Dosto, Ratio (anupat) ka matlab hai do cheezon ki aapas mein tulna — "A ke paas B se kitna guna hai". Proportion (samanupat) matlab do ratios barabar hona. Aur Partnership (saajhedari) matlab profit ko investment ke hisaab se baantna.
CET Group D mein isse 1–2 questions aate hain. Par asli fayda ye hai ki ratio ka concept Time & Work, Speed-Distance, Mixture, Average — har jagah kaam aata hai. Ratio strong = aadha Maths strong!
Chapter 4 (Percentage) ka fraction wala kaam yahan bhi lagega — 25% ka matlab hai ratio 1:4. Dono ek hi sikke ke do pehlu hain. Chalo shuru! 💪
1. Ratio (Anupat) ki Basics
- a : b ka matlab hai a/b. Yahan a = antecedent (pehla), b = consequent (dusra).
- Ratio ki koi unit nahi hoti — dono cheezein ek hi unit mein honi chahiye (5 kg : 500 g nahi, pehle 5000 g : 500 g = 10:1 karo).
- Ratio ke dono terms ko same number se multiply ya divide karo — value nahi badalti. 36:63 = 4:7 (dono ÷9).
- Ratio ko fraction/% mein: 3:5 → 3/8 aur 5/8 → 37.5% aur 62.5%.
2. Ratio mein Baantna (sabse zyada aane wala sawal)
⭐ Share = (apna part ÷ total parts) × Total quantity
Solved Example: ₹1200 ko 3 : 5 mein baanto.
Step 1: Total parts = 3 + 5 = 8
Step 2: Pehla = 1200 × 3/8 = 450 · Dusra = 1200 × 5/8 = 750
Check: 450 + 750 = 1200 ✓ (answer ka check hamesha karo!)
Step 1: Total parts = 3 + 5 = 8
Step 2: Pehla = 1200 × 3/8 = 450 · Dusra = 1200 × 5/8 = 750
Check: 450 + 750 = 1200 ✓ (answer ka check hamesha karo!)
3. Do Ratios ko Jodna (a:b aur b:c diya ho)
✏️ Solved Example — a:b = 2:3, b:c = 4:5. a:c = ?
Step 1: b dono mein alag hai (3 aur 4) — dono ko barabar karo. LCM(3,4) = 12
Step 2: a:b = 2:3 ko ×4 → 8:12 | b:c = 4:5 ko ×3 → 12:15
Step 3: a : b : c = 8 : 12 : 15 → a : c = 8 : 15 ✓
Fast trick: a:c = (2×4) : (3×5) = 8:15 — seedha cross multiply!
Step 2: a:b = 2:3 ko ×4 → 8:12 | b:c = 4:5 ko ×3 → 12:15
Step 3: a : b : c = 8 : 12 : 15 → a : c = 8 : 15 ✓
Fast trick: a:c = (2×4) : (3×5) = 8:15 — seedha cross multiply!
4. Compound, Duplicate & Sub-duplicate Ratio
| Naam | Formula | Example |
|---|---|---|
| Compound (mila hua) | a:b aur c:d → ac : bd | 3:4 & 8:9 → 24:36 = 2:3 |
| Duplicate | a² : b² | 3:4 → 9:16 |
| Sub-duplicate | √a : √b | 9:16 → 3:4 |
| Triplicate | a³ : b³ | 2:3 → 8:27 |
| Inverse (ulta) | b : a | 3:5 → 5:3 |
5. Proportion (Samanupat)
- a : b :: c : d ka matlab a/b = c/d. Yahan a, d = extremes (bahar wale) aur b, c = means (beech wale).
- Golden rule: Product of extremes = Product of means → a × d = b × c (cross multiply).
| Kya poocha | Formula | Example |
|---|---|---|
| Mean proportional (a aur b ke beech) | √(a × b) | 4, 9 → √36 = 6 |
| Third proportional (a, b ka) | b² / a | 4, 8 → 64/4 = 16 |
| Fourth proportional (a, b, c ka) | (b × c) / a | 4, 6, 10 → 60/4 = 15 |
6. Direct aur Inverse Proportion
| Type | Matlab | Example |
|---|---|---|
| Direct (seedha) | Ek badhe to dusra bhi badhe | Zyada pen → zyada daam |
| Inverse (ulta) | Ek badhe to dusra ghate | Zyada aadmi → kam din lagenge |
✏️ Solved Example — 12 pen ka daam ₹96 to 7 pen ka?
