✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 24 aur 36 ka HCF kya hai?
- 4 aur 6 ka LCM kya hai?
- 2 : 3 :: 8 : x. x nikaalo.
Answers dekho
- 24 = 12 × 2, 36 = 12 × 3, to HCF = 12
- 4 ke multiples: 4, 8, 12… 6 ke: 6, 12… LCM = 12
- x = (3 × 8) ÷ 2 = 12
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 1 — Proportion Ka Recap ek baar dekh lijiye, phir yahan wapas aaiye.
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🎬 Kahani: Kabir Ki Bhel Ka Raaz
School mele mein Kabir ki bhel puri sabse hit thi. Uski recipe (wazan ke hisaab se) thi: murmura : sev : moongfali = 4 : 2 : 1.
Agle saal mele ke liye usne 5 kg murmura khareed liya. Ab sawaal: sev aur moongfali kitni lein?
Uska dost Sunny bola, “Arre, sev sabko pasand hai, sev double kar do, bhel aur tasty banegi!” Kabir soch mein pad gaya. Agar sirf sev badhaayi to kya woh wahi famous bhel rahegi? 🤔
Ab tak humne do cheezon ka ratio dekha tha. Aaj seekhenge teen ya zyada cheezon wala ratio, aur Kabir ki shopping list bhi banayenge. 😉
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📖 Concept: Teen Cheezein, Ek Ratio
Step 1: a : b : c ka matlab
Murmura : sev : moongfali = 4 : 2 : 1 ka matlab: har 4 hisse murmura ke saath 2 hisse sev aur 1 hissa moongfali.
Isse teen jodiyan (pairs) bhi milti hain: murmura : sev = 4 : 2, sev : moongfali = 2 : 1, murmura : moongfali = 4 : 1.
WHY yeh kaam ka hai? Ek hi line mein teeno ka rishta likh diya. Teen alag ratios yaad rakhne ki zaroorat nahi.
Step 2: Simplest form: SAB terms ko same number se divide
12 : 18 : 24 → teeno ka HCF = 6 → 2 : 3 : 4.
WHY sab terms? Agar sirf do terms ko divide karo (12 : 18 : 24 → 2 : 3 : 24), to teesri cheez ka rishta bigad jaata hai. Ratio tabhi wahi rehta hai jab har term same factor se badle.
Step 3: Scale karna: SAB terms ko same number se guna
4 : 2 : 1 ko 3 se guna → 12 : 6 : 3. Dono ek hi ratio hain.
Kabir ka sawaal: Murmura 4 hisse = 5 kg, to 1 hissa = 5 ÷ 4 = 1.25 kg.
- Sev = 2 hisse = 2 × 1.25 = 2.5 kg
- Moongfali = 1 hissa = 1.25 kg
WHY 1 hissa nikaala? Ek hisse ki value pata ho to baaki sab uske guna se nikal aate hain. Yeh wahi unitary method hai, bas teen cheezon ke saath.
Step 4: Decimal ya fraction terms ko whole numbers banao
1.5 : 2 : 2.5 → sab ko 2 se guna → 3 : 4 : 5.
1/2 : 1/3 : 1/4 → denominators ka LCM = 12 → sab ko 12 se guna → 6 : 4 : 3.
WHY LCM se? LCM se guna karne pe saare denominators cancel ho jaate hain aur chhote se chhote whole numbers milte hain.
Step 5: Do ratios ko jodo: common term barabar karo
Agar A : B = 2 : 3 aur B : C = 4 : 5 ho, to A : B : C kya hai?
- B dono mein hai: pehle mein 3, doosre mein 4. Inhe barabar karo. LCM(3, 4) = 12.
- A : B = 2 : 3 → × 4 → 8 : 12
- B : C = 4 : 5 → × 3 → 12 : 15
- Ab B dono mein 12 hai: A : B : C = 8 : 12 : 15
$A:B = 2:3,\ B:C = 4:5 \Rightarrow A:B:C = 8:12:15$
WHY B barabar? B ek hi cheez hai. Uski ek hi value ho sakti hai. Jab tak B ko dono ratios mein same number nahi banaate, tab tak A aur C ko ek line mein nahi rakh sakte.
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🔐 Deep Concept: Sunny Ki “Sirf Sev Double” Wali Galti
Sunny ne kaha sirf sev double karo: 4 : 2 : 1 → 4 : 4 : 1. Dekho kya hua:
| Murmura : Sev | Sev : Moongfali | Murmura : Moongfali | |
|---|---|---|---|
| Asli 4 : 2 : 1 | 2 : 1 | 2 : 1 | 4 : 1 |
| Sunny 4 : 4 : 1 ❌ | 1 : 1 | 4 : 1 | 4 : 1 |
Ek term badla, aur do jodiyan badal gayin! Murmura : moongfali wahi raha, par baaki sab alag. Isliye yeh naya recipe hai, Kabir ki famous bhel nahi.
