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