Step 1: Ye direct proportion hai (pen badhe, daam badha)
Step 2: 1 pen = 96/12 = ₹8 → 7 pen = 8 × 7 = ₹56 ✓
Step 2: 1 pen = 96/12 = ₹8 → 7 pen = 8 × 7 = ₹56 ✓
7. Partnership (Saajhedari)
⭐ Profit ka batwara
Simple partnership (sabka paisa same time ke liye lagta hai): Profit ratio = Investment ratio
Compound partnership (time alag-alag): Profit ratio = (Investment × Time) ka ratio
Compound partnership (time alag-alag): Profit ratio = (Investment × Time) ka ratio
✏️ Solved Example 1 (simple)
A ne ₹5000, B ne ₹7000 lagaye. ₹3600 profit hua. B ka hissa?
Step 1: Ratio = 5000 : 7000 = 5 : 7, total parts = 12
Step 2: B = 3600 × 7/12 = ₹2100 ✓ (A = 3600 × 5/12 = ₹1500)
Step 1: Ratio = 5000 : 7000 = 5 : 7, total parts = 12
Step 2: B = 3600 × 7/12 = ₹2100 ✓ (A = 3600 × 5/12 = ₹1500)
✏️ Solved Example 2 (time alag — compound)
A ne ₹8000, 9 mahine ke liye; B ne ₹12000, 5 mahine ke liye. ₹2200 profit. A ka hissa?
Step 1: A = 8000 × 9 = 72000 · B = 12000 × 5 = 60000
Step 2: Ratio = 72000 : 60000 = 6 : 5 (total 11 parts)
Step 3: A = 2200 × 6/11 = ₹1200 ✓ (B = ₹1000)
Step 1: A = 8000 × 9 = 72000 · B = 12000 × 5 = 60000
Step 2: Ratio = 72000 : 60000 = 6 : 5 (total 11 parts)
Step 3: A = 2200 × 6/11 = ₹1200 ✓ (B = ₹1000)
📌 Partner ke 2 types (theory question)
Active/Working partner = business chalata hai (kabhi-kabhi salary bhi leta hai) · Sleeping partner = sirf paisa lagata hai, kaam nahi karta.
✏️ Type 1 — "Ratio 5:7 hai, dono mein 8 jodne par 7:9 ho jata hai"
Step 1: Numbers ko 5x aur 7x maano (x = common factor)
Step 2: (5x + 8)/(7x + 8) = 7/9 → cross multiply: 9(5x+8) = 7(7x+8)
Step 3: 45x + 72 = 49x + 56 → 4x = 16 → x = 4
Step 4: Numbers = 20 aur 28 ✓ (check: 28/36 = 7/9 ✓)
Step 2: (5x + 8)/(7x + 8) = 7/9 → cross multiply: 9(5x+8) = 7(7x+8)
Step 3: 45x + 72 = 49x + 56 → 4x = 16 → x = 4
Step 4: Numbers = 20 aur 28 ✓ (check: 28/36 = 7/9 ✓)
✏️ Type 2 — "A:B:C = 2:3:5, total ₹4000, C ka hissa?"
Step 1: Total parts = 2+3+5 = 10
Step 2: C = 4000 × 5/10 = ₹2000 ✓ (A = 800, B = 1200)
Step 2: C = 4000 × 5/10 = ₹2000 ✓ (A = 800, B = 1200)
✏️ Type 3 — Income–Expenditure–Saving
Do aadmi ki income 4:5, kharcha 3:4, aur dono ₹800 bachate hain. Pehle ki income?
Step 1: Income = 4x, 5x · Kharcha = 3y, 4y
Step 2: 4x − 3y = 800 aur 5x − 4y = 800
Step 3: Solve karo → x = 800, y = 800
Step 4: Incomes = ₹3200 aur ₹4000 ✓ (kharche 2400, 3200 — dono ki bachat 800 ✓)
Step 1: Income = 4x, 5x · Kharcha = 3y, 4y
Step 2: 4x − 3y = 800 aur 5x − 4y = 800
Step 3: Solve karo → x = 800, y = 800
Step 4: Incomes = ₹3200 aur ₹4000 ✓ (kharche 2400, 3200 — dono ki bachat 800 ✓)
✏️ Type 4 — Mixture / Alloy ka ratio
40 litre ke mixture mein doodh:paani = 3:1. Kitna paani milayein ki 3:2 ho jaye?
Step 1: Doodh = 40 × 3/4 = 30 L, paani = 10 L
Step 2: Doodh wahi rahega (30). Naya ratio 3:2 → paani = 30 × 2/3 = 20 L
Step 3: Extra paani = 20 − 10 = 10 litre ✓
Step 1: Doodh = 40 × 3/4 = 30 L, paani = 10 L
Step 2: Doodh wahi rahega (30). Naya ratio 3:2 → paani = 30 × 2/3 = 20 L
Step 3: Extra paani = 20 − 10 = 10 litre ✓
✏️ Type 5 — Partnership mein "kaun kitne mahine"
Jab investment same par time alag ho → sirf time ka ratio chalega. Jab time same par paisa alag ho → sirf paise ka ratio. Dono alag hon to paisa × time. Bas yahi ek line pura partnership hai!
