Understanding Ratio and Proportion

Ratio is the comparison between two quantities of the same kind. It is expressed as a:b or a/b.

Proportion means two ratios are equal. If a:b = c:d, then a, b, c, d are in proportion.

Important Formulas

1. Ratio Basics

If a:b = 2:3, then:

• a/b = 2/3
• a = 2x and b = 3x (where x is common factor)

2. Proportion Rules

If a:b = c:d, then:

• Product of Extremes = Product of Means → a × d = b × c
• Alternendo: a:c = b:d
• Invertendo: b:a = d:c
• Componendo: (a+b):b = (c+d):d
• Dividendo: (a-b):b = (c-d):d

3. Compound Ratio

If a:b and c:d are two ratios, then: Compound Ratio = (a×c):(b×d)

4. Third/Fourth Proportional

If a:b = b:c, then c is the third proportional to a and b c = b²/a

If a:b = c:d, then d is the fourth proportional d = (b×c)/a

5. Mean Proportional

If a:b = b:c, then b is the mean proportional between a and c.

b = √(a × c)

Solved Examples

Example 1: Basic Ratio

Question: Divide ₹850 in the ratio 3:2.

Solution:

Sum of ratio parts = 3 + 2 = 5 First part = (3/5) × 850 = ₹510 Second part = (2/5) × 850 = ₹340

Answer: ₹510 and ₹340

Example 2: Proportion Problem

Question: If 15:x = 25:40, find x. Solution:

Using: Product of extremes = Product of means 15 × 40 = 25 × x 600 = 25x x = 600/25 = 24

Answer: x = 24

Example 3: Ages in Ratio (SSC Favorite!)

Question: The ratio of ages of A and B is 3:5. After 4 years, the ratio becomes 2:3. Find their present ages.

Solution:

Let present ages be 3x and 5x

After 4 years: (3x + 4)/(5x + 4) = 2/3

Cross multiply: 3(3x + 4) = 2(5x + 4) 9x + 12 = 10x + 8 x = 4

Present ages: 3×4 = 12 years and 5×4 = 20 years

Answer: 12 years and 20 years

Important Tricks & Shortcuts

Trick 1: Quick Ratio Comparison

To compare ratios a:b and c:d:

• Calculate a/b and c/d
• Larger fraction = Larger ratio

Trick 2: Ratio Increase/Decrease

If a ratio a:b is increased by adding x to both:

• New ratio = (a+x):(b+x)
• This ratio will be closer to 1:1 than original

Trick 3: Finding Unknown from Ratio

If a:b = 3:4 and a+b = 140:

• Quick Method: a = (3/7)×140 = 60, b = (4/7)×140 = 80

Related Concepts

Understanding Ratio and Proportion helps you master:

📖 Next Steps

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