# Suppose a certain item increased in price by 18% a total of 5 times and then decreased in price by 10% a total of 2 times. by what overall percent did the price increase?

Suppose a certain item increased in price by 18% a total of 5 times and then decreased in price by 10% a total of 2 times. By what overall percent did the price increase?
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Step 1
Let, initial price of the item was \$x
Now, we have the formulae $A=p\left(1+p{\right)}^{t}\to \left(1\right)$
$P=\text{Initial price}$
$A=\text{Amount after t times}$
$p=$ Rate of charge
Now, when the price of the item increased by 18% a total of 5 times. Then
$p=x,p=0.18,t=5$
From (1) Amount $={A}_{1}=x\left(1+0.18{\right)}^{5}\phantom{\rule{0ex}{0ex}}=\left(1.18{\right)}_{x}^{5}$
Step 2
When the price is decreased by 10% a total of 2 times then
$p-\left(1.18{\right)}^{5}x,p=-0.10\left(decrease\right),t=2$
From (1), Amount $={A}_{2}=\left(1.18{\right)}_{x}^{5}\left(1-0.10{\right)}^{2}\phantom{\rule{0ex}{0ex}}=\left(1.18{\right)}^{5}x\left(0.9{\right)}^{2}\phantom{\rule{0ex}{0ex}}=1.8531x$
Overall charge in percentage
$=\frac{{A}_{2}-p}{p}×100\mathrm{%}\phantom{\rule{0ex}{0ex}}=\frac{1.8531x}{x}×100\mathrm{%}\phantom{\rule{0ex}{0ex}}=\frac{0.8531x}{x}×100\mathrm{%}\phantom{\rule{0ex}{0ex}}=85.31\mathrm{%}$