✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 4 : 6 simplest?
- 10 : 15 simplest?
- Simplest form ka niyam (Post 3 — Ratio Ka Simplest Form)?
Answers dekho
- 2 : 3
- 2 : 3
- HCF 1 tak bhaag
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 3 — Ratio Ka Simplest Form dekh lijiye.
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🎬 Kahani: Do Jug, Ek Swaad?
Mela mein do stall. Stall A: 4 glass paani mein 6 chammach sharbat. Stall B: 10 glass paani mein 15 chammach. 🥤
Ravi: “B mein zyada sharbat, B meetha!” Neha: “Par B mein paani bhi zyada…”
Dono ka swaad same hai ya alag? Yahi sawaal hai proportion ka.
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📖 Concept: Proportion = Do Ratio Barabar (a : b :: c : d)
💡 Is post ki ek badi baat: Jab a : b aur c : d ek hi ratio family mein hon, kehte hain a : b :: c : d (“a is to b as c is to d”). Matlab: dono mein same rishta, chahe size alag.
Step 1: Tareeka 1: Dono simplify
4 : 6 = 2 : 3, 10 : 15 = 2 : 3 → 4 : 6 :: 10 : 15. Swaad same! ✅
WHY? Same simplest form = same family = same rishta. Kab aasaan: chhote numbers.
Step 2: Tareeka 2: Per glass (unit rate)
A: 6/4 = 1.5 chammach/glass. B: 15/10 = 1.5. Same! ✅
Kab aasaan: jab “har ek pe kitna” natural lage, ya kai options ki tulna (Post 5 — Ratio Se Tulna: Strong Ya Light?).
Step 3: Tareeka 3: Cross product
4 × 15 = 60, 6 × 10 = 60 → barabar → proportion ✅.
WHY kaam karta? 4/6 = 10/15. Dono taraf 6 × 15 se guna: 4 × 15 = 10 × 6. Fraction hata diye, barabari bachi! Kab aasaan: bade/ajeeb numbers, ek missing term.
Step 4: Extremes aur means
a : b :: c : d mein a, d = extremes (bahar), b, c = means (andar). Niyam: extremes ka guna = means ka guna.
Step 5: Missing term
3 : 5 :: x : 40 → 3 × 40 = 5 × x → x = 24.
Doosra tareeka: 5 se 40 = × 8, to 3 × 8 = 24. Same factor wali soch! Check: 24 : 40 = 3 : 5 ✅
🪜 Atak Gaye? Aise Socho
- Dono ratio alag likho.
- Simplify karo, same?
- Ya cross product: a × d vs b × c.
- Missing term: factor dhoondo ya cross guna.
- Check: naya ratio simplify karke dekho.
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🔐 Deep Concept: Proportion Ka Asli Matlab: “Ek Hi Factor Dono Pe”
4 : 6 se 10 : 15 tak: 4 × 2.5 = 10, 6 × 2.5 = 15. Dono terms ko same factor 2.5 mila. Yahi proportion hai!
Do nazariye, ek sach:
- Andar ka factor: har ratio mein pehla = doosra × 2/3 (dono mein same).
- Beech ka factor: stall A se B tak har cheez × 2.5.
Isliye missing term do tareeke se nikalta: “andar” (3 : 5 → x = 40 × 3/5) ya “beech” (5 → 40 = × 8).
Galat proportion kaisa? Agar B mein 10 glass + 2 = 12? Nahi, paani 4 → 10 hua (+6), sharbat 6 → 15 (+9): jod same nahi, guna same. Yahi Post 1 — Proportional Badlaav Kya Hai? ka central idea: proportional = same factor, same farak nahi. 🎯
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✏️ Solved Examples
Example 1 (Easy): Kya 6 : 8 :: 15 : 20?
Cross: 6 × 20 = 120, 8 × 15 = 120 → haan.
Check: dono = 3 : 4 ✅
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Example 2 (Medium): Kya 5 : 7 :: 10 : 15?
5 × 15 = 75, 7 × 10 = 70 → nahi.
Check: 10 : 14 hota tab haan ✅
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Example 3 (NCERT style): 7 : x :: 21 : 36. x?
7 → 21 = × 3, to x × 3 = 36 → x = 12.
Check: 7 × 36 = 252 = 12 × 21 ✅
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Mela ka faisla: 4 : 6 :: 10 : 15, swaad bilkul same. Ravi ne zyada sharbat dekha, par paani bhi utne hi guna zyada tha. 🥤
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🏠 Proportion Roz Ki Zindagi Mein
- 🥤 Sharbat/chai: zyada log, same swaad.
