Ratio and Proportion: Comprehensive Guide

1. Definitions

Ratio is a comparison of two quantities by division. It represents the relation that one quantity bears to the other.

If a and b are two quantities of the same kind, then a/b is known as the ratio of a and b.

Denoted as: a : b, where:
- a is called the antecedent (first term).
- b is called the consequent (second term).
Example: The ratio of 30 kg to 50 kg is 3 : 5.

Proportion is the equality of two ratios. Four quantities a, b, c, and d are said to be in proportion if:

a : b = c : d or a : b :: c : d.

Extremes: First and fourth terms (a and d).
Means: Second and third terms (b and c).
Key Property: Product of extremes = Product of means (a × d = b × c).

2. Types of Ratios

Duplicate Ratio: a² : b².
Triplicate Ratio: a³ : b³.
Sub-duplicate Ratio: √a : √b.
Sub-triplicate Ratio: ³√a : ³√b.
Compound Ratio: ab : cd.

3. Continued Proportion

Four quantities a, b, c, and d are in continued proportion if:

a : b = b : c = c : d.

Three quantities a, b, and c are in continued proportion if:

a : b = b : c or a × c = b².

Here, b is the mean proportional between a and c.
c is the third proportional to a and b.

4. Important Formulas

Cross-Multiplication:
If a/b = c/d, then a × d = b × c.
Componendo and Dividendo:
If a/b = c/d, then (a + b)/(a - b) = (c + d)/(c - d).
Fourth Proportional:
If a : b = c : d, then d = (b × c)/a.
Mean Proportional:
If a : b = b : c, then b = √(a × c).
Third Proportional:
If a : b = b : c, then c = (b²)/a.

5. Tricks and Tips to Solve Questions

Cross-Multiplication: Use cross-multiplication to solve proportion problems quickly.
Unitary Method: Find the value of one unit and scale it up or down.
Simplify Ratios: Always simplify ratios to their lowest terms.
Use Variables: Assign variables to unknown quantities and set up equations.
Proportional Division: Divide a quantity in a given ratio using:
Part = (Ratio / Total Ratio) × Total Quantity.

6. Variation

Direct Variation: If x increases, y increases proportionally.
x = k × y (where k is a constant).
Inverse Variation: If x increases, y decreases proportionally.
x = k / y (where k is a constant).
Joint Variation: If x varies with both y and z:
x = k × y × z (where k is a constant).

7. Partnership

When two or more people invest in a business together, they are called partners.

Types of Partners:
- Working Partner: Manages the business.
- Inactive Partner: Only invests money.
Profit Sharing:
- Simple Partnership: Investments are for the same time period.
  Profit Ratio = Investment Ratio.
- Compound Partnership: Investments are for different time periods.
  Profit Ratio = Investment × Time.

8. Solved Examples

Example 1: If a/b = 3/4 and b/c = 5/6, what is a/c?

Solution: Multiply the two ratios:

a/b × b/c = 3/4 × 5/6 = 15/24 = 5/8.

Answer: 5/8.

Example 2: Divide 200 in the ratio 3 : 2.

Solution: Total ratio = 3 + 2 = 5.

First part = (3/5) × 200 = 120. Second part = (2/5) × 200 = 80.

Answer: 120, 80.

Example 3: If 40, x, x, 40 are in proportion, find x.

Solution: Product of means = Product of extremes:

x × x = 40 × 40 ⇒ x² = 1600 ⇒ x = 40.

Answer: 40.

9. MCQ Examples

Question 1: If 2 : 3 = x : 9, what is x?

A) 4
B) 6
C) 8
D) 10

Solution: Cross-multiply:

2 × 9 = 3 × x ⇒ 18 = 3x ⇒ x = 6.

Answer: B) 6.

Question 2: A and B started a business investing Rs. 90,000 and Rs. 20,000 respectively. In what ratio should the profit be divided after 2 years?

A) 9:2
B) 3:2
C) 18:20
D) 18:4

Solution: Profit ratio = Investment ratio = 90000 : 20000 = 9 : 2. Answer: A) 9:2.