📊 Ratio & Proportion
The foundation of comparative mathematics used in partnership distributions, mixtures, ages, and more.
🌟 Golden Rule:
a : b = a/b |
a : b :: c : d ⇒ a×d = b×c
1. Core Concepts
| Concept | Definition | Example |
|---|---|---|
| Ratio | Comparative relation between two quantities | 3:2 means 3 parts to 2 parts |
| Proportion | Equality of two ratios | 3:2 :: 6:4 (since 3×4=2×6) |
| Continued Proportion | a : b : c where a/b = b/c | 4 : 8 : 16 (common ratio 2) |
| Fourth Proportional | For a:b :: c:d, d is fourth proportional | 2:3 :: 4:6 → 6 is fourth proportional |
Key Formulas:
- Compound Ratio: (a:b) & (c:d) → ac:bd
- Duplicate Ratio: a²:b²
- Triplicate Ratio: a³:b³
- Sub-duplicate Ratio: √a:√b
- Inverse Ratio: b:a (reciprocal)
2. Problem Types & Solutions
| Problem Type | Solution Approach | Example |
|---|---|---|
| Simplification | Divide both terms by HCF | 15:25 → 3:5 (divided by 5) |
| Equivalent Ratios | Multiply/divide both terms by same number | 3:4 = 6:8 = 9:12 |
| Comparing Ratios | Cross-multiply to compare a:b vs c:d | 3:5 vs 2:3 → 9 vs 10 → 3:5 < 2:3 |
| Dividing Quantities | Sum of ratio parts → divide accordingly | Divide ₹120 in 3:2 → 3+2=5 → ₹72 & ₹48 |
Proportion Shortcuts:
- If a:b :: b:c, then b² = ac (b is mean proportional)
- If a:b :: c:d, then (a+b):(a-b) :: (c+d):(c-d) (Componendo-Dividendo)
- Third proportional to a,b is b²/a
3. Applications in Real Problems
A. Mixture Problems
| Scenario | Solution |
|---|---|
| Mix two items in ratio 3:2 costing ₹10 & ₹15/kg | Cost price = (3×10 + 2×15)/5 = ₹12/kg |
| Alcohol:Water = 5:1, total 42L. Add water to make 1:1 | Alcohol = (5/6)×42 = 35L → Need 35L water → Add 35-7=28L |
B. Age Problems
Key Principle: Age ratios change over time, but age difference remains constant.
Example: Present ages A:B = 5:3. 10 years ago = 7:3. Find current ages.
Let current ages = 5x, 3x
(5x-10)/(3x-10) = 7/3 → 15x-30 = 21x-70 → x=20/3
Ages = 100/3 ≈ 33.3 yrs, 20 yrs
Example: Present ages A:B = 5:3. 10 years ago = 7:3. Find current ages.
Let current ages = 5x, 3x
(5x-10)/(3x-10) = 7/3 → 15x-30 = 21x-70 → x=20/3
Ages = 100/3 ≈ 33.3 yrs, 20 yrs
C. Partnership Problems
| Investment Scenario | Profit Sharing Ratio |
|---|---|
| A invests ₹x for m months, B invests ₹y for n months | Profit ratio = x×m : y×n |
| Three partners invest ₹10k, ₹15k, ₹20k for 6, 4, 2 months | Ratio = 10×6 : 15×4 : 20×2 = 60:60:40 = 3:3:2 |
🔥 Previous Year UGC NET Questions - Ratio & Proportion:
- If a:b=2:3 and b:c=4:5, then a:b:c=? (Ans: 8:12:15)
- ₹782 divided in 1/2 : 2/3 : 3/4 ratio. Largest share? (Convert to 6:8:9 → 23 parts → Largest=9/23×782=₹306)
- Two numbers ratio 3:5. If each increased by 10, ratio becomes 5:7. Numbers? (Let 3x,5x → (3x+10)/(5x+10)=5/7 → x=5 → 15 & 25)
- In mixture, copper:zinc=7:5. If 4kg zinc added, ratio becomes 7:6. Original quantity? (Let 7x,5x → 7x/(5x+4)=7/6 → x=4 → 28kg,20kg)
🚀 Combined Problems (Ratio + Time/Speed)
Integrated Problem Solving:
- Speed Ratios: If A's speed:B's speed = 3:4, then time ratio for same distance = 4:3 (inverse)
- Work Efficiency: If A:B efficiency ratio = 5:3, time ratio for same work = 3:5
- Distance Coverage: Two cyclists start together with speed ratio 4:5 → When faster completes 50km, how far is slower? (Distance ratio = Speed ratio → 4:5 = x:50 → x=40km)
⏱️ Time-Saving Tips for Ratio Problems:
- Always reduce ratios to simplest form first
- For age problems: Focus on difference which remains constant
- In mixture problems: Track the unchanged component
- For partnership: Calculate capital × time for each partner