Trick 1: 'x' laga do — sab aasan
Ratio a:b → numbers ax aur bx
Jab bhi "ratio 5:7 hai" likha ho, numbers ko 5x aur 7x maan lo. Ek equation se x mil jayega, aur sab answer nikal aayenge.
Ratio a:b → numbers ax aur bx
Jab bhi "ratio 5:7 hai" likha ho, numbers ko 5x aur 7x maan lo. Ek equation se x mil jayega, aur sab answer nikal aayenge.
Trick 2: a:b aur b:c jodne ka cross-trick
a:c = (a×c-wala) : (b×b-wala)
a:b = 2:3, b:c = 4:5 → a:c = (2×4):(3×5) = 8:15. Beech wale b ko barabar karne ki mehnat khatam!
a:c = (a×c-wala) : (b×b-wala)
a:b = 2:3, b:c = 4:5 → a:c = (2×4):(3×5) = 8:15. Beech wale b ko barabar karne ki mehnat khatam!
Trick 3: Difference wala shortcut
1 part = antar ÷ parts ka antar
"A aur B 3:5 mein hain aur antar 400 hai" → parts ka antar = 2, to 1 part = 400/2 = 200 → A = 600, B = 1000. Total nikalne ki zaroorat hi nahi!
1 part = antar ÷ parts ka antar
"A aur B 3:5 mein hain aur antar 400 hai" → parts ka antar = 2, to 1 part = 400/2 = 200 → A = 600, B = 1000. Total nikalne ki zaroorat hi nahi!
Trick 4: Partnership — mahine wala paisa
Paisa × Time = "asli hissa"
Har partner ka paisa × mahine karo, phir ratio banao. Ratio ko chhota karke (simplify) hi aage badho — calculation aadhi ho jati hai.
Paisa × Time = "asli hissa"
Har partner ka paisa × mahine karo, phir ratio banao. Ratio ko chhota karke (simplify) hi aage badho — calculation aadhi ho jati hai.
Trick 5: Ratio ↔ Percentage jodo
3:1 = 75% : 25%
Ratio ko % mein badal ke socho, Chapter 4 wale shortcuts lag jayenge. 1:3 = 25%:75%, 2:3 = 40%:60%, 1:4 = 20%:80%.
3:1 = 75% : 25%
Ratio ko % mein badal ke socho, Chapter 4 wale shortcuts lag jayenge. 1:3 = 25%:75%, 2:3 = 40%:60%, 1:4 = 20%:80%.
❌ Galti 1: Total parts bhoolna
₹1200 ko 3:5 mein baantna hai to pehla hissa 1200 × 3/8 = 450 hai, na ki 1200 × 3/5. Neeche parts ka TOTAL aata hai, akela part nahi.
❌ Galti 2: Alag-alag unit mein ratio banana
5 kg : 500 g ka ratio 5:500 NAHI — pehle unit same karo: 5000 g : 500 g = 10 : 1. Metre-cm, ghanta-minute — sab mein yahi rule.
❌ Galti 3: Third aur Fourth proportional mix karna
Third proportional (a, b ka) = b²/a — do numbers diye hote hain. Fourth proportional (a, b, c ka) = bc/a — teen numbers. Kitne numbers diye hain, wahi se pehchano!
❌ Galti 4: Partnership mein time ignore karna
8000 (9 mahine) aur 12000 (5 mahine) ka ratio 8:12 NAHI — 72000 : 60000 = 6:5 hota hai. Jab tak "same time" na likha ho, time zaroor multiply karo.
❌ Galti 5: Ratio ko seedha jod dena
a:b = 2:3 aur b:c = 4:5 se a:c = 2:5 NAHI hota. Pehle b ko barabar karo (8:12:15) — tab a:c = 8:15.
