# Tax Computation Question

Say a product cost a and a tax of x% is applied to it, then another tax of y% is applied to the total. Is there a way for me to somehow "combine" the values of x% and y% so that I can do something like this:

a + (a * z%)


where z% is the combined value?

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When you add $p\%$ to $a$ you get $a+(a*p/100)=a*(1+\frac{p}{100})$, so you multiply $a$ with $(1+\frac{p}{100})$. Do that twice. First with $p=x$, then with $p=y$, and you get ... –  Jyrki Lahtonen Dec 13 '11 at 6:54
a * ( 1 + x/100) + a * ( 1 + y/100)??? I don't think it's right. For one, I think the expression is probably something like this: (a + (a * x) ) + ((a + (a * x) ) * y) where x and y have already been divided by 100. I want to know if it's possible to get rid of x and y and represent them as a combined value. –  StackOverflowNewbie Dec 13 '11 at 7:22
No, that’s not what Jyrki suggested. First you multiply by $1+\frac{x}{100}$ to get $a\left(1+\frac{x}{100}\right)$, which is an $x$% increase over $a$. Then you multiply that by $1+\frac{y}{100}$ to get $$a\left(1+\frac{x}{100}\right)\left(1+\frac{y}{100}\right)\,,$$ a $y$% increase over the $x$% increase. –  Brian M. Scott Dec 13 '11 at 7:41

Assuming one tax is applied to the other as well as to the original item, it is easier to start with something like a * (1 + z%) as the form of your tax.

So your double tax is a * (1 + x%) * (1 + y%).

So the combined tax rate is z% = x% + y% + (x% * y%), i.e. one tax is applied to the other as well as to the original item as in the assumption.

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Jyrki's answer is correct. First, take the percentage, rewrite it as a decimal, and add 1 to it. So if the tax increases the price by 5% for example, we multiply the price by 1.05. If the other tax is 10% (and this is a strange tax that taxes tax...), we multiply it again by 1.1. So if the original price is $a$, our new cost is

$1.1(1.05a)=(1.1\times1.05)a=1.155a$

Now, to get the percent increase, we just subtract out 1 again and write it as a percent. The increase in cost is $.155a$, so a 5% tax and 10% tax will increase the cost by 15.5%

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