SSC CGL Preparation – Day 10

Quantitative Aptitude – Ratio and Proportion

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📘 1. Introduction

Ratio and Proportion are fundamental concepts in Arithmetic. SSC CGL frequently includes 2–3 questions based on these topics due to their wide application in time, work, mixture, speed, and profit problems.

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🔹 2. Ratio – Basic Definition

A ratio is a comparison of two quantities of the same kind, by division.

📌 Formula:
If a and b are two quantities, then their ratio is written as:
a : b or a/b

✅ Example:
Ratio of 4 pens to 6 pens = 4:6 = 2:3

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🔹 3. Key Points on Ratios

• Ratios must be between quantities of the same unit.
• Ratios can be simplified like fractions.
• If a : b = 2 : 3, then a = 2x, b = 3x for some common multiple x.

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🔹 4. Proportion – Basic Definition

When two ratios are equal, they form a proportion.

📌 Formula:
If a : b = c : d, we write it as:
a : b :: c : d

Here, a, b, c, d are in proportion.

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🔹 5. Types of Proportion

• Direct Proportion: Increase in one leads to increase in the other.
• Inverse Proportion: Increase in one leads to decrease in the other.
• Continued Proportion: If a : b = b : c, then a, b, c are in continued proportion.

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🔹 6. Properties of Proportion

1. Product of Extremes = Product of Means
     a : b = c : d ⟹ a × d = b × c
2. Invertendo: If a : b = c : d ⟹ b : a = d : c
3. Alternendo: If a : b = c : d ⟹ a : c = b : d
4. Componendo: If a : b = c : d ⟹ (a + b) : b = (c + d) : d
5. Dividendo: If a : b = c : d ⟹ (a − b) : b = (c − d) : d
6. Componendo and Dividendo:
     If a : b = c : d ⟹ (a + b) : (a − b) = (c + d) : (c − d)

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🔹 7. Compound Ratio

The ratio of the product of antecedents to the product of consequents.

✅ Example:
If a : b = 2 : 3 and c : d = 4 : 5
Then compound ratio = (2×4) : (3×5) = 8 : 15

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🔹 8. Duplicate, Triplicate, Sub-duplicate Ratios

Type	Meaning
Duplicate Ratio	Square of given ratio
Triplicate Ratio	Cube of given ratio
Sub-duplicate Ratio	Square root of given ratio

✅ Example:
Ratio = 4 : 9 →
Duplicate = 16 : 81
Triplicate = 64 : 729
Sub-duplicate = 2 : 3

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🔹 9. Application-Based Problems

✅ Ratio Division:

To divide an amount A in the ratio m : n
Share of first = mm+n×A\frac{m}{m + n} \times Am+nm​×A
Share of second = nm+n×A\frac{n}{m + n} \times Am+nn​×A

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✅ Combined Ratio (Successive Ratios):

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

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✅ Increase/Decrease in Quantity:

If a quantity increases in the ratio x : y, then percentage increase =
(y−xx)×100%\left(\frac{y – x}{x}\right) \times 100 \%(xy−x​)×100%

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🧠 10. Tricks for SSC CGL

• Convert all quantities into same units before finding the ratio.
• When a question says “in the ratio of A : B”, assume actual values as Ax and Bx to solve.
• Use Componendo and Dividendo to simplify proportion-based equations quickly.
• Master successive ratios and direct/inverse problems using tables.

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📌 11. Examples

Example 1:

Divide ₹1200 in the ratio 2 : 3.
✅ Solution:
Sum of ratios = 2 + 3 = 5
First part = (2/5) × 1200 = ₹480
Second part = ₹720

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Example 2:

If A : B = 3 : 4 and B : C = 2 : 5, find A : B : C.
✅ Solution:
Make B common in both:
A : B = 3 : 4
B : C = 4 : 10 → (Multiply both by 2.5)
Now A : B : C = 3 : 4 : 10

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📝 12. SSC CGL Previous Year Example

Q: If a sum is divided between A, B, and C in the ratio 3 : 4 : 5 and C gets ₹1500, what is the total amount?
Solution:
C’s share = 5 parts = ₹1500 ⟹ 1 part = 300
Total = (3 + 4 + 5) × 300 = ₹3600
✅ Answer: ₹3600