Ratio and Proportion (விகிதம் மற்றும் விகிதாசாரம்)

Ratio and Proportion is one of the most fundamental topics in quantitative aptitude that appears in almost every competitive exam. Mastering this topic helps you solve problems related to mixtures, partnerships, time and work, and even geometry problems faster. It's the backbone of mathematical reasoning! 🎯

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🧠 Foundation: Understanding the Core Concept (அடிப்படை கருத்துகள்)

Ratio (விகிதம்): A comparison between two or more quantities of the same kind. Think of it as "how many times" one quantity is compared to another.

Proportion (விகிதாசாரம்): When two ratios are equal, they form a proportion. It shows the relationship between quantities.

Real-world examples:

• 🍕 Recipe: 2 cups flour to 1 cup water (2:1 ratio)
• 🗺️ Map scale: 1 cm represents 10 km (1:1000000 ratio)
• 💰 Salary distribution: Manager gets ₹60,000, Assistant gets ₹40,000 (3:2 ratio)

The Core Formula

Ratio: a:b = a/b where a and b are quantities Proportion: If a:b = c:d, then a×d = b×c (Cross multiplication)

விகிதம்: a:b = a/b இங்கே a மற்றும் b அளவுகள் விகிதாசாரம்: a:b = c:d எனில், a×d = b×c (குறுக்கு பெருக்கல்)

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Level 0: Formula Playground

Formula Type	English	Tamil	Example
Simple Ratio	a:b = a/b	விகிதம் a:b = a/b	12:8 = 3:2
Proportion	a:b = c:d ⟹ a×d = b×c	விகிதாசாரம்	4:6 = 2:3
Third Proportional	If a:b = b:c, then c = b²/a	மூன்றாம் விகிதம்	For 4:6, third proportional = 36/4 = 9
Mean Proportional	If a:x = x:b, then x = √(a×b)	நடு விகிதம்	Between 4 and 9: x = √36 = 6

Practice Drills:

1. Simplify the ratio 45:75

   • Solution: 45:75 = 45÷15 : 75÷15 = 3:5

2. Find x if 3:5 = x:20

   • Solution: 3×20 = 5×x ⟹ 60 = 5x ⟹ x = 12

3. Are 6:9 and 8:12 in proportion?

   • Solution: 6×12 = 72 and 9×8 = 72. Yes, they are in proportion.

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⚙️ Unit Conversion Mastery (அலகு மாற்றம்)

Common Conversion Mistakes:

• Mixing different units in ratios (₹:paisa, hours:minutes)
• Not converting to same base before comparison
• Forgetting to convert back to required units in answers

Key Conversion Factors:

• 1 rupee = 100 paisa
• 1 hour = 60 minutes
• 1 kg = 1000 grams
• 1 meter = 100 cm

Conversion Practice Problems:

1. Express the ratio of 2 hours to 45 minutes

   • Convert to same unit: 2 hours = 120 minutes
   • Ratio = 120:45 = 8:3

2. Find the ratio of ₹5 to 75 paisa

   • Convert: ₹5 = 500 paisa
   • Ratio = 500:75 = 20:3

3. What is the ratio of 1.5 kg to 250 grams?

   • Convert: 1.5 kg = 1500 grams
   • Ratio = 1500:250 = 6:1

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Level 1: Problem Reading Strategy (கேள்வியை அணுகும் முறை)

G-F-V Framework:

• Given (கொடுக்கப்பட்டது): What information is provided?
• Find (கண்டுபிடிக்க வேண்டியது): What are we looking for?
• Verify (சரிபார்த்தல்): Does our answer make logical sense?

Example 1: Two numbers are in the ratio 3:4. If their sum is 35, find the numbers.

• Given: Ratio = 3:4, Sum = 35
• Find: The two numbers
• Solution: Let numbers be 3x and 4x. Then 3x + 4x = 35 ⟹ 7x = 35 ⟹ x = 5
• Answer: Numbers are 15 and 20
• Verify: 15:20 = 3:4 ✓ and 15+20 = 35 ✓

Example 2: Divide ₹120 between A and B in the ratio 2:3.