Gehri baat: 3-term ratio asal mein 3 jodiyon ka ek “package” hai. Package tabhi wahi rehta hai jab teeno terms ko ek hi factor se guna ya divide karo. Isi wajah se 3-term ratio ko ek akele fraction (jaise 4/2/1) mein nahi likh sakte.
sirf charumam.com readers ke liye 🔐
✏️ Solved Examples
Example 1 (Easy): 15 : 25 : 40 ko simplest form mein likho.
15, 25, 40 ka HCF = 5.
15 ÷ 5 : 25 ÷ 5 : 40 ÷ 5 = 3 : 5 : 8.
WHY sahi hai? 3, 5, 8 ka HCF 1 hai, to aur chhota nahi ho sakta ✅
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Example 2 (Medium): Ek mistri concrete mein cement : ret : gitti = 1 : 2 : 4 rakhta hai. Uske paas 6 bori cement hai. Ret aur gitti kitni bori chahiye?
1 hissa = 6 bori.
Ret = 2 × 6 = 12 bori, Gitti = 4 × 6 = 24 bori.
WHY sahi hai? 6 : 12 : 24 → 6 se divide → 1 : 2 : 4 ✅
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Example 3 (NCERT style): A : B = 3 : 4 aur B : C = 6 : 7 hai. A : B : C nikaalo.
B ko barabar karo: LCM(4, 6) = 12.
A : B = 3 : 4 → × 3 → 9 : 12. B : C = 6 : 7 → × 2 → 12 : 14.
A : B : C = 9 : 12 : 14.
WHY sahi hai? Check: 9 : 12 = 3 : 4 ✅, 12 : 14 = 6 : 7 ✅
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Ab Kabir bhi expert: 5 kg murmura, 2.5 kg sev, 1.25 kg moongfali, aur bhel ka swaad bilkul wahi! 🎉
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🏠 Real Life Mein Teen Terms Wala Ratio
- 🧱 Concrete: Cement : ret : gitti = 1 : 2 : 4 ek bahut common mix hai.
- 🌾 Khaad (NPK): Bori pe likha 10 : 26 : 26 batata hai nitrogen : phosphorus : potassium ka hissa.
- 🎨 Paint ka shade: Laal : peela : safed = 2 : 3 : 1. Ek bhi galat hua to rang badal jaata hai.
- 🍪 Nankhatai: Maida : ghee : cheeni jaise recipe ratios se hi har baar same swaad aata hai.
- 📚 Padhaai ka plan: Maths : Science : English = 3 : 2 : 1 ghante.
Har jagah ek hi niyam: badlo to sab ko ek saath, same factor se.
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Ratio simplify karna hai | ✅ Sab terms ka HCF lo, sab ko divide karo | Har term same factor se badle |
| Ek quantity di hai, baaki chahiye | ✅ 1 hisse ki value nikaalo, phir guna | Unitary method |
| A : B aur B : C diye hain | ✅ B ko LCM se barabar karo | B ki ek hi value ho sakti hai |
| Sirf ek term badalna | ❌ Ratio wahi nahi rahega | Baaki jodiyan bigad jaati hain |
| A : B aur B : C ko seedha jodna (2 : 3 aur 4 : 5 → 2 : 3 : 5) | ❌ Kabhi nahi | B do alag values pe atak jaata hai |
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❌→✅ Worked Mistake
Sawaal: Riya : Siya ki pocket money = 2 : 3 aur Siya : Tiya = 4 : 5. Teeno ka ratio kya hai?
❌ Galat: “Riya : Siya : Tiya = 2 : 3 : 5.”
Galti kya hai? Pehle ratio mein Siya = 3 aur doosre mein Siya = 4. Seedha jodne se Siya ka 4 gayab ho gaya. Check: 3 : 5 kya 4 : 5 ke barabar hai? 3 × 5 = 15, 5 × 4 = 20. Nahi! ❌
✅ Sahi:
- Siya ko barabar karo: LCM(3, 4) = 12
- Riya : Siya = 2 : 3 → × 4 → 8 : 12
- Siya : Tiya = 4 : 5 → × 3 → 12 : 15
- Riya : Siya : Tiya = 8 : 12 : 15
Check: 8 : 12 = 2 : 3 ✅, 12 : 15 = 4 : 5 ✅. Yaad rakho: Common wala term pehle barabar karo.
