### Theory:

Basic Concepts:
1. To express $$x$$ as a percentage of $$y$$; percentage $$=$$ $\left[\frac{x}{y}×100\right]%$

2. If $$x$$$$\%$$ of a quantity is $$y$$, then the whole quantity $$=$$ $\left[\frac{y}{x}×100\right]%$
Fundamental Formulae:
1. Increase/Decrease in quantity:

(i) If quantity increases by $$R\%$$, then [Where $$R$$ denotes the rate of change in percentage]

New quantity $$=$$ Original quantity $$+$$ Increases in the quantity

$$=$$ Original quantity $$+$$ $$R$$$$\%$$ of Original quantity

$$=$$ Original quantity $$+$$ $\frac{R}{100}$ of Original quantity

$$=$$ $\left[1+\frac{R}{100}\right]$ $$\times$$ Original quantity

New quantity $$=$$ $\left[\frac{\mathit{100}+R}{100}\right]$$$\times$$ Original quantity.

(ii) Similarly, if quantity decreases by $$R$$$$\%$$, then, the new quantity $$=$$ $\left[\frac{\mathit{100}-R}{100}\right]$$$×$$ $$\text{Original quantity}$$

2. Population:

(i) If a population of a city increases by $$R$$$$\%$$ per annum, then the population after '$$n$$' years $$=$$ ${\left(1+\frac{R}{100}\right)}^{n}$ of the original population.

Population after '$$n$$' years $$=$$ ${\left(1+\frac{R}{100}\right)}^{n}$$$×$$ $$\text{Original quantity}$$

(ii) Population '$$n$$' years ago $$=$$ $$\frac{\text{original population}}{(1 + \frac{R}{100})^n}$$

(iii) If the population is increased from $$X$$ to $$Y$$, then the percentage of increase is: $$=$$ $$(Y - X)$$ $$×100$$

3. Rate is more/less than another:

(i) If a number $$x$$ is $$R$$$$\%$$ more than $$y$$, then $$y$$ is less than $$x$$ by $\left(\frac{R}{100+R}×100\right)\phantom{\rule{0.147em}{0ex}}%$

(ii) If a number $$x$$ is $$R$$$$\%$$ less than $$y$$, then $$y$$ is more than $$x$$ by $\left(\frac{R}{100-R}×100\right)\phantom{\rule{0.147em}{0ex}}%$

4. Prices of a commodity Increase/Decrease by $$R\%$$:

(i) If the price of a commodity increase by $$R\%$$, then there is a reduction in consumption, so as not to increase the expenditure. $\left[\frac{x}{y}×100\right]%$.

(ii) If the price of a commodity decreases by $$R\%$$, then increases in consumption, so as not to increase the expenditure. $\left[\frac{y}{x}×100\right]%$.

If a quantity is increased or decreases by $$x\%$$ and another quantity is increased or decreased by $$y\%$$, the percent $$\%$$ change on the product of both the quantity is given by require $$\%$$ change $$=$$ $\frac{R}{100}$.

Note: For increasing use ($$+$$)ve sign and for decreasing use ($$-$$)ve sign.