Ratio and Proportion - TNPSC Tutorial

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📊 Rate and Proportion - Basics

A mathematical method of comparing two or more quantities!

📚 What is ratio?

Ratiois to compare two quantities of the same type.

🎯 Code:a : b (reading: "for a and b")
📌 Meaning:a : b means a/b
💡 Example:3 : 5 means 3/5

⭐ Ratio parts

Part	Explanation	Example (3:5)
Antecedent	The first element of the ratio	3
Consequent	The second element of the ratio	5

📐 Ratio types

Type	Explanation	example
simple ratio	Simplified ratio	6:9 = 2:3
Duplicate	a² : b²	Double of 2:3 = 4:9
Triplicate	a³ : b³	Triple of 2:3 = 8:27
Sub-duplicate	√a : √b	Complement of 4:9 = 2:3
Sub-triplicate	∛a : ∛b	Sub-triple of 8:27 = 2:3
Inverse Ratio	b : a	Inverse of 2:3 = 3:2

📊 What is Proportion?

Proportionis the condition where two ratios are equal.

🎯 Code:a : b :: c : d (or) a : b = c : d
📌 Reading Method:"The ratio of a to b is equal to the ratio of c to d"
💡 Meaning:a/b = c/d

🔢 Elements of Proportion

organ	Explanation	In a:b::c:d
Extremes	First and last	a and d
Means	The ones in the middle	b and c

🔑 Key Rule:Product of end elements = product of middle elements
a × d = b × c

📈📉 Direct & Inverse Proportions

📈 Direct Proportion (Direct)

If one increases, so will the other

x ∝ y → x = ky

Eg: More work, more pay

📉 Inverse Proportion

If one increases, the other decreases

x ∝ 1/y → xy = k

Eg: High speed, low time

🎯 Rate characteristics

Multiplying or dividing both elements of a ratio by the same number does not change the ratio
a : b = ka : kb (k ≠ 0)
The order of the ratio is important: 2:3 ≠ 3:2
The ratio will not have a unit

📐 Ratio and Proportion - Key Formulas

All Formulas Essential for TNPSC Exam!

🔢 Basic Ratio Formulas

formula	Explanation
a : b = a/b	Converting a ratio to a fraction
a : b = ka : kb	Equal ratio (k ≠ 0)
a : b : c = a/c : b/c : 1	Three ratio simplification

📊 Proportional Formulas

Type	formula	Note
Basic proportions	a : b :: c : d → a × d = b × c	Cross multiplication
The fourth is proportionality	d = (b × c) / a	a : b :: c : ? If
The third is proportionality	c = b² / a	a : b :: b : ? If
Average Proportion	x = √(a × b)	If a : x :: x : b

💰 Allocation Formulas

condition	formula
Bisector (a:b)	First Share = Total × a/(a+b) Second Part = Total × b/(a+b)
Tripartite (a:b:c)	First Share = Total × a/(a+b+c) Second Part = Total × b/(a+b+c) Third share = Total × c/(a+b+c)

🔄 Ratio conversion formulas

Change	Original	After changing
Invertendo	a : b = c : d	b : a = d : c
Alternendo	a : b = c : d	a : c = b : d
Componendo	a : b = c : d	(a+b) : b = (c+d) : d
Dividendo	a : b = c : d	(a-b) : b = (c-d) : d
Componendo-Dividendo	a : b = c : d	(a+b):(a-b) = (c+d):(c-d)

📈 Mixture - Alligation

🎯 Rule of Alligation:
Ratio of two goods = (price difference)

Formula:
Lowest Price : Highest Price = (Highest Price - Average) : (Average - Lowest Price)

condition	formula
mixing ratio	a : b = (d₂ - d) : (d - d₁)
Average price	M = (a×d₁ + b×d₂) / (a + b)

👥 Partnership formulas

Type	formula
Simple partnership	Rate of return = Rate of investment
Joint partnership	Rate of return = (Investment × Duration) ratio A : B = (P₁ × T₁) : (P₂ × T₂)

📊 Ratio comparison formulas

comparison	method
Compare a:b and c:d	a×d > b×c → a:b > c:d a×d < b×c → a:b < c:d a×d = b×c → a:b = c:d
Integrating ratios	a:b and b:c → a:b:c (Equating b)

📝 Rate and Proportion - 10 Key Examples

Frequently Asked Question Types in TNPSC Exam!