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📝 10 MCQs
Q1. 6 : 9 : 12 ka simplest form kya hai?
Solution dekho
Sahi jawab: (a)
- (a) sahi: HCF 3 se divide: 2 : 3 : 4 ✅
- (b) galat: Sirf 2 se divide kiya, decimal aa gaya. Simplest form mein whole numbers chahiye.
- (c) galat: 6 se divide karne pe decimal aaya. Whole numbers wala chhota form 2 : 3 : 4 hai.
- (d) galat: Order ulta ho gaya.
Q2. Ratio 4 : 2 : 1 mein pehli quantity 20 hai. Baaki do?
Solution dekho
Sahi jawab: (b)
- (a) galat: 20 mein se 2 aur 3 ghata diye, ratio aise nahi chalta.
- (b) sahi: 1 hissa = 20 ÷ 4 = 5. Doosri = 2 × 5 = 10, teesri = 5 ✅
- (c) galat: Guna kar diya (20 × 2), jabki 20 to 4 hisse hain.
- (d) galat: Order ulta: doosri 10 hai, teesri 5.
Q3. A : B = 1 : 2 aur B : C = 2 : 3 ho, to A : B : C = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: B dono mein pehle se 2 hai, seedha 1 : 2 : 3 ✅
- (b) galat: B ko jod diya (2 + 2), galat.
- (c) galat: A ko badal diya.
- (d) galat: C ko galat guna kiya.
Q4. A : B = 2 : 3 aur B : C = 4 : 5 ho, to A : B : C = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: B ko 12 banaya: 8 : 12 aur 12 : 15 ✅
- (b) galat: B ke do alag values (3 aur 4) ko ignore kiya.
- (c) galat: Sirf B badla, A aur C ko nahi.
- (d) galat: 12 : 20 = 3 : 5, jabki B : C = 4 : 5 chahiye.
Q5. Concrete 1 : 2 : 4 (cement : ret : gitti). 3 bori cement ho to gitti kitni?
Solution dekho
Sahi jawab: (b)
- (a) galat: 6 to ret hai (2 × 3).
- (b) sahi: Gitti = 4 × 3 = 12 bori ✅
- (c) galat: 3 + 4 = 7, jodna galat hai.
- (d) galat: 4 sirf hisse ka number hai, bori nahi.
Q6. Kaunsa ratio 3 : 5 : 7 ke barabar hai?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sab ko 2 se guna ✅
- (b) galat: Har term mein 1 joda, jodne se ratio badalta hai.
- (c) galat: Sirf pehle do terms double kiye.
- (d) galat: Teesra term 3 guna (21) hona chahiye tha, 28 nahi.
Q7. Ratio 2 : 3 : 5 mein pehli aur teesri quantity ka ratio kya hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: Yeh pehli aur doosri ka ratio hai.
- (b) sahi: Pehla term 2, teesra 5 → 2 : 5 ✅
- (c) galat: Yeh doosri aur teesri ka ratio hai.
- (d) galat: 5 ko 2 se guna kar diya, koi wajah nahi.
Q8. Paint mein laal : peela : safed = 2 : 3 : 1. Peela 9 litre ho, to laal aur safed?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 1 hissa = 9 ÷ 3 = 3 litre. Laal = 6, safed = 3 ✅
- (b) galat: 9 mein se ghataya, galat soch.
- (c) galat: Guna kiya (9 × 2), jabki 9 to 3 hisse hain.
- (d) galat: Order ulta.
Q9. Triangle ki bhujaayein 3 : 4 : 5 mein hain. Sabse chhoti bhuja 6 cm hai. Sabse lambi?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 1 hissa = 6 ÷ 3 = 2 cm. Lambi = 5 × 2 = 10 cm ✅
- (b) galat: 8 cm beech wali bhuja hai.
- (c) galat: 6 + 5 = 11, jodna galat hai.
- (d) galat: 6 × 5 = 30, 6 ko 1 hissa maan liya.
Q10. 1/2 : 1/3 : 1/4 ko whole numbers mein likho.
Solution dekho
Sahi jawab: (b)
- (a) galat: Denominators utha liye, ulta ho gaya. 1/2 sabse bada hai.
- (b) sahi: LCM 12 se guna: 6 : 4 : 3 ✅
- (c) galat: Yeh order bhi galat hai.
- (d) galat: Numerators sab 1 hain, par values alag hain.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Kya 3-term ratio ko fraction mein likh sakte hain?
Ek akele fraction mein nahi. 4 : 2 : 1 asal mein teen jodiyan hai: 4/2, 2/1, 4/1. Haan, har cheez ka poore mein hissa fraction mein likh sakte ho: murmura = 4/7, sev = 2/7, moongfali = 1/7 (Post 4).
