# Successive Percentage Change Important Formulas:

#### Overview:

 Topic Included: Formulas, Definitions & Exmaples. Main Topic: Quantitative Aptitude. Quantitative Aptitude Sub-topic: Percentage Aptitude Notes & Questions. Questions for practice: 10 Questions & Answers with Solutions.

#### 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.