📌 Example 1: Calculation of simple ratio

Question:What is the simple ratio of 75 to 125?

Solution:
Step 1: 75 : 125
Step 2: Divide both by 25
Step 3: 75 ÷ 25 : 125 ÷ 25
Step 4: 3: 5

Answer: 3 : 5

📌 Example 2: Division by ratio

Question:If Rs.630 is divided among A, B, C in the ratio 2:3:4, what is the share of each?

Solution:
Total parts = 2 + 3 + 4 = 9

A's share = 630 × 2/9 = 140
B's share = 630 × 3/9 = 210
C's share = 630 × 4/9 = 280

Answer: A = ₹140, B = ₹210, C = ₹280

📌 Example 3: Fourth Proportionality

Question:3 : 5 :: 12 : ? - See Fourth Proportion.

Solution:
Formula: a : b :: c : d → d = (b × c) / a

d = (5 × 12) / 3
d = 60 / 3
d = 20

Answer: 20

📌 Example 4: Third Proportion

Question:What is the third proportional of 4 and 16?

Solution:
Formula: a : b :: b : c → c = b² / a

c = 16² / 4
c = 256 / 4
c = 64

Answer: 64

📌 Example 5: Mean Proportionality

Question:What is the mean proportional of 9 and 16?

Solution:
Formula: a : x :: x : b → x = √(a × b)

x = √(9 × 16)
x = √144
x = 12

Answer: 12

📌 Example 6: Combining Proportions

Question:If A : B = 2 : 3 and B : C = 4 : 5, then A : B : C = ?

Solution:
Let B be equal.

A : B = 2 : 3 → 8 : 12 (multiply by 4)
B : C = 4 : 5 → 12 : 15 (multiply by 3)

A : B : C = 8 : 12 : 15

Answer: 8 : 12 : 15

📌 Example 7: Distribution of Partnership Profits

Question:A invests Rs.50,000 for 12 months and B invests Rs.60,000 for 10 months. How will they divide the profit of Rs.22,000?

Solution:
A's share = 50,000 × 12 = 6,00,000
Share of B = 60,000 × 10 = 6,00,000

Ratio = 6,00,000 : 6,00,000 = 1 : 1

A's profit = 22,000 × 1/2 = 11,000
B's profit = 22,000 × 1/2 = 11,000

Answer: A = ₹11,000, B = ₹11,000

📌 Example 8: Calculation of age in proportion

Question:Age ratio of father and son is 7:2. After 5 years the ratio is 8:3. What is their current age?

Solution:
Current age = 7x and 2x etc

After 5 years:
(7x + 5) / (2x + 5) = 8 / 3

3(7x + 5) = 8(2x + 5)
21x + 15 = 16x + 40
5x = 25
x = 5

Father's age = 7 × 5 = 35 years
Age of son = 2 × 5 = 10 years

Answer: Father = 35 years, son = 10 years

📌 Example 9: Mixture calculation

Question:Mixing ₹40/kg tea and ₹50/kg tea in what ratio gives a mixture of ₹44/kg?

Solution:
Method of Alligation:

Lowest Price : Highest Price = (Highest Price - Average) : (Average - Lowest Price)

= (50 - 44) : (44 - 40)
= 6 : 4
= 3 : 2

Answer: 3 : 2

📌 Example 10: Componendo-Dividendo

Question:If (x + 3) / (x - 3) = 5/3, what is the value of x?

