Successive Percentage Change Important Formulas:

Overview:

Successive Percentage Change Case-2:

If the present value of an item is \(k\) and the value of the item changes \(x \ \%\) in first year and \(y \ \%\) in 2nd year, successively then overall percent \((\%)\) change of the value and final value of \(k\) will be-

Overall percent \((\%)\) change, $$ \left[\pm x \pm y \pm \frac{xy}{100}\right]$$

Final value of \(k\), $$ \left[k \left(1 \pm \frac{x}{100}\right) \left(1 \pm \frac{y}{100}\right) \left(1 \pm \frac{z}{100}\right)\right]$$

Note: If the value of the item increases by \(x \ \%\) and \(y \ \%\) then use Positive \((+)\) sign, and if the value of the item decreases by \(x \ \%\) and \(y \ \%\) then use Negative \((-)\) sign.

Example (1): If the present price of the book is \(100 \ Rs.\) and the price of the book increases \(10 \ \%\) in first year, and \(20 \ \%\) in second yaer, successively. Find the overall percent change of price of the book, and the final price of the book after \(2\) years?

Solution: Given values, present price of the book \(k = 100 \ Rs.\), \(\%\) change in first year \(x = 10 \ \%\) and \(\%\) change in second year \(y = 20 \ \%\), then

according to equation \((1)\), Overall percent change $$ = \left[10 + 20 + \frac{10 \times 20}{100} \right] $$ $$ = \left[30 + \frac{200}{100}\right] = [30 + 2] \% $$ $$ = 32 \ \% \ (Answer)$$

now, according to equation \((2)\), price of the book after \(2\) years, $$ = \left[100 \left(1 + \frac{10}{100}\right) \left(1 + \frac{20}{100}\right) \right] $$ $$ = \left[100 \times \frac{11}{10} \frac{6}{5} \right] $$ $$ = 132 \ Rs. \ (Answer) $$

We have used positive sign \((+)\) here, because price is increasing successively.

Example (2): If the present price of the sugar is \(50 \ Rs/kg\) and the price of the sugar decreases \(5 \ \%\) in first year, and \(10 \ \%\) in second yaer, successively. Find the overall percent change of price of the sugar, and the final price of the sugar after \(2\) years?

Solution: Given values, present price of the sugar \(k = 50 \ Rs/kg\), \(\%\) change in first year \(x = 5 \ \%\) and \(\%\) change in second year \(y = 10 \ \%\), then

according to equation \((1)\), Overall percent change $$ = \left[- 5 - 10 - \frac{5 \times 10}{100} \right] $$ $$ = \left[- 15 - \frac{50}{100}\right] = [- 15 - 0.5] \% $$ $$ = - 15.5 \ \% \ (Answer)$$

now, according to equation \((2)\), price of the sugar after \(2\) years, $$ = \left[50 \left(1 - \frac{5}{100}\right) \left(1 - \frac{10}{100}\right) \right] $$ $$ = \left[50 \times \frac{19}{20} \frac{9}{10} \right] $$ $$ = 42.75 \ Rs/kg \ (Answer) $$

We have used negative sign \((-)\) here, because price is decreasing successively.