Ratio & Proportion is one of the most important topics for competitive exams. It forms the base for questions of partnership, mixtures, profit & loss, time & work, time–speed–distance, and more.
In this chapter, we will understand basics, shortcut tricks, conversions, and high-level exam questions in the simplest way.

1. Ratio –

What is a Ratio?

A ratio compares two quantities of the same kind (same units).
Example:

5 kg : 2 kg
3 boys : 4 boys
20 minutes : 10 minutes

Ratios show how many times one quantity contains another.

Forms of Ratios

A ratio can be written in three forms:

a : b
a / b
a to b

All three represent the same comparison.

Key Characteristics

(a) Ratios have no units

Because units cancel out.
Example: 10 kg : 5 kg → 10/5 = 2 → ratio = 2 : 1

(b) Ratios can be simplified (like fractions)

You can divide both sides by the GCD or HCF.

Example:
36 : 54
GCD = 18 → 36/18 : 54/18 = 2 : 3

(c) Equivalent Ratios

If you multiply or divide both sides by the same number, the ratio remains the same.

Example:
2 : 5
Multiply both by 3 → 6 : 15
Divide both by 2 → 1 : 2.5

Ratio in Real Life

Speed comparison
Age comparison
Mixture problems (milk : water)
Financial investment (A invests : B invests)
Recipe proportions

Internal & External Ratios

Internal Ratio:

Dividing a quantity internally into parts.
Example: ₹300 divided in ratio 1 : 2 →
Total parts = 3
→ 300 × 1/3 = 100
→ 300 × 2/3 = 200

External Ratio:

Used in geometry (dividing line segments externally).

Conversion Between Ratio & Fraction

Ratio → Fraction

a : b = a/b
Example: 3 : 4 = 3/4

Fraction → Ratio

a/b → a : b
Example: 0.75 = 75/100 → divide by 25 → 3 : 4

Shortcut Ratio Tricks

Trick 1: From Percentage to Ratio

45% : 55%
Remove % → 45 : 55
Divide by 5 → 9 : 11

Trick 2: Fraction to Ratio Quickly

3/8
Ratio = 3 : 8
If two fractions:
(3/4) : (5/6)
LCM of denominators = 12
→ 3/4 = 9/12
→ 5/6 = 10/12
Ratio = 9 : 10

Trick 3: Ratio Scaling (Very Useful in Exams)

Given A : B = 3 : 7
B : C = 14 : 5
Make B equal:
7 × 2 = 14
So A : B = 6 : 14
Hence A : B : C = 6 : 14 : 5

✔ Convert Ratio → Fraction

Ratio a : b
Fraction of a = a / (a + b)

Example:
Ratio = 2 : 3
Fraction of 2 = 2/5
Fraction of 3 = 3/5

✔ Convert Fraction → Ratio

Example:
Fraction = 3/7
Ratio = 3 : (7–3) = 3 : 4

✔ Convert Ratio → Percentage

a : b → a/(a+b) × 100

2. Proportion – Complete, Deep Explanation

What is Proportion?

A proportion states that two ratios are equal.
Example:
2/3 = 4/6 → This is a proportion
2 : 3 :: 4 : 6

Types of Proportion

(A) Direct Proportion

When one quantity increases, the other increases.
Example:
More hours worked → More salary
Mathematically:
a/b = c/d

(B) Inverse Proportion

One quantity increases while the other decreases.
Example:
Speed ↑ → Time ↓
Workers ↓ → Days ↑
Formula:
a × b = c × d

(C) Continued Proportion

If a : b = b : c, then a, b, c are in continued proportion.

(D) Compound Proportion

Combination of multiple proportions.
Example:
If 12 men do a job in 15 days, how many days will 20 men take?

Rules in Proportion

1. Product Rule

In a proportion:
a : b = c : d
So,
a × d = b × c
This is called the Cross-multiplication rule.

2. Mean & Extreme Terms

a : b = c : d
a, d → Extreme terms
b, c → Mean terms
Product of extremes = Product of means

3. Fourth Proportion

a : b = c : x
x = (b × c) / a

4. Third Proportion

a : b = b : x
x = (b²)/a

5. Mean Proportion (Geometric Mean)

a : x = x : b
x² = ab → x = √ab

⭐ Exam-Level Applications (Important for SSC, Banking, CET, Railways, etc.)

📌 1. Ratio-based distribution

Example:
₹1200 is divided between A and B in ratio 3 : 5.

