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:
$\cssId{s17}{(10.3)(1.19x)}
\cssId{s18}{= (0.7)(1.19)x}
\cssId{s19}{= 0.83x}$
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 backtoback 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{5018}{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