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 $\,19\%\,$, and then decreases by $\,30\%\,$.
What is the resulting percent increase or decrease?
Solution:
$(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 $\,x\,$ 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 $\,x = 1x\,$ 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 $\,40\%\,$, and then increases by $\,40\%\,$.
What is the resulting percent increase or decrease?
Solution:
$(1 + 0.4)(1 - 0.4)x = (1.4)(0.6)x = 0.84x = (1 - 0.16)x\,$;
$16\%\,$ 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:
 $(1 + 0.4)(1 - 0.4)x$ 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:
$(1 - 0.5)(1 + 0.5)x = (0.5)(1.5)x = 0.75x = (1 - 0.25)x\,$;
$25\%\,$ decrease
Question:
Suppose the price of an item increases by $\,30\%\,$, and then decreases by $\,10\%\,$.
What is the resulting percent increase or decrease?
Solution:
$(1 - 0.1)(1 + 0.3)x = (0.9)(1.3)x = 1.17x = (1 + 0.17)x\,$;
$17\%\,$ 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:
$(1 + 0.5)(1 + 0.5)x = (1.5)(1.5)x = 2.25x = (1 + 1.25)x\,$;
$125\%\,$ increase
Question:
Suppose an item costs $\,\$50\,$. 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:$\,(0.3)(1.2)(\$50) = \$18$It was an overall decrease. The percent decrease is:$\displaystyle \frac{50-18}{50} = 0.64 = 64\% $Second approach: You don't need the original price at all! Just denote it by$\,x\,$:$(0.3)(1.2)x = 0.36x = (1 - 0.64)x\,$;$64\%\,\$ decrease
Master the ideas from this section

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