• Given: Total amount = ₹120, Ratio = 2:3
• Find: A's share and B's share
• Solution: Total parts = 2+3 = 5. A gets (2/5)×120 = ₹48, B gets (3/5)×120 = ₹72
• Verify: 48:72 = 2:3 ✓ and 48+72 = 120 ✓

Example 3: The ratio of boys to girls in a class is 5:7. If there are 20 boys, how many girls are there?

• Given: Ratio boys:girls = 5:7, Boys = 20
• Find: Number of girls
• Solution: 5:7 = 20:x ⟹ 5x = 7×20 ⟹ x = 28
• Verify: 20:28 = 5:7 ✓

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Level 2: Pattern Recognition (கணக்குகளின் வகையை அறிதல்)

Pattern	English Keywords	Core Concept	Tamil Term
Simple Division	divide, share, distribute	Split amount in given ratio	எளிய பங்கீடு
Ages Problem	ages, years ago/hence	Apply ratio to time contexts	வயது கணக்கு
Mixture Problems	mix, combine, alloy	Ratio of different components	கலவை கணக்கு
Partnership	profit, investment, capital	Share profits based on investment ratio	கூட்டாண்மை

Pattern 1: Simple Division (எளிய பங்கீடு)

Keywords: divide, share, split, distribute in ratio Core Insight: Total amount × (individual ratio part)/(sum of all parts)

Visual: Total Amount = 100 Ratio = 2:3 Parts = 2+3 = 5

A's Share = (2/5) × 100 = 40 B's Share = (3/5) × 100 = 60

Worked Example: Divide 450 candies among three children in the ratio 2:3:4.

• Solution: Total parts = 2+3+4 = 9
• Child 1: (2/9) × 450 = 100 candies
• Child 2: (3/9) × 450 = 150 candies
• Child 3: (4/9) × 450 = 200 candies

Pattern 2: Ages Problem (வயது கணக்கு)

Keywords: ages, years ago, years hence, present age Core Insight: Ratio changes with time, but the difference remains constant

Mixture = Component A + Component B If A:B = 2:3 in 50 liters Then A = (2/5) × 50 = 20 liters B = (3/5) × 50 = 30 liters

Worked Example: A mixture contains milk and water in ratio 4:1. If total mixture is 60 liters, find quantity of milk.

• Solution: Total parts = 4+1 = 5
• Milk quantity = (4/5) × 60 = 48 liters
• Water quantity = (1/5) × 60 = 12 liters

Pattern 4: Partnership (கூட்டாண்மை)

Keywords: profit, loss, investment, capital, business Core Insight: Profit sharing ratio = Investment ratio × Time ratio

A invests ₹3000 for 12 months B invests ₹4000 for 9 months Investment × Time ratio = (3000×12):(4000×9) = 36000:36000 = 1:1

Worked Example: A and B start a business. A invests ₹6000 and B invests ₹9000. They earn ₹2500 profit. Find A's share.

• Solution: Investment ratio = 6000:9000 = 2:3
• Total parts = 2+3 = 5
• A's share = (2/5) × 2500 = ₹1000

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Level 3: Advanced Scenarios (சிக்கலான கணக்குகள்)

Compound Ratios: When multiple ratios are combined

• If A:B = 2:3 and B:C = 4:5, then A:B:C = (2×4):(3×4):(3×5) = 8:12:15

Inverse Proportion: When one quantity increases, other decreases

• Speed ∝ 1/Time (for constant distance)
• Workers ∝ 1/Time (for constant work)

Common Advanced Traps

• Mixing direct and inverse proportions
• Not adjusting for time differences in partnerships
• Forgetting to account for compound ratios correctly

Advanced Example 1: If 6 men can complete work in 15 days, how many days will 10 men take?

