The concepts of ratio and proportion find their applications in problems based on the linear equations, speed & distance, partnership, time & work and many more. Here in this chapter have all the concepts and important formulas.

**RATIO:**

The comparative relation between two quantities of the same type by division is called ratio. In other words, ratio means what part of one quantity is of another.

Ratio always occurs between the same units as kg: kg, hours: hours, liters: liters e.t.c.

Let us consider two quantities x and y, their ratio is x:y or x/y or x÷y.

Here the two quantities that are being compared are called terms. The first quantity ‘x’ is called antecedent. The second quantity ‘y’ is called consequent.

**Note On Concepts of Ratio And Proportion :**

- A ratio is a number so, when comparing two quantities they must be expressed in the same units.
- A ratio does not change even though it is multiplied or divided by the same number.

i,e 3÷ 5 = 6 ÷ 10 = 9 ÷ 15 ……..etc.

#### Proportion

If two ratios are equal to each other then, they are called proportional.

\begin{align*}

i,e \ \dfrac {a}{b}=\dfrac {c}{d} \end{align*}

are in proportion and is represented by **a:b::c:d**

#### Directly proportional

Two quantities x and y are said to be directly proportional to each other if increase/decrease in one quantity(x) produces increase/decrease in other quantity (y) respectively.

\begin{align*} It \ is \ represented \ by \ x\propto y \end{align*}

#### Inversely proportional

Two quantities x and y are said to be inversely proportional to each other if increase/decrease in one quantity(x) produces decrease /increase in other quantity (y) respectively.

\begin{align*} It \ is \ represented \ by \ x\propto \dfrac {1}{y}

\end{align*}

#### Mean Proportion

Let x be the mean/second proportion between a and b for ratio a:x :: x:b

\begin{align*}

\therefore \dfrac {a}{x}=\dfrac {x}{b}\Rightarrow x^{2}=ab\Rightarrow x=\sqrt {ab} \end{align*}

#### First Proportion

Let x be the first proportion between a,b and c for x:a :: b:c \begin{align*}

\therefore \dfrac {x}{a}=\dfrac {b}{c}\Rightarrow x=\dfrac {ab}{c}

\end{align*}

### Third Proportion

Let x be the third proportion between a and b for ratio a:b :: b:x

\begin{align*}

\therefore \dfrac {a}{b}=\dfrac {b}{x}\Rightarrow x=\dfrac {b^{2}}{a}

\end{align*}

#### Fourth Proportion

Let x be the fourth proportion between a,b and c for a:b :: c:x \begin{align*}

\therefore \dfrac {a}{b}=\dfrac {c}{x}\Rightarrow x=\dfrac {bc}{a}

\end{align*}

## Types Of Ratios

#### Mixed Ratio

Let x:y and a:b are two ratios then, ax: by is called as Mixed Ratio

#### Duplicate Ratio

The ratio of squares of two numbers (let a, b) is called the dulpicate ratio of those numbers.

\begin{align*} Duplicate \ of \ a:b \ is \ a^{2}:b^{2} \end{align*}

#### Triplicate Ratio

The ratio of cubes of two numbers (let a, b) is called the triplicate ratio of those numbers.

\begin{align*}

Triplicate \ of \ a:b \ is \ a^{3}:b^{3} \end{align*}

#### Sub Duplicate Ratio

The ratio of square roots of two numbers (let a, b) is called the sub duplicate ratio of those numbers.

\begin{align*} Sub \ Duplicate \ of \ a:b \ is \ \sqrt {a}:\sqrt {b}

\end{align*}

#### Sub Triplicate Ratio

The ratio of cube roots of two numbers (let a, b) is called sub tripicate ratio of those numbers.

\begin{align*}

Sub \ Triplicate \ of \ a:b \ is \ \sqrt [3] {a}:\sqrt [3] {b}

\end{align*}

### Formulas Of Ratio And Proportion Concepts

\begin{align*}

If \ \dfrac {a}{b}=\dfrac {c}{d} \ (or) \ a:b::c:d \\ are \ in \ proportion\ then, \end{align*}

- \begin{align*}

its \ Invertendo \ is \ \dfrac {b}{a}=\dfrac {d}{c} \end{align*} - \begin{align*}

its \ Alternendo \ is \ \dfrac {a}{c}=\dfrac {b}{d} \end{align*} - \begin{align*}

Componendo \ is \ \begin{aligned}\dfrac {a+b}{b}=\dfrac {c+d}{d}

\end{aligned} \end{align*} \begin{align*} \Rightarrow \left[ \dfrac {a}{b}+1=\dfrac {c}{d}+1\right] \end{align*} - \begin{align*}

Dividendo \ is \ \begin{aligned}\dfrac {a-b}{b}=\dfrac {c-d}{d}

\end{aligned} \end{align*} \begin{align*} \Rightarrow \left[ \dfrac {a}{b}-1=\dfrac {c}{d}-1\right] \end{align*} - \begin{align*}

If \ \begin{aligned}\dfrac {a}{b}=\dfrac {c}{d}=\dfrac {e}{f}=\ldots \end{aligned} \end{align*} then each ratio

\begin{align*} = \dfrac {a+c+e+\ldots }{b+d+f+\ldots } \\ = \ \dfrac {Sum \ Of \ Numerators}{Sum \ Of \ Denominators} \end{align*}