Ratio & Proportion
SSC-CGL Exams
1. Introduction
Ratio and Proportion form the foundation of many SSC CGL questions — including partnership, income sharing, and mixtures. Mastering this chapter helps in solving Time–Speed–Distance, Work, and Allegation problems too.
2. Concept of Ratio
A ratio compares two or more quantities of the same kind in the same units.
Ratio of a to b = a/b or a : b
Key Points
Ratio has no unit
Multiplying or dividing both terms by the same number doesn't change the ratio
2:3 = 4:6 = 6:9
The sum of ratio parts can be used to find shares
A's share = (A's part / Sum of parts) × Total
Example 1
If the ratio of A and B's salaries is 3:5 and the total salary is ₹40000, find A's salary.
A = 3/(3+5) × 40000 = 3/8 × 40000 = ₹15000
A = ₹15000, B = ₹25000
3. Concept of Proportion
When two ratios are equal, they are said to be in proportion.
a:b = c:d ⇒ a/b = c/d
Here, a, b, c, d are in proportion. The extremes are a and d, and the means are b and c.
Product of extremes = Product of means
a × d = b × c
Example 2
If 3:9 = x:27, find x.
3×27 = 9×x ⇒ x = 9
x = 9
4. Types of Variation
| Type | Description | Example |
|---|---|---|
| Direct Variation | If one quantity increases, the other also increases | More money → More goods |
| Inverse Variation | If one increases, the other decreases | More men → Less time |
| Joint Variation | One quantity varies with two or more quantities | Cost ∝ Quantity × Rate |
Formulas
Direct: x₁/y₁ = x₂/y₂
Inverse: x₁ × y₁ = x₂ × y₂
Example 3 (Direct)
If 5 pens cost ₹25, how much will 8 pens cost?
5/25 = 8/x ⇒ x = 40
₹40
Example 4 (Inverse)
If 6 men can finish a work in 10 days, how many men can finish it in 5 days?
6×10 = x×5 ⇒ x = 12
12 men
5. Partnership & Income Distribution
In a partnership, profit is divided in the ratio of money × time invested.
Profit Ratio = (P₁ × T₁) : (P₂ × T₂) : (P₃ × T₃)
Example 5
A invests ₹12000 for 12 months, and B invests ₹8000 for 9 months. Find their profit-sharing ratio.
A:B = (12000×12):(8000×9) = 144000:72000 = 2:1
Profit Ratio = 2:1
Example 6
If A and B invest in the ratio 3:5 and earn ₹16000 total profit, find each share.
A = 3/(3+5)×16000 = 6000
B = 5/(3+5)×16000 = 10000
A = ₹6000, B = ₹10000
Example 7 (New Partner)
A and B start a business. After 4 months, C joins. A's investment = ₹8000, B's investment = ₹12000, C's investment = ₹16000. Find profit ratio at the end of 1 year.
A:B:C = (8000×12):(12000×12):(16000×8) = 96000:144000:128000 = 3:4.5:4
Ratio = 6:9:8
6. Shortcut Tips
| Case | Trick |
|---|---|
| Sum-based ratio | Total = sum of ratio parts |
| Partnership | Profit ∝ Money × Time |
| Direct variation | Use proportion (x₁/y₁ = x₂/y₂) |
| Inverse variation | Use product (x₁y₁ = x₂y₂) |
7. Practice Section
Q1. Divide ₹900 in the ratio 2:3:4.
View Answer
Sum = 2+3+4 = 9
Parts = 200, 300, 400
₹200, ₹300, ₹400
Q2. If 4:5 = 8:x, find x.
View Answer
4×x = 5×8 ⇒ x = 10
x = 10
Q3. If 6 men finish a work in 8 days, in how many days will 12 men finish it?
View Answer
6×8 = 12×x ⇒ x = 4
4 days
Q4. A and B start a business investing ₹6000 and ₹8000 respectively. Find their profit ratio.
View Answer
A:B = 6000:8000 = 3:4
3:4
Q5. A invests ₹20000 for 10 months and B invests ₹25000 for 8 months. Find their profit ratio.
View Answer
A:B = 20000×10 : 25000×8 = 200000 : 200000 = 1:1
Equal Profit
Q6. A, B, and C start a business. A invests ₹10000 for 12 months, B invests ₹8000 for 6 months, C invests ₹12000 for 4 months. Find profit ratio.
View Answer
A:B:C = (10000×12):(8000×6):(12000×4) = 120000:48000:48000 = 5:2:2
5:2:2
Q7. If 3 men can do a work in 12 days, how many days will 6 men take?
View Answer
3×12 = 6×x ⇒ x = 6
6 days
8. Quick Summary
| Concept | Formula | Example |
|---|---|---|
| Ratio | a:b | 2:3 |
| Proportion | a:b = c:d ⇒ ad = bc | 2:4 = 3:6 |
| Direct Variation | x₁/y₁ = x₂/y₂ | pens & cost |
| Inverse Variation | x₁y₁ = x₂y₂ | men & time |
| Partnership | Profit ∝ Money × Time | A:B = P₁T₁ : P₂T₂ |
You've completed Article 5: Ratio & Proportion!
Courage Tip: Always read carefully whether the problem says "joined after" or "withdrew after." These time factors are the real game changers in partnership questions.
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