Ratio
The ratio of two numbers $x$ and $y$, is the fraction $\frac{x}{y}$ and we write it as $x:y$.
In the ratio $x:y$, $x$ is the $1^{st}$ term or antecedent and $y$, the $2^{nd}$ or consequent.
Eg. The ratio $2:3$ represents $\frac{2}{3}$ with antecedent = 2, consequent = 3.
Rule: If each term of a ratio is multiplied or divided by the same non-zero number then it does not affect the ratio.
Eg. $2:4$ $= 4:8$ $= 8:16$. Also, $4:6$ $= 2:3$.
Proportions:
If two ratios (or fractions are equal) then we say that they are in proportion.
If $P:Q=R:S$, we write $P:Q::R:S$ and we say that $P$, $Q$, $R$, $S$ are in propotion.
Here $P$ and $S$ are called extrems, $Q$ and $R$ called mean terms.
Product of means = Product of extremes.
Thus, $P:Q::R:S$ $\Leftrightarrow (Q\times R) = (P\times S)$
Fourth Proportional:
If $P:Q=R:S$, then $S$ is called the $4^{th}$ proportional to $P$, $Q$, $R$.
Third Proportional:
$P:Q=R:S$, then $R$ is called the $3^{rd}$ proportional to $P$ and $Q$.
Mean Proportional:
Mean proportional between $P$ and $Q$ is $\sqrt{PQ}$.
Comparison of Ratios:
$(P:Q)>(R:S)$ $\Leftrightarrow \frac{P}{Q}>\frac{R}{S}$.
Duplicate Ratios:
Duplicate ratio of $(P:Q)$ is $(P^{2}:Q^{2})$.
Sub-duplicate ratio of $\left(P:Q\right)$ is $\left(\sqrt{P}:\sqrt{Q}\right)$.
Triplicate ratio of $(P:Q)$ is $(P^{\frac{1}{3}}:Q^{\frac{1}{3}})$.
If $\frac{P}{Q}=\frac{R}{S}$, then
$\frac{P+Q}{P-Q}=\frac{R+S}{R-S}$
Variations:
$x$ is directly proportional to $y$ means, if $x=ky$, where $k$ is some constant and we write, $x\propto y$
$x$ is inversely proportional to $y$ means, if $xy=k$, where $k$ is some constant and we write, $x\propto \frac{1}{y}$.