Total parts = 3+5 = 8
A = 1200 × 3/8 = ₹450
B = 1200 × 5/8 = ₹750

📌 2. Changing Ratio Questions

Boys : Girls = 4 : 5
If boys increase by 20, new ratio becomes 6 : 5.
Find girls.

Let original boys = 4k
Original girls = 5k

4k + 20 = 6k
→ 20 = 2k
→ k = 10
→ Girls = 50

📌 3. Exam Trick – Ratios with Difference

Ratio = 5 : 8
Difference = 12

Unit difference = 8 – 5 = 3
Value of 1 unit = 12/3 = 4
Values:
A = 5×4 = 20
B = 8×4 = 32

📌 4. Partnership (Direct Application of Ratios)

A invests ₹4000 for 12 months
B invests ₹6000 for 8 months
Profit ratio = (4000×12) : (6000×8)
= 48000 : 48000
= 1 : 1

📌 5. Mixture & Alligation (Ratio Shortcut)

Ratio = (High value – Mean) : (Mean – Low value)

Example:
Mix milk ₹60/litre with water ₹0/litre to get mixture of ₹40/litre.

Ratio = (60–40) : (40–0) = 20 : 40 = 1 : 2

Deeper Algebra: Solving Proportional Equations

Example:
If $x : y = 4 : 7$ and $x + y = 33$ , find $x, y$ .

Let $x = 4 k$ , $y = 7 k$ . Sum: $4 k + 7 k = 11 k = 33 \Rightarrow k = 3$ . So $x = 12$ , $y = 21$ .

Why this works: Any ratio $a : b$ can be represented as $a \cdot k : b \cdot k$ .

🔷 Advanced Tips & Pitfalls

Always simplify early. Cancelling common factors reduces arithmetic errors.
Watch units. If ratios compare lengths and times, ensure consistent units.
When multiple ratios are chained, find a common term and scale appropriately.
For inverse proportion problems, convert them to direct proportionality by reciprocals.

🔷 Practice Problems (with answers)

Solve these — do them without a calculator to build speed.

Divide ₹600 in the ratio $3 : 7$ .
If $a : b = 5 : 8$ and $b : c = 4 : 9$ , find $a : b : c$ .
If $\frac{x}{12} = \frac{5}{18}$ , find $x$ .
Three people invest ₹2000, ₹3000, ₹5000. Profit ₹6000 distributed in ratio of investments. Find each share.
If $x$ is inversely proportional to $y$ and $x = 6$ when $y = 4$ , find $x$ when $y = 12$ .

Answers (work shown briefly)

Sum = $3 + 7 = 10$ . Parts: $600 \times \frac{3}{10} = 180$ , $600 \times \frac{7}{10} = 420$ .
Make B common: $a : b = 5 : 8$ → multiply by 4 → 20:32. $b : c = 4 : 9$ → multiply by 8 → 32:72. So $a : b : c = 20 : 32 : 72$ . Simplify dividing by 4 → $5 : 8 : 18$ .
$x / 12 = 5 / 18 \Rightarrow x = 12 \times \frac{5}{18} = 12 \times \frac{5}{18} = \frac{60}{18} = \frac{10}{3} \approx 3.333...$ . (Or $x = \frac{10}{3}$ .)
Investment ratio = $2000 : 3000 : 5000 = 2 : 3 : 5$ . Total parts = 10. Profit per part = $6000 / 10 = 600$ . Shares: 2×600=1200, 3×600=1800, 5×600=3000.
$x \propto 1 / y$ . Constant $k = x \times y = 6 \times 4 = 24$ . When $y = 12$ , $x = 24 / 12 = 2$ .

⭐ Quick Revision Table

Concept	Meaning	Shortcut
Ratio	Comparison	Simplify with HCF
Proportion	Equality of ratios	Cross multiplication
Ratio → Fraction	a/(a+b)	Direct
Ratio combine	A:B & B:C	Equalize middle term
Exam trick	Ratios with difference	Find 1 unit value

⭐ Key Takeaways

Ratio & Proportion is the foundation for many other chapters.
Always simplify ratios using HCF.
Use cross-multiplication to confirm proportion.
For any distribution, convert ratio to “total parts → unit value”.
Learn to combine ratios – this is the most common exam skill.
Practice speed techniques daily for perfection.

⏭ Next Chapter Preview: – Percentage

In the next chapter we will study:

Basics of percentage
Increase/Decrease shortcuts
Successive percentage tricks
Exam-level percentage questions
Smart shortcuts used in banking & CET exams

Link to:

Ratio & Proportion – Concepts, Tricks, Conversions & Exam-Level Applications