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**Basic Concepts:**

**1**. To express \(x\) as a percentage of \(y\); percentage \(=\) $[\frac{x}{y}\times 100]\%$

**2**. If \(x\)\(\%\) of a quantity is \(y\), then the whole quantity \(=\) $[\frac{y}{x}\times 100]\%$

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

\(=\) $[1+\frac{R}{100}]$ \(\times\) Original quantity

**New quantity**\(=\) $[\frac{\mathit{100}+R}{100}]\hspace{0.17em}$\(\times\) Original quantity.

**(ii)**

**Similarly,**if quantity decreases by \(R\ \)\(\%\), then, the

**new quantity**\(=\) $[\frac{\mathit{100}-R}{100}]\hspace{0.17em}$\(×\) \(\text{Original quantity}\)

**2**.

**Population**:

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

**Population after**'\(n\)'

**years**\(=\) ${(1+\frac{R}{100})}^{n}$\(×\) \(\text{Original quantity}\)

**(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 $(\frac{R}{100+R}\times 100)\phantom{\rule{0.147em}{0ex}}\%$

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

**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. $[\frac{x}{y}\times 100]\%$.

**(ii)**If the price of a commodity decreases by \(R\%\), then increases in

**consumption**, so as not to increase the expenditure. $[\frac{y}{x}\times 100]\%$.

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.