﻿ Problems involving percent increase and decrease
PROBLEMS INVOLVING PERCENT INCREASE AND DECREASE

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:$(0.7)(1.19)(\$50) = \$41.65$Why? To increase any amount by$\,19\%\,$, just multiply by$\,1.19\,$:$\,x + 0.19x = 1x + 0.19x = 1.19x\,$Notice that when you increase, you multiply by a number greater than$\,1\,$. If you decrease any amount by$\,30\%\,$, then$\,70\%\,$remains:$x - 0.3x = 1x - 0.3x = 0.7x\,$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:$(1 + 0.85)(1 - 0.38)(x) = (1.85)(0.62)x = 1.15x$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:$(1.5)(0.5)(x) = 0.75x$Question: Suppose an item costs$\,x\,$. If the price increases by$\,50\%\,$, and then increases by$\,50\%\,$, the new price is: Answer:$(1.5)(1.5)(x) = 2.25x$Question: Suppose an item costs$\,\$100\,$.
If the price decreases by $\,50\%\,$, and then decreases by $\,50\%\,$, the new price is:
$(0.5)(0.5)(x) = \$25.00\$
Master the ideas from this section