• Solution: This is inverse proportion (Men ∝ 1/Days)
• 6 × 15 = 10 × x ⟹ x = 90/10 = 9 days

Advanced Example 2: A, B, C are partners. A:B = 2:3, B:C = 4:5. If total profit is ₹94,000, find each partner's share.

• Solution: A:B:C = (2×4):(3×4):(3×5) = 8:12:15
• Total parts = 8+12+15 = 35
• A's share = (8/35) × 94,000 = ₹21,600
• B's share = (12/35) × 94,000 = ₹32,400
• C's share = (15/35) × 94,000 = ₹40,000

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⚠️ Common Beginner Mistakes (பொதுவான தவறுகள்)

1. Unit Mixing Error

   • ❌ Wrong: Comparing ₹5 with 50 paisa directly as 5:50
   • ✅ Correct: Convert to same unit first: 500 paisa : 50 paisa = 10:1

2. Cross Multiplication Confusion

   • ❌ Wrong: If a:b = c:d, thinking a×b = c×d
   • ✅ Correct: a×d = b×c (cross multiply diagonally)

3. Ratio vs Fraction Mistake

   • ❌ Wrong: Thinking 3:5 means 3/5 of total goes to first person
   • ✅ Correct: First person gets 3/(3+5) = 3/8 of total

4. Direct vs Inverse Proportion

   • ❌ Wrong: Assuming all relationships are direct proportions
   • ✅ Correct: Check if quantities increase/decrease together (direct) or oppositely (inverse)

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🚀 Key Takeaways (முக்கிய குறிப்புகள்)

• Always simplify ratios to their lowest terms for easier calculation
• Convert to same units before comparing quantities
• Use cross multiplication to solve proportion problems quickly
• Remember the G-F-V framework for systematic problem solving
• Practice identifying patterns - 80% of problems follow the 4 main patterns above

Success Formula

Practice + Pattern Recognition + Unit Awareness = Ratio & Proportion Mastery 🎯

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📝 Problems to Practice (பயிற்சிக் கேள்விகள்)

Pattern 1: Simple Division (எளிய பங்கீடு)

Problem 1: Divide ₹3600 among A, B, and C in the ratio 2:3:4. Pattern Identification: Simple division in given ratio Solution:

• Total parts = 2+3+4 = 9
• A's share = (2/9) × 3600 = ₹800
• B's share = (3/9) × 3600 = ₹1200
• C's share = (4/9) × 3600 = ₹1600 Answer: A gets ₹800, B gets ₹1200, C gets ₹1600

Problem 2: Two numbers are in ratio 5:8. If their sum is 91, find the larger number. Pattern Identification: Simple division with total given Solution:

• Let numbers be 5x and 8x
• Sum: 5x + 8x = 91 ⟹ 13x = 91 ⟹ x = 7
• Numbers are 35 and 56 Answer: Larger number is 56

Problem 3: A sum of money is divided among A, B, C such that A gets 2/5 of what B gets, and B gets 1/4 of what C gets. If C gets ₹800 more than A, find the total sum. Pattern Identification: Complex ratio relationship Solution:

• Let C get 4y, then B gets y, and A gets (2/5)y = (2y/5)
• C - A = 4y - 2y/5 = 18y/5 = 800 ⟹ y = 800×5/18 = ₹222.22
• Total = 4y + y + 2y/5 = 27y/5 = 27×222.22/5 = ₹1200 Answer: Total sum is ₹1200

Pattern 2: Ages Problem (வயது கணக்கு)

Problem 4: The ratio of present ages of A and B is 3:5. After 6 years, their ages will be in ratio 2:3. Find their present ages. Pattern Identification: Age ratio changing over time Solution:

• Let present ages be 3x and 5x
• After 6 years: (3x+6):(5x+6) = 2:3
• Cross multiply: 3(3x+6) = 2(5x+6) ⟹ 9x+18 = 10x+12 ⟹ x = 6
• Present ages: 18 and 30 years Answer: A is 18 years old, B is 30 years old

