Theory:

Basic Concepts:
1. To express \(x\) as a percentage of \(y\); percentage \(=\) [xy×100]%
 
2. If \(x\)\(\%\) of a quantity is \(y\), then the whole quantity \(=\) [yx×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 \(+\) R100 of Original quantity
 
\(=\) [1+R100] \(\times\) Original quantity
  
New quantity \(=\) [100+R100]\(\times\) Original quantity.
 
(ii) Similarly, if quantity decreases by \(R \)\(\%\), then, the new quantity \(=\) [100R100]\(×\) \(\text{Original quantity}\)
 
  
2. Population:
 
(i) If a population of a city increases by \(R \)\(\%\) per annum, then the population after '\(n\)' years \(=\) (1+R100)n of the original population.
  
Population after '\(n\)' years \(=\) (1+R100)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 (R100+R×100)%
 
(ii) If a number \(x\) is \(R\)\(\%\) less than \(y\), then \(y\) is more than \(x\) by (R100R×100)%
 
 
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. [xy×100]%.
 
(ii) If the price of a commodity decreases by \(R\%\), then increases in consumption, so as not to increase the expenditure. [yx×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 \(=\) R100.
  
Note: For increasing use (\(+\))ve sign and for decreasing use (\(-\))ve sign.