Ratio a:b mein hissa kaise nikale?tap 👆
(apna part ÷ total parts) × total
a:b :: c:d mein golden rule?tap 👆
a × d = b × c (extremes = means)
Mean proportional of a, b?tap 👆
√(ab) — 4,9 → 6
Third proportional to a, b?tap 👆
b² / a — 4,8 → 16
Fourth proportional to a, b, c?tap 👆
(b × c) / a — 4,6,10 → 15
Compound ratio of a:b aur c:d?tap 👆
ac : bd
Duplicate ratio of 3:4?tap 👆
9 : 16 (a² : b²)
Sub-duplicate ratio of 9:16?tap 👆
3 : 4 (√a : √b)
Triplicate ratio of 2:3?tap 👆
8 : 27 (a³ : b³)
a:b = 2:3, b:c = 4:5 → a:c?tap 👆
8 : 15 (cross multiply)
Simple partnership mein profit ratio?tap 👆
= Investment ka ratio
Time alag ho to profit ratio?tap 👆
= (Investment × Time) ka ratio
Sleeping partner kaun?tap 👆
Sirf paisa lagata hai, kaam nahi karta
Ratio ki unit hoti hai?tap 👆
Nahi — par dono terms ki unit same honi chahiye
36 : 63 simplest form?tap 👆
4 : 7 (dono ÷ 9)
Inverse proportion ka matlab?tap 👆
Ek badhe → dusra ghate (aadmi ↑, din ↓)
Ratio 3:1 percentage mein?tap 👆
75% : 25%
Antar (difference) se 1 part kaise?tap 👆
Antar ÷ parts ka antar
| Cheez | Formula / Rule |
|---|---|
| Ratio a:b mein baantna | Share = (part / total parts) × total |
| Proportion a:b::c:d | ad = bc |
| Mean proportional | √(ab) |
| Third proportional (a, b) | b²/a |
| Fourth proportional (a, b, c) | bc/a |
| Compound ratio | ac : bd |
| Duplicate / Sub-duplicate | a²:b² / √a:√b |
| a:b & b:c jodna | a:c = (a×c) : (b×b) cross |
| Simple partnership | Profit ∝ investment |
| Compound partnership | Profit ∝ investment × time |
| Ratio 1:3 / 2:3 / 1:4 | 25:75 / 40:60 / 20:80 (%) |
| Unit rule | Dono terms same unit mein karo |
Practice (exam-style)
Q1. The ratio 36 : 63 in its simplest form is:
💡 Dono ko HCF 9 se divide: 36÷9 = 4, 63÷9 = 7.
Practice (exam-style)
Q2. ₹1200 is divided between two persons in the ratio 3 : 5. The larger share is:
💡 Total parts 8 → bada hissa = 1200 × 5/8 = 750 (chhota 450).
Practice (exam-style)
Q3. If a : b = 2 : 3 and b : c = 4 : 5, then a : c is:
💡 a:b:c = 8:12:15 → a:c = 8:15 (cross: 2×4 : 3×5).
Practice (exam-style)
Q4. The mean proportional between 4 and 9 is:
💡 √(4×9) = √36 = 6.
Practice (exam-style)
Q5. The third proportional to 4 and 8 is:
💡 b²/a = 64/4 = 16.
Practice (exam-style)
Q6. The fourth proportional to 4, 6 and 10 is:
💡 (b×c)/a = (6×10)/4 = 15.
Practice (exam-style)
Q7. Two numbers are in the ratio 5 : 7. If 8 is added to each, the ratio becomes 7 : 9. The smaller number is:
💡 5x,7x → 9(5x+8) = 7(7x+8) → x = 4 → numbers 20, 28.
Practice (exam-style)
Q8. ₹4000 is divided among A, B and C in the ratio 2 : 3 : 5. C's share is:
💡 Total parts 10 → C = 4000 × 5/10 = 2000.
Practice (exam-style)
Q9. A invests ₹5000 and B invests ₹7000 in a business for the same period. If the profit is ₹3600, B's share is:
💡 Ratio 5:7 → B = 3600 × 7/12 = 2100.
Practice (exam-style)
Q10. A invests ₹8000 for 9 months and B invests ₹12000 for 5 months. If the total profit is ₹2200, A's share is:
💡 72000 : 60000 = 6:5 → A = 2200 × 6/11 = 1200.
Practice (exam-style)
Q11. The compound ratio of 3 : 4 and 8 : 9 is:
💡 (3×8) : (4×9) = 24 : 36 = 2 : 3.
Practice (exam-style)
Q12. The duplicate ratio of 3 : 4 is:
💡 Duplicate = a² : b² = 9 : 16 (27:64 triplicate hota hai).
Practice (exam-style)
Q13. If 12 pens cost ₹96, then the cost of 7 pens is:
💡 1 pen = 96/12 = ₹8 → 7 pens = ₹56.
Practice (exam-style)
Q14. The incomes of two persons are in the ratio 4 : 5 and their expenditures in the ratio 3 : 4. If each saves ₹800, the income of the first person is:
💡 4x−3y = 800, 5x−4y = 800 → x = 800 → income = 4×800 = ₹3200.
Ratio ke sawal mein hamesha 'x' laga do (5:7 → 5x, 7x) — ek equation se sab nikal aayega. Baantne ka formula: (part ÷ TOTAL parts) × total. Do ratios jodne ke liye beech wale ko barabar karo, ya cross-multiply shortcut (2:3 & 4:5 → 8:15). Proportion mein ad = bc; mean = √(ab), third = b²/a, fourth = bc/a. Partnership mein profit investment × time ke ratio mein baantta hai — time bhoolna sabse badi galti hai. Aur ratio ke dono terms same unit mein hone chahiye. 🚀