Ratio and Proportion: Introduction


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 \(27 : 64 = \sqrt[3] {27} : \sqrt[3] {64} = 3 : 4\)

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 $$ K = \sqrt {AB} $$ $$ K = \sqrt {4 \times 5} = \sqrt {20} $$




Ratio and Proportion

Chapter 1: Introduction

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