (1 mazdoor kitne din).
- Beech mein badlaav: pehle hua kaam ghatao, bache pe lagao.
- Ghante bhi badlein to mazdoor × din × ghante fixed.
- Sense check: ek badha to doosra ghata?
- Mazdoor/kursi jaise jawab poore number mein socho.
- Jo kaam baanta nahi ja sakta, wahan inverse nahi.
— charumam.com 🎯
👉 Aage Kya?
Ab sabse mazedaar challenge: mixed sawaal! Direct hai ya inverse, khud pehchaano. Saath mein do nal ek saath, tricky cases aur sabse common galtiyan. Chapter ka final boss! 🏆
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