Problem 5: Father's age is 3 times his son's age. After 15 years, father's age will be twice the son's age. Find their present ages. Pattern Identification: Age relationship with future condition Solution:

• Let son's present age = x, father's age = 3x
• After 15 years: 3x+15 = 2(x+15) ⟹ 3x+15 = 2x+30 ⟹ x = 15
• Present ages: Son = 15, Father = 45 Answer: Son is 15 years old, Father is 45 years old

Problem 6: 5 years ago, the ratio of A's age to B's age was 1:2. Present ratio is 2:3. Find their ages after 5 years. Pattern Identification: Past and present age ratios given Solution:

• Let present ages be 2x and 3x
• 5 years ago: (2x-5):(3x-5) = 1:2
• Cross multiply: 2(2x-5) = 1(3x-5) ⟹ 4x-10 = 3x-5 ⟹ x = 5
• Present ages: 10 and 15
• After 5 years: 15 and 20 Answer: After 5 years, A will be 15 and B will be 20

Pattern 3: Mixture Problems (கலவை கணக்கு)

Problem 7: A mixture of 40 liters contains milk and water in ratio 3:1. How much water must be added to make the ratio 1:1? Pattern Identification: Mixture ratio change by adding component Solution:

• Current: Milk = 30L, Water = 10L
• Let x liters water be added
• New ratio: 30:(10+x) = 1:1 ⟹ 30 = 10+x ⟹ x = 20L Answer: 20 liters of water must be added

Problem 8: Two alloys contain copper and tin in ratios 2:3 and 3:4 respectively. In what ratio should these alloys be mixed to get an alloy with copper and tin in ratio 5:7? Pattern Identification: Mixing two different ratio mixtures Solution:

• Alloy 1: Copper = 2/5, Tin = 3/5
• Alloy 2: Copper = 3/7, Tin = 4/7
• Final: Copper = 5/12, Tin = 7/12
• Using alligation: Ratio = (5/12 - 3/7):(2/5 - 5/12) = 1/84:1/60 = 5:7 Answer: Mix in ratio 5:7

Problem 9: A vessel contains 60L of pure milk. 12L is removed and replaced with water. This process is repeated. Find the final ratio of milk to water. Pattern Identification: Repeated replacement mixture Solution:

• After 1st operation: Milk = 60×(48/60) = 48L, Water = 12L
• After 2nd operation: Milk = 48×(48/60) = 38.4L, Water = 21.6L
• Ratio = 38.4:21.6 = 16:9 Answer: Final ratio is 16:9

Pattern 4: Partnership (கூட்டாண்மை)

Problem 10: A invests ₹5000 for 6 months, B invests ₹8000 for 4 months. If profit is ₹1300, find B's share. Pattern Identification: Investment with different time periods Solution:

• Investment ratio = (5000×6):(8000×4) = 30000:32000 = 15:16
• B's share = (16/31) × 1300 = ₹672.58 ≈ ₹673 Answer: B's share is ₹673

Problem 11: A, B, C invest in ratio 3:4:5. A invests for whole year, B for 8 months, C for 6 months. If profit is ₹7400, find A's share. Pattern Identification: Different investment amounts and time periods Solution:

• Investment×Time ratio = (3×12):(4×8):(5×6) = 36:32:30 = 18:16:15
• Total parts = 49
• A's share = (18/49) × 7400 = ₹2720.41 ≈ ₹2720 Answer: A's share is ₹2720

Problem 12: Two partners A and B start a business. A invests ₹12000 initially. After 3 months, B joins with ₹15000. At the end of year, profit is ₹8000. Find the profit sharing ratio. Pattern Identification: Partners joining at different times Solution:

• A's investment×time = 12000×12 = 144000
• B's investment×time = 15000×9 = 135000
• Ratio = 144000:135000 = 16:15 Answer: A:B profit sharing ratio is 16:15