- 🗺️ Map: 1 cm : 5 km :: 7 cm : 35 km.
- 🛒 Daam: 3 pen ₹45 :: 5 pen ₹75.
- 📷 Photo enlarge: 4 : 6 :: 8 : 12, shakal same.
- 💊 Dawai: vajan ke proportion mein dose.
Photo ko 4 × 6 se 8 × 12 karo: proportion. 4 × 6 se 6 × 8 (dono + 2): chehra kheencha hua lagega!
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Chhote numbers | ✅ Simplify | 4 : 6, 10 : 15 |
| Har ek pe kitna | ✅ Unit rate | 1.5/glass |
| Bade numbers | ✅ Cross product | a × d = b × c |
| Missing term, saaf factor | ✅ Beech ka factor | × 8 |
| Missing term, ajeeb factor | ✅ Cross guna | x = bc/a |
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❌→✅ Worked Mistake
Sawaal: Kya 2 : 3 :: 5 : 6? (“dono mein +3”)
❌ Galat: “2 se 5 = +3, 3 se 6 = +3, to haan.”
✅ Sahi: cross: 2 × 6 = 12, 3 × 5 = 15 → nahi. Check: 2/3 ≈ 0.67, 5/6 ≈ 0.83 ✅
🧠 Rule of thumb: Proportion mein guna ka factor same hona chahiye, jod nahi. Shak ho to cross product.
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📝 10 MCQs
Q1. 4 : 6 :: 10 : 15?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: Dono 2 : 3 ✅
- (b) galat: Cross 60 = 60, isliye yeh galat.
- (c) galat: Check karke pata chalta.
- (d) galat: Hamesha same rahega, isliye yeh galat.
Q2. 3 : 5 :: x : 40, x?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Bilkul sahi jawab: × 8 ✅
- (b) galat: Jod wali galat soch, isliye yeh option sahi nahi.
- (c) galat: 5 × 3 galat jodi, isliye yeh option sahi nahi.
- (d) galat: 40 × 5/8 galat hai, isliye yeh option sahi nahi.
Q3. Extremes in a : b :: c : d?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: Bahar wale ✅
- (b) galat: Yeh means hain, isliye yeh galat.
- (c) galat: Yeh pehla ratio hai, isliye yeh galat.
- (d) galat: Yeh doosra ratio hai, isliye yeh galat.
Q4. 2 : 3 :: 5 : 6?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Bilkul sahi jawab: 12 ≠ 15 ✅
- (b) galat: Cross product alag hai.
- (c) galat: Jod se proportion nahi.
- (d) galat: Cross se pakka nahi, isliye yeh galat.
Q5. 6 : 8 :: 15 : ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Bilkul sahi jawab: × 2.5 ✅
- (b) galat: Jod wali soch galat, isliye yeh option sahi nahi.
- (c) galat: Ulta factor laga diya.
- (d) galat: 6 : 8 ≠ 15 : 24 hota, isliye yeh galat.
Q6. Cross product kyun?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: Fraction hat jaate ✅
- (b) galat: Iska kaaran hai, ratta nahi.
- (c) galat: Guna se aata, jod se nahi.
- (d) galat: Hamesha chalta hai, isliye yeh galat.
Q7. Unit rate 6/4 aur 15/10?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Bilkul sahi jawab: Same ✅
- (b) galat: 15/10 = 1.5 hai, isliye yeh galat.
- (c) galat: 6/4 = 1.5 hai, isliye yeh galat.
- (d) galat: Bhaag galat kiya, isliye yeh option sahi nahi.
Q8. 9 : x :: 3 : 7, x?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Bilkul sahi jawab: × 3 ✅
- (b) galat: Jod wali soch, isliye yeh galat.
- (c) galat: Ulta factor laga, isliye yeh galat.
- (d) galat: 9 × 7 bina bhaag, isliye yeh galat.
Q9. Photo 4 × 6 se 6 × 8: shakal?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 4 : 6 ≠ 6 : 8 ✅
- (b) galat: Ratio badal gaya, isliye yeh galat.
- (c) galat: Double = 8 × 12, isliye yeh galat.
- (d) galat: Jod hai, guna nahi, isliye yeh galat.
