MORE PROBLEMS INVOLVING PERCENT INCREASE AND DECREASE

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

EXAMPLES:
Question:
Suppose the price of an item increases by [beautiful math coming... please be patient] $\,19\%\,$, and then decreases by $\,30\%\,$.
What is the resulting percent increase or decrease?
Solution:
[beautiful math coming... please be patient] $(0.7)(1.19)x = 0.83x = (1 - 0.17)x\ $;
$17\%\,$ decrease
Why?
As discussed in Problems Involving Percent Increase and Decrease,
a price [beautiful math coming... please be patient] $\,x\,$ changes to $\,1.19x\,$ after the $\,19\%\,$ increase.
After the subsequent $\,30\%\,$ decrease, only $\,70\%\,$ of this remains:
$(1-0.3)(1.19x) = (0.7)(1.19)x = 0.83x$

The price started at $\,x\,$. It ended at $\,0.83x\,$.
So, the overall change was a decrease (note that $\,0.83 \lt 1\,$).

How much of a decrease was there in going from [beautiful math coming... please be patient] $\,x = 1x\,$ to $\,0.83x\,$?
Answer:   $\,1x - 0.83x = .17x$
That is, $\,17\%\,$ of $\,x\,$ was ‘lost’ in the process.
The combined effect of the back-to-back increase/decrease was a $\,17\%\,$ decrease.
Question:
Suppose the price of an item decreases by [beautiful math coming... please be patient] $\,40\%\,$, and then increases by $\,40\%\,$.
What is the resulting percent increase or decrease?
Solution:
[beautiful math coming... please be patient] $(1 + 0.4)(1 - 0.4)x = (1.4)(0.6)x = 0.84x = (1 - 0.16)x\,$;
$16\%\,$ decrease

Pause for a moment and appreciate the power in renaming an expression!
There are four names for the same expression given above, and each has its strength:
[beautiful math coming... please be patient] $(1 + 0.4)(1 - 0.4)x$ this name makes it clear that we're doing a $\,40\%\,$ decrease (the $\,1 - 0.4\,$)
and a $\,40\%\,$ increase (the $\,1 + 0.4\,$)
$(1.4)(0.6)x$ this name is a whole lot easier to plug into a calculator
$0.84x$ this name, as compared to the original $\,1x\,$, shows that the overall effect was a decrease
$(1 - 0.16)x$ this name shows that it was a $\,16\%\,$ decrease
Question:
Suppose the price of an item increases by $\,50\%\,$, and then decreases by $\,50\%\,$.
What is the resulting percent increase or decrease?
Solution:
[beautiful math coming... please be patient] $(1 - 0.5)(1 + 0.5)x = (0.5)(1.5)x = 0.75x = (1 - 0.25)x\,$;
$25\%\,$ decrease
Question:
Suppose the price of an item increases by [beautiful math coming... please be patient] $\,30\%\,$, and then decreases by $\,10\%\,$.
What is the resulting percent increase or decrease?
Solution:
[beautiful math coming... please be patient] $(1 - 0.1)(1 + 0.3)x = (0.9)(1.3)x = 1.17x = (1 + 0.17)x\,$;
$17\%\,$ increase
Question:
Suppose the price of an item increases by $\,50\%\,$, and then increases by $\,50\%\,$ again.
What is the resulting percent increase or decrease?
Solution:
[beautiful math coming... please be patient] $(1 + 0.5)(1 + 0.5)x = (1.5)(1.5)x = 2.25x = (1 + 1.25)x\,$;
$125\%\,$ increase
Question:
Suppose an item costs [beautiful math coming... please be patient] $\,\$50\,$.
The price increases by $\,20\%\,$, and then decreases by $\,70\%\,$.
What is the resulting percent increase or decrease?
Solution:
There are two good approaches. You choose!
First approach:
Compute new price, then compute percent change:
new price is: [beautiful math coming... please be patient] $\,(0.3)(1.2)(\$50) = \$18$
It was an overall decrease.
The percent decrease is:
[beautiful math coming... please be patient] $\displaystyle \frac{50-18}{50} = 0.64 = 64\% $
Second approach:
You don't need the original price at all! Just denote it by $\,x\,$:
[beautiful math coming... please be patient] $(0.3)(1.2)x = 0.36x = (1 - 0.64)x\,$;
$64\%\,$ decrease
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Locating Points in Quadrants and on Axes

 
 
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
All answers are rounded to two decimal places.