2. Order kitna zaroori hai?
Bahut! 4 : 2 : 1 aur 1 : 2 : 4 bilkul alag recipes hain. Isliye hamesha label likho: “murmura : sev : moongfali”.
3. Kya 4 ya 5 terms wala ratio bhi hota hai?
Haan. Jaise practice ka time: warm-up : batting : bowling : fielding = 2 : 4 : 3 : 3. Niyam wahi: simplify ya scale karte waqt sab terms ko same number se.
4. Do ratios jodte waqt LCM hi kyun lete hain?
LCM se common term ki sabse chhoti barabar value milti hai, to numbers chhote rehte hain. Koi bhi common multiple chalega (jaise 24), par baad mein simplify karna padega.
5. Agar A : B aur A : C diye hon (B : C nahi) to?
Same tareeka, bas ab A common hai. A : B = 3 : 4 aur A : C = 2 : 5 → A ko LCM(3, 2) = 6 banao → 6 : 8 aur 6 : 15 → A : B : C = 6 : 8 : 15.
6. Kya ratio ke terms mein 0 ho sakta hai?
Mixture mein agar koi cheez daali hi na ho, to uska hissa 0 hai (jaise 2 : 0 : 3). Par phir woh asal mein 2 cheezon ka mix hai. Aam taur pe sawaalon mein saare terms positive hote hain.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy 8 : 12 : 20 ko simplest form mein likho.
Solution dekho
HCF = 4 → 2 : 3 : 5.
Q2. Easy 2 : 3 : 4 ko 5 guna karke likho.
Solution dekho
10 : 15 : 20.
Q3. Easy Murmura : sev : moongfali = 4 : 2 : 1. Murmura 800 g ho to baaki?
Solution dekho
1 hissa = 800 ÷ 4 = 200 g. Sev = 400 g, moongfali = 200 g.
Q4. Easy Kya 4 : 6 : 10 aur 6 : 9 : 15 barabar ratios hain?
Solution dekho
4 : 6 : 10 → ÷ 2 → 2 : 3 : 5. 6 : 9 : 15 → ÷ 3 → 2 : 3 : 5. Haan, barabar.
Q5. Medium 0.5 : 1.5 : 2 ko whole numbers mein likho.
Solution dekho
Sab ko 2 se guna: 1 : 3 : 4.
Q6. Medium 1/3 : 1/2 : 1/6 ko whole numbers mein likho.
Solution dekho
LCM(3, 2, 6) = 6. Sab ko 6 se guna: 2 : 3 : 1.
Q7. Medium A : B = 3 : 5 aur B : C = 5 : 2. A : B : C?
Solution dekho
B pehle se barabar (5): 3 : 5 : 2.
Q8. Medium A : B = 2 : 5 aur B : C = 3 : 4. A : B : C?
Solution dekho
LCM(5, 3) = 15. A : B = 6 : 15, B : C = 15 : 20. A : B : C = 6 : 15 : 20.
Check: 6 : 15 = 2 : 5 ✅, 15 : 20 = 3 : 4 ✅
Q9. NCERT style Concrete 1 : 2 : 4 mein gitti 20 bori hai. Cement aur ret kitni?
Solution dekho
1 hissa = 20 ÷ 4 = 5 bori. Cement = 5 bori, ret = 10 bori.
Q10. NCERT style Rangoli ke rang laal : peela : hara = 5 : 3 : 2. Peela 450 g hai. Laal aur hara?
Solution dekho
1 hissa = 450 ÷ 3 = 150 g. Laal = 5 × 150 = 750 g, hara = 2 × 150 = 300 g.
Q11. NCERT style Meena ka roz ka time padhaai : khel : TV = 4 : 3 : 1 hai. Khel 1.5 ghanta hai. Padhaai aur TV?
Solution dekho
1 hissa = 1.5 ÷ 3 = 0.5 ghanta. Padhaai = 4 × 0.5 = 2 ghante, TV = 0.5 ghanta (30 minute).
Q12. NCERT style Bhujaayein 3 : 4 : 5 ratio mein rakh ke banaye gaye saare triangles kya congruent (bilkul ek jaise) honge?
Solution dekho
Nahi. 3 cm, 4 cm, 5 cm wala aur 6 cm, 8 cm, 10 cm wala dono 3 : 4 : 5 mein hain. Shape same hai par size alag, isliye congruent nahi.
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. Kya bhujaon ka ratio 1 : 3 : 5 rakh ke triangle ban sakta hai?
