# Ratio and Proportion Aptitude Formulas, Definitions, & Examples:

#### Overview:

 Topic Included: Formulas, Definitions & Exmaples. Main Topic: Quantitative Aptitude. Quantitative Aptitude Sub-topic: Ratio and Proportion Aptitude Notes & Questions. Questions for practice: 10 Questions & Answers with Solutions.

#### What is Ratio:

Comparision of two numbers in the form of division is called ratio.

Example: Comparision of $$x$$ and $$y = x : y = \frac{x}{y}$$

#### Types of Ratio:

(1). Compound Ratio: The multiplication of two or more ratios is called compounded ratio.

Example: Find the compounded ratio of $$2 : 3$$ and $$4 : 5$$?

Solution: compound ratio of $$2 : 3$$ and $$4 : 5$$ is $$2 : 3 \times 4 : 5$$ $$= \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$$

(2). Duplicate Ratio: Square of any ratio is called duplicate ratio and square-root of any ratio is called sub-duplicate ratio.

Example (1): Find the duplicate ratio of $$3 : 4$$?

Solution: duplicate ratio of $$3 : 4 = 3^2 : 4^2 = 9 : 16$$

Example (2): Find the sub-duplicate ratio of $$4 : 9$$?

Solution: sub-duplicate ratio of $$4 : 9 = \sqrt 4 : \sqrt 9 = 2 : 3$$

(3). Triplicate Ratio: Cube of any ratio is called triplicate ratio and cube-root of any ratio is called sub-triplicate ratio.

Example (1): Find the triplicate ratio of $$2 : 3$$?

Solution: triplicate ratio of $$2 : 3 = 2^3 : 3^3 = 8 : 27$$

Example (2): Find the sub-triplicate ratio of $$27 : 64$$?

Solution: sub-triplicate ratio of $$27 : 64 = \sqrt[3] {27} : \sqrt[3] {64} = 3 : 4$$

#### What is Proportion:

Proportion is generated from the word proportional which means equivalent. When two ratios are equivalent to each other then it is called proportion. The sign of proportion is (::).

Example: $$2 : 3 :: 4 : 9 \Rightarrow \frac{2}{3} = \frac{4}{9}$$, Here $$\frac{2}{3}$$ is equivalent to $$\frac{4}{9}$$

#### Types of Proportion:

(1). Third Proportion: Let proportion $$A : B :: B : K$$, then $$K$$ is called third proportion of $$A$$ and $$B$$, and mathematically will be expressed as-$$Proportion = A : B :: B : K$$ $$\frac{A}{B} = \frac{B}{K} \Rightarrow \bbox[5px,border:2px solid #800000] {K = \frac{B^2}{A}}$$

Example: Find the third proportion of $$2$$ and $$3$$?

Solution: Given values, $$A = 2$$ and $$B = 3$$ then, Third proportion of $$2$$ and $$3$$ will be $$K = \frac{B^2}{A}$$ $$K = \frac{3^2}{2} = \frac{9}{2} = 4.5$$

(2). Fourth Proportion: Let proportion $$A : B :: C : K$$, then $$K$$ is called fourth proportion of $$A$$, $$B$$ and $$C$$, and mathematically will be expressed as-$$Proportion = A : B :: C : K$$ $$\frac{A}{B} = \frac{C}{K} \Rightarrow \bbox[5px,border:2px solid #800000] {K = \frac{B \times C}{A}}$$

Example: Find the fourth proportion of $$2$$, $$3$$ and $$4$$?

Solution: Given values, $$A = 2$$, $$B = 3$$ and $$C = 4$$ then, Fourth proportion of $$2$$, $$3$$ and $$4$$ will be $$K = \frac{B \times C}{A}$$ $$K = \frac{3 \times 4}{2} = \frac{12}{2} = 6$$

(3). Mean Proportion: Let proportion $$A : K :: K : B$$, then $$K$$ is called mean proportion of $$A$$ and $$B$$, and mathematically will be expressed as-$$Proportion = A : K :: K : B$$ $$\frac{A}{K} = \frac{K}{B}$$ $$K^2 = A \times B \Rightarrow \bbox[5px,border:2px solid #800000] {K = \sqrt {AB}}$$

Example: Find the mean proportion between $$4$$ and $$5$$?

Solution: Given values, $$A = 4$$ and $$B = 5$$ then, mean proportion of $$4$$ and $$5$$ will be $$K = \sqrt {AB}$$ $$K = \sqrt {4 \times 5} = \sqrt {20}$$