We looked at some basic concepts in our earlier published material.

Ratio

If $\displaystyle a \text{ and } b$ are two quantities of the same kind and in the same units such that $\displaystyle b \neq 0$ then the quotient $\displaystyle \frac{a}{b}$ is called the ratio between $\displaystyle a \text{ and } b$.

Points to remember:

1. Ratio $\displaystyle \frac{a}{b}$ has no units and can be written as $\displaystyle a:b$.
2. In the ratio $\displaystyle a:b$, we call $\displaystyle a$ as the first term or antecedent and $\displaystyle b$ as the second term or consequent. The second term of the ratio cannot be zero.
3. In a ratio $\displaystyle a:b , b \neq 0$ cannot be zero. Similarly, In a ratio $\displaystyle b:a , a \neq 0$ cannot be zero.
4. If both terms of the ratio are multiplied by or divided by the same number, the ratio does not change. The same is not true if we were to add or subtract the same number from terms of the ratio.
5. A ratio must always be represented in the lowest terms. If the H.C.F of both the terms is 1, then we can say that the ratio of both the terms is the lowest.
6. Also $\displaystyle a:b$ and $\displaystyle b:a$ are equal only if $\displaystyle a = b$. What that means is that the order of the terms of the ratio is important.

Increase or decrease in the ratio

If the quantity increases or decreases in the ratio $\displaystyle a:b$, then the new resulting quantity would be $\displaystyle \frac{b}{a}$ times the original quantity. Let us say that the original quantity was $\displaystyle x$, then the new quantity $\displaystyle = \frac{b}{a} \times x$

Commensurable and incommensurable quantities:

If the ratio between any two quantities of the same units can be expressed in the ratios of integers, then the quantities are said to be commensurable, or else they are incommensurable quantities. So for example, $\displaystyle \frac{3}{7}$ are commensurable quantities while $\displaystyle \frac{\sqrt{3}}{7}$ are inconsumable quantities.

Composition of Ratios

Compound Ratio: When two or more ratios are multiplied term-wise, the ratio thus obtained is called compound ratio. Example:

For ratios $\displaystyle a:b$ and $\displaystyle c:d$, the compound ratio is $\displaystyle (a \times c):(b \times d)$.

Similarly if there were three ratios, $\displaystyle a:b$, $\displaystyle c:d$ and $\displaystyle e:f$ , then the compound ratio would be $\displaystyle (a \times c \times e):(b \times d \times f)$.

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Duplicate Ratio: It is the compound ratio of two equal ratios.

Duplicate ratio of $\displaystyle a:b$

= Compound ratio of $\displaystyle a:b \text{ and } a:b$

= $\displaystyle (a \times a):(b \times b) = a^2:b^2$

Triplicate Ratio: It is the compound ratio of three equal ratios.

Triplicate ratio of $\displaystyle a:b$

= Compound ratio of $\displaystyle a:b , a:b \text{ and } a:b$

= $\displaystyle (a \times a \times a):(b \times b \times b) = a^3:b^3$

Sub-duplicate Ratio: For any ratio $\displaystyle a:b$, its sub-duplicate ratio is $\displaystyle \sqrt{a}: \sqrt{b}$

Sub-triplicate Ratio: For any ratio $\displaystyle a:b$, its sub-triplicate ratio is $\displaystyle \sqrt[3]{a}: \sqrt[3]{b}$

Reciprocal Ratio: For any ratio $\displaystyle a:b$, where $\displaystyle a, b \neq 0$, its reciprocal ratio $\displaystyle = \frac{1}{a} : \frac{1}{b} = b:a$

Proportion

Four non zero quantities, $\displaystyle a, b, c, \text{ and } d$, are said to be in proportion if $\displaystyle a:b=c:d$

In the above case, $\displaystyle a$ is the first term, $\displaystyle b$ is the second term, $\displaystyle c$ is the third and $\displaystyle d$ is the fourth term.

$\displaystyle a$ and $\displaystyle d$ are called the extremes and $\displaystyle b$ and $\displaystyle c$ are called means (middle terms).

When

$\displaystyle a:b=c:d \Rightarrow \frac{a}{b} = \frac{c}{d} \Rightarrow a \times d = b \times c$

$\displaystyle \Rightarrow \text{product of extremes = product of means}$

In $\displaystyle a:b=c:d$ quantities $\displaystyle a \text{ and } b$ should be of the same units and $\displaystyle c \text{ and } d$ should be of the same units. Example $\displaystyle 5 m : 10 m = 10 Rs. : 20 Rs.$

$\displaystyle a, b, \text{ and } c$ are in continued proportion if $\displaystyle a:b=b:c$. Here $\displaystyle b$ is the mean proportion between $\displaystyle a \text{ and } c$. And $\displaystyle c$ is the third proportion between $\displaystyle a \text{ and } b$.

Important Properties of Proportions

$\displaystyle \text{If } \frac{a}{b} = \frac{c}{d} \text{ then the following propertied hold }$

$\displaystyle \text{By Invertendo: } \frac{b}{a} = \frac{d}{c}$

$\displaystyle \text{By Alternendo: } \frac{a}{c} = \frac{b}{d}$

$\displaystyle \text{By Componendo: } \frac{a+b}{b} = \frac{c+d}{d}$

$\displaystyle \text{By Dividendo: } \frac{a-b}{b} = \frac{c-d}{d}$

$\displaystyle \text{By Componendo and Dividendo: } \frac{a+b}{a-b} = \frac{c+d}{c-d}$