Ratio & Proportion - Theory & Concepts
⚖️ Ratio & Proportion - Complete Theory
Foundation for Alligation, Partnership, Time & Work, and more!
🎯 What is Ratio?
Ratio compares two quantities of the same kind.
- Expressed as a:b or a/b
- Example: If 2 apples and 3 oranges → Ratio = 2:3
Important: Both quantities must be in same unit!
📐 Basic Formulas
Ratio Basics
If a:b = m:n, then:
a/b = m/n
an = bm (cross multiplication)
From Ratio to Actual Values
If ratio is a:b and sum is S:
First part = [a/(a+b)] × S
Second part = [b/(a+b)] × S
Example: Ratio 3:2, Sum = 500
First = (3/5) × 500 = 300
Second = (2/5) × 500 = 200
🔄 Types of Ratios
1. Compounded Ratio
(a:b) and (c:d) = ac:bd
Example: (2:3) and (4:5) = 8:15
2. Duplicate Ratio
a:b → a²:b²
Example: 3:4 → 9:16
3. Triplicate Ratio
a:b → a³:b³
4. Sub-duplicate Ratio
a:b → √a:√b
5. Inverse Ratio
a:b → b:a
🎯 Proportion
Proportion: Equality of two ratios
- a:b = c:d or a/b = c/d
- Read as “a is to b as c is to d”
Types of Proportions
1. Mean Proportional:
If a/b = b/c, then b² = ac
b = √(ac)
b is the mean proportional between a and c
2. Third Proportional:
If a/b = b/c, then c is third proportional
c = b²/a
3. Fourth Proportional:
If a/b = c/d, then d is fourth proportional
d = bc/a
⚡ Important Properties
Property 1: Invertendo
If a/b = c/d, then b/a = d/c
Property 2: Alternendo
If a/b = c/d, then a/c = b/d
Property 3: Componendo
If a/b = c/d, then (a+b)/b = (c+d)/d
Property 4: Dividendo
If a/b = c/d, then (a-b)/b = (c-d)/d
Property 5: Componendo-Dividendo ⭐
If a/b = c/d, then:
(a+b)/(a-b) = (c+d)/(c-d)
Most useful for solving equations quickly!
💡 Solved Examples
Example 1: Finding Actual Values
Q: Divide ₹750 in ratio 2:3.
Solution:
Total parts = 2 + 3 = 5
First = (2/5) × 750 = ₹300
Second = (3/5) × 750 = ₹450
Example 2: Third Proportional
Q: Find third proportional to 4 and 6.
Solution:
If a/b = b/c, then c = b²/a
c = 6²/4 = 36/4 = 9
Answer: 9
Example 3: Componendo-Dividendo
Q: If (x+y)/(x-y) = 4/3, find x/y.
Solution:
Using Componendo-Dividendo in reverse:
(x+y)/(x-y) = 4/3
Adding & subtracting:
2x/2y = (4+3)/(4-3) = 7/1
x/y = 7/1
Answer: 7:1
Example 4: Income-Expenditure
Q: Incomes of A and B are in ratio 5:4. Expenditures in ratio 3:2. If both save ₹1,000, find A’s income.
Solution:
Let incomes = 5x and 4x
Let expenditures = 3y and 2y
Savings:
5x - 3y = 1000 ... (1)
4x - 2y = 1000 ... (2)
From (2): 2x - y = 500
y = 2x - 500
Substitute in (1):
5x - 3(2x - 500) = 1000
5x - 6x + 1500 = 1000
-x = -500
x = 500
A's income = 5x = 5 × 500 = ₹2,500
🔄 Ratio Applications
Application 1: Age Problems
Present ages in ratio a:b
After n years: (a+n):(b+n)
Before n years: (a-n):(b-n)
Application 2: Speed-Time-Distance
If speeds are in ratio a:b:
Times taken are in ratio b:a (inverse!)
Distances in same time are in ratio a:b
Application 3: Mixing Solutions
Links to Alligation & Mixture!
⚡ Quick Shortcuts
Shortcut 1: Ratio to Percentage
Ratio a:b
Total = a + b
a's percentage = [a/(a+b)] × 100%
b's percentage = [b/(a+b)] × 100%
Example: 3:7
Total = 10
First = 30%, Second = 70%
Shortcut 2: Increasing Ratio Terms
If a:b, multiply both by same number:
a:b = 2a:2b = 3a:3b (all same ratio!)
Shortcut 3: Combining Ratios
If A:B = 2:3 and B:C = 4:5
Make B common:
A:B = 8:12 (multiply by 4)
B:C = 12:15 (multiply by 3)
A:B:C = 8:12:15
📝 Practice Problems
Level 1:
- Divide 560 in ratio 3:5
- Find mean proportional between 9 and 16
- If a/b = 3/4, find (a+b)/(a-b)
Level 2:
- Ages of A and B are 3:5. After 6 years, 2:3. Find present ages.
- If (2x+3y)/(3x-2y) = 7/4, find x:y
- Three numbers in ratio 2:3:4, sum = 108. Find numbers.
Level 3:
- Incomes in ratio 4:3, expenditures 3:2. Both save ₹500. Find incomes.
- If A:B = 2:3, B:C = 4:5, C:D = 6:7, find A:D
- Wine and water in ratio 3:2. After adding 5L water, ratio becomes 3:4. Find original quantity.
🔗 Related Topics
Uses Ratio & Proportion:
Practice:
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