﻿ More Problems Involving Percent Increase and Decrease
MORE PROBLEMS INVOLVING PERCENT INCREASE AND DECREASE
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!

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:
$\cssId{s9}{(0.7)(1.19)x} \cssId{s10}{= 0.83x} \cssId{s11}{= (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:
$\cssId{s17}{(1-0.3)(1.19x)} \cssId{s18}{= (0.7)(1.19)x} \cssId{s19}{= 0.83x}$

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:   $\,1x - 0.83x = 0.17x$
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:
$\cssId{s33}{(1 + 0.4)(1 - 0.4)x} \cssId{s34}{= (1.4)(0.6)x} \cssId{s35}{= 0.84x} \cssId{s36}{= (1 - 0.16)x}\,$;
$16\%\,$ 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:
$\cssId{s53}{(1 - 0.5)(1 + 0.5)x} \cssId{s54}{= (0.5)(1.5)x} \cssId{s55}{= 0.75x} \cssId{s56}{= (1 - 0.25)x}\,$;
$25\%\,$ 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:
$\cssId{s62}{(1 - 0.1)(1 + 0.3)x} \cssId{s63}{= (0.9)(1.3)x} \cssId{s64}{= 1.17x} \cssId{s65}{= (1 + 0.17)x}\,$;
$17\%\,$ 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:
$\cssId{s71}{(1 + 0.5)(1 + 0.5)x} \cssId{s72}{= (1.5)(1.5)x} \cssId{s73}{= 2.25x} \cssId{s74}{= (1 + 1.25)x}\,$;
$125\%\,$ 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 \cssId{s89}{\frac{50-18}{50}} \cssId{s90}{= 0.64} \cssId{s91}{= 64\%} $Second approach: You don't need the original price at all! Just denote it by$\,x\,$:$\cssId{s95}{(0.3)(1.2)x} \cssId{s96}{= 0.36x} \cssId{s97}{= (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

CONCEPT QUESTIONS EXERCISE:
All answers are rounded to two decimal places.
PROBLEM TYPES:
 1 2
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