Solution dekho
Bhujaayein k, 3k, 5k lo. Triangle ke liye kinhi do bhujaon ka jod teesri se bada hona chahiye.
k + 3k = 4k, jo 5k se chhota hai. Isliye triangle nahi ban sakta, chahe k kuch bhi ho.
Q2. A : B : C = 2 : 3 : 4. Teeno mein 10 jod diya. Kya naya ratio bhi 2 : 3 : 4 rahega?
Solution dekho
Example: 10 : 15 : 20 (k = 5) → 20 : 25 : 30 = 4 : 5 : 6. Nahi rahega.
WHY: Jodne se chhote term ka hissa zyada badhta hai. Ratio sirf guna-bhaag se bachta hai.
Q3. A : B = 2 : 3, B : C = 4 : 5 aur C : D = 6 : 7. A : B : C : D nikaalo.
Solution dekho
Pehle A : B : C = 8 : 12 : 15. Ab C ko barabar karo: LCM(15, 6) = 30.
A : B : C → × 2 → 16 : 24 : 30. C : D → × 5 → 30 : 35.
A : B : C : D = 16 : 24 : 30 : 35. Check: 16 : 24 = 2 : 3, 24 : 30 = 4 : 5, 30 : 35 = 6 : 7 ✅
Q4. Bhel mein murmura : sev : moongfali = 4 : 2 : 1. Riya kehti hai, “Moongfali poori bhel ka 1/4 hai, kyunki 1 aur 4 hai.” Kya Riya sahi hai?
Solution dekho
Nahi. Poori bhel = 4 + 2 + 1 = 7 hisse. Moongfali = 1 hissa = 1/7 (lagbhag 14.3%).
1 : 4 to moongfali aur murmura ki aapas mein tulna hai, poore se nahi. Yahi baat Post 4 ka base hai.
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🏆 Olympiad Corner
Q1. a : b : c = 2 : 3 : 5 ho, to (a + b) : (b + c) : (c + a) kya hoga?
Solution dekho
a = 2k, b = 3k, c = 5k. a + b = 5k, b + c = 8k, c + a = 7k.
Ratio = 5 : 8 : 7.
Q2. Agar 2A = 3B = 4C ho, to A : B : C kya hai?
Solution dekho
Maan lo 2A = 3B = 4C = 12k (2, 3, 4 ka LCM 12).
A = 6k, B = 4k, C = 3k → A : B : C = 6 : 4 : 3.
Check: 2 × 6 = 12, 3 × 4 = 12, 4 × 3 = 12 ✅
Q3. A : B = 3 : 4 aur A : C = 2 : 5. A : B : C?
Solution dekho
A ko barabar karo: LCM(3, 2) = 6. A : B = 6 : 8, A : C = 6 : 15.
A : B : C = 6 : 8 : 15.
Q4. Teen numbers 2 : 3 : 4 ke ratio mein hain aur unke squares ka jod 725 hai. Numbers nikaalo.
Solution dekho
Numbers 2k, 3k, 4k. 4k² + 9k² + 16k² = 29k² = 725 → k² = 25 → k = 5.
Numbers = 10, 15, 20. Check: 100 + 225 + 400 = 725 ✅
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💗 CharuMam Motivation
Kabir ki bhel isliye famous thi kyunki usne har baar santulan rakha. Ek cheez zyada, baaki kam, to swaad bigad jaata. 🍛
Aapki zindagi ka bhi ek ratio hai: padhaai : khel : neend. Sirf padhaai double kar di aur neend aadhi, to “recipe” bigad jaati hai. Teeno ko santulan mein rakho. Thoda focus, roz ki mehnat, aur bahut sa sabr, yahi success ka perfect ratio hai! 🌟
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🟢 Master Recap
- a : b : c = teen cheezon ka ek saath ratio. Isme teen jodiyan chhupi hain.
- Simplify: sab terms ka HCF lo aur sab ko divide karo.
- Scale: sab terms ko same number se guna karo.
- Decimal/fraction terms → LCM ya 10 se guna karke whole numbers.
- 1 hisse ki value nikaalo, phir baaki terms se guna.
- A : B aur B : C jodna: B ko LCM se barabar karo (2 : 3 aur 4 : 5 → 8 : 12 : 15).
- Sirf ek term badalna = naya ratio (Sunny ki galti).
- Order hamesha label ke saath likho.
— charumam.com 🎯
👉 Aage Kya?
Riya wale HOTS mein dekha ki moongfali poori bhel ka 1/7 hai. Ab socho, agar ₹700 ko teen bhai-behno mein 4 : 2 : 1 mein baantna ho to kisko kitna milega? Agle post mein seekhenge ratio mein baantna! 💰
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