Q10. Proportion ka central idea?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Bilkul sahi jawab: Guna ✅
- (b) galat: Farak se nahi chalta, isliye yeh galat.
- (c) galat: Jod se nahi chalta, isliye yeh galat.
- (d) galat: Unit alag ho sakti, isliye yeh galat.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. :: aur = mein farak?
Same matlab; :: purana tareeka, = bhi chalta (a/b = c/d).
2. Kya ratio ka order proportion mein?
Haan, a : b :: c : d mein a-c aur b-d same cheezein.
3. Continued proportion?
a : b :: b : c, jaise 2 : 6 :: 6 : 18 (b² = ac).
4. Kya 4 ratios?
Haan, a : b :: c : d :: e : f, sab same family.
5. Rate aur proportion?
Same rate = proportion: 60 km/h pe 2 h : 120 km :: 3 h : 180 km.
6. Aage?
Kaun sa strong? Post 5 — Ratio Se Tulna: Strong Ya Light?.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy 5 : 8 :: 15 : 24?
Solution dekho
Haan. Check: 120 = 120 ✅
Q2. Easy 3 : 4 :: 12 : 15?
Solution dekho
Nahi. Check: 45 ≠ 48 ✅
Q3. Easy 2 : 9 :: x : 45?
Solution dekho
10. Check ✅
Q4. Easy Extremes, means: 1 : 2 :: 3 : 6?
Solution dekho
Extremes 1, 6; means 2, 3. Check: 6 = 6 ✅
Q5. Medium x : 12 :: 15 : 36?
Solution dekho
5. Check: 5 × 36 = 180 = 12 × 15 ✅
Q6. Medium 4 notebook ₹100: 7 ka?
Solution dekho
₹175. Check: 4 : 100 :: 7 : 175 ✅
Q7. Medium 1 cm : 4 km. 6.5 cm?
Solution dekho
26 km. Check ✅
Q8. Medium 8 : 14 :: 20 : ?
Solution dekho
35. Check ✅
Q9. NCERT style Continued proportion 4, 10, x?
Solution dekho
100 = 4x → 25. Check: 4 : 10 :: 10 : 25 ✅
Q10. NCERT style (x + 1) : 5 :: 6 : 10?
Solution dekho
x = 2. Check: 3 : 5 ✅
Q11. NCERT style Kya 1.2 : 1.8 :: 4 : 6?
Solution dekho
Haan. Check: 7.2 = 7.2 ✅
Q12. NCERT style Photo 5 × 7 se chaudai 15 to lambai?
Solution dekho
21. Check ✅
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. a : b :: c : d to (a + b) : b :: (c + d) : d?
Solution dekho
Haan: a/b + 1 = c/d + 1. Check: 4 : 6 → 10 : 6 :: 25 : 15 ✅
Q2. Kaunsa number har term mein jodo ki 5, 9, 1, 3 proportion bane?
Solution dekho
(5 + x)(3 + x) = (9 + x)(1 + x) → 15 + 8x = 9 + 10x → x = 3. Check: 8 : 12 :: 4 : 6 ✅
Q3. a : b :: b : c, a = 3, c = 27. b?
Solution dekho
9. Check: 3 : 9 :: 9 : 27 ✅
Q4. Kya 2 : 3 :: 3 : 2?
Solution dekho
Nahi: 4 ≠ 9. Check ✅
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🏆 Olympiad Corner
Q1. 3 : x :: x : 12?
Solution dekho
6.
Q2. 2.5 : 4 :: 5 : x?
Solution dekho
8.
Q3. a : b = 3 : 4, a + b = 49?
Solution dekho
21, 28.
Q4. Fourth proportional of 4, 7, 12?
Solution dekho
21.
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💗 CharuMam Motivation
Ab mela mein koi bhi stall ka swaad pehle hi bata doge! Proportion = same rishta, alag size. 🟰
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🟢 Master Recap
- 💡 Chaabi: Jab a : b aur c : d ek hi ratio family mein hon, kehte hain a : b :: c : d (“a is to b as c is to d”). Matlab: dono mein same rishta, chahe size alag.
- a : b :: c : d = same family.
- Simplify, unit rate, ya cross product.
- Extremes × = means ×.
- Cross product kyun: bd se guna.
- Missing term: factor ya cross.
- Guna same, jod nahi.
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👉 Aage Kya?
Agar ratio barabar na ho, to kaunsa strong? Post 5 — Ratio Se Tulna: Strong Ya Light?.
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