Solution:
To use Componendo-Dividendo:
(a+b)/(a-b) = (c+d)/(c-d)

Here: x+3 / x-3 = 5/3

Componendo-Dividendo:
[(x+3)+(x-3)] / [(x+3)-(x-3)] = (5+3) / (5-3)
2x / 6 = 8 / 2
2x / 6 = 4
2x = 24
x = 12

Answer: x = 12

📊 Important notes

Always write the ratio in simple form
Calculate the total parts first in division questions
Investment × duration ratio is important in partnership
Alligation method is very useful for compound questions

⚡ Rate and Proportion - Crossroads

Super Tricks to Save Time in TNPSC Exam!

🚀 Trick 1: Distribution Quick Method

Divide in the ratio a : b:

First Share = Total × a ÷ (a+b)
Second Part = Total × b ÷ (a+b)

Example:To divide ₹500 in the ratio 2:3

First share = 500 × 2 ÷ 5 =₹200

Second share = 500 × 3 ÷ 5 =₹300

🚀 Trick 2: Proportional fourth element

a : b :: c : ? If:

? = (b × c) ÷ a

Example:2 : 5 :: 6 : ?

? = (5 × 6) ÷ 2 = 30 ÷ 2 =15

🚀 Trick 3: Mean Proportionality - Square Root

Mean Proportional of a and b:

= √(a × b)

Example:Average ratio of 4 and 25

= √(4 × 25) = √100 =10

🚀 Trick 4: Third Proportion - Square

Third Proportional of a and b:

= b² ÷ a

Example:Third proportionality of 3 and 6

= 6² ÷ 3 = 36 ÷ 3 =12

🚀 Trick 5: To combine ratios - LCM method

If A:B = m:n and B:C = p:q:

A:B:C = mp : np : nq
(or find LCM of B)

Example:A:B = 2:3, B:C = 5:7

LCM(3,5) = 15 of B

A:B = 10:15, B:C = 15:21

A:B:C =10:15:21

🚀 Trick 6: Alligation - X shape

Price 1 (d₁) Price 2 (d₂)
\ /
\ /
Mean (M)
/ \
/ \
(d₂-M) (M-d₁)
Ratio = (d₂-M) : (M-d₁)

Example:Mix ₹30 and ₹40 rice to make ₹36 mix

Ratio = (40-36) : (36-30) = 4 : 6 =2:3

🚀 Trick 7: Partnership - Multiplication Ratio

Investment × Duration = Equity

A : B = (P₁ × T₁) : (P₂ × T₂)

Example:A: ₹1000 × 6 months, B: ₹1500 × 4 months

A : B = 6000 : 6000 =1 : 1

🚀 Trick 8: Age ratio questions

In the ratio a:b, after n years c:d but:

Difference = n × (ad - bc) / [(c-a)(d-b) difference]

Shortcut:Find the value of x by taking the change of ratio difference

🚀 Trick 9: Componendo-Dividendo direct formula

If (a+b)/(a-b) = (c+d)/(c-d):

a/b = (c+d+c-d)/(c+d-c+d) = 2c/2d = c/d

Example:(x+4)/(x-4) = 3/2

CD: (x+4+x-4)/(x+4-x+4) = (3+2)/(3-2)

2x/8 = 5/1 → x =20

🚀 Trick 10: Percentage change in ratio

x% increase/decrease in a:b ratio:

New ratio = a(1±x/100) : b(1±x/100)

Quick Tip:If both elements of the ratio are changed by the same percentage, the ratio will not change!

📊 TNPSC Frequently Asked Categories

Question type	Crossroads
Distribution Questions	Total × Part ÷ Total Parts
proportionality	Cross multiply ad = bc
composition	Alligation X method
partnership	Investment × Duration
age	Putting the ratio as x, Eq

💡 Important memory tips

Ratio = Fraction:3:5 = 3/5
Proportionality = Cross Multiplication:ad = bc
Mean = square root:√(ab)
Third = Square ÷ First:b²/a
Direct Ratio:One is ↑ and the other is ↑
Counter Ratio:One is ↑ and the other is ↓

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