﻿ Problems involving percent increase and decrease

# Problems involving percent increase and decrease

Want more practice with percents and related concepts?

Here, you will practice solving problems involving percent increase and decrease. You may use a calculator for these exercises.

## Examples

Question: Suppose an item costs $\,\$50\,.$If the price increases by$\,19\%\,,$and then decreases by$\,30\%\,,$the new price is: Solution: $$\cssId{s8}{(0.7)(1.19)(\50) = \41.65}$$ Why? To increase any amount by$\,19\%\,,$just multiply by$\,1.19\,$: $$\cssId{s11}{x + 0.19x} \cssId{s12}{= 1x + 0.19x} \cssId{s13}{= 1.19x}$$ Notice that when you increase, you multiply by a number greater than$\,1\,.$If you decrease any amount by$\,30\%\,,$then$\,70\%\,$remains: $$\cssId{s16}{x - 0.3x} \cssId{s17}{= 1x - 0.3x} \cssId{s18}{= 0.7x}$$ Thus, to decrease any amount by$\,30\%\,,$just multiply by$\,0.7\,.$Notice that when you decrease, you multiply by a number less than$\,1\,.$Combining these ideas: $\$50$ original amount $(1.19)(\$50)$new amount, after the$\,19\%\,$increase$(0.7)\cdot (1.19)(\$50)$ new amount, after the $\,30\%\,$ decrease $(0.7)(1.19)(\$50) = \$41.65$ round dollar amounts (as needed) to two decimal places

What if we switch the order of applying the increase/decrease?

 $\$50$original amount$(0.7)(\$50)$ new amount, after the $\,30\%\,$ decrease $(1.19)\cdot (0.7)(\$50)$new amount, after the$\,19\%\,$increase$(1.19)(0.7)(\$50) = \$41.65$round dollar amounts (as needed) to two decimal places Same result! Since$\,(1.19)(0.7) = (0.7)(1.19)\,,$you can do the multiplication in whatever order you prefer. Question: Suppose an item costs$\,x\,$. If the price decreases by$\,38\%\,,$and then increases by$\,85\%\,,the new price is: Answer: \begin{align} &\cssId{s46}{(1 + 0.85)(1 - 0.38)(x)}\cr\cr &\qquad \cssId{s47}{= (1.85)(0.62)x}\cr\cr &\qquad \cssId{s48}{= 1.15x} \end{align} In this exercise, all answers are rounded to two decimal places. Question: Suppose an item costs\,x\,.$If the price decreases by$\,50\%\,$and then increases by$\,50\%\,,$the new price is: Answer: $$\cssId{s54}{(1.5)(0.5)(x) = 0.75x}$$ Question: Suppose an item costs$\,x\,$. If the price increases by$\,50\%\,,$and then increases by$\,50\%\,,$the new price is: Answer: $$\cssId{s59}{(1.5)(1.5)(x) = 2.25x}$$ Question: Suppose an item costs$\,\$100\,.$ If the price decreases by $\,50\%\,,$ and then decreases by $\,50\%\,,$ the new price is:
$$\cssId{s64}{(0.5)(0.5)(x) = \25.00}$$