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### Order of Discounts and Taxes

```Date: 04/22/2002 at 20:34:08
From: Kim
Subject: Percents

Hi,

I'm wondering if you could tell me why it doesn't matter in
determining the cost of an item whether you take off the tax before or
after you take off the saving rate.

For instance, if I buy an item that originally costs 100 dollars
during a 20% off sale and the tax is 10% (for sake of simplicity),
intuitively I think that I want to take the 20% off before I calculate
the tax, so I will be taxed on a lower amount, but it doesn't seem to
matter whether I take the tax off the 100 dollars first and then
calculate the savings, or I take the savings first and then calculate
the tax. Am I right that order doesn't matter, and is there an
explanation?

Kim
```

```
Date: 04/23/2002 at 10:35:18
From: Doctor Ian
Subject: Re: Percents

Hi Kim,

As with most things, it's often easier to see what's going on if
you use variables instead of actual numbers.

Suppose the normal price of an item is P, and the tax rate is T.
Then the amount you'd normally pay for the item is

P(1 + T)

Now, suppose you discount the price by some percentage, D.  If
you do it before the tax, you end up paying

[P(1 - D)](1 + T)

But if you apply the discount to the after-tax amount, you end up
paying

[P(1 + T)](1 - D)

Since you're just multiplying things together, the order doesn't
matter; so you end up paying the same thing in both cases.

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 04/24/2002 at 01:43:59
From: Kim
Subject: Percents

Thank you for your reply. It took me a while to figure out what the
one was doing in the formula, but now I see how it allows you to get
the price minus price times discount, or price plus price times tax.
It makes a lot of sense. Thanks again.
```
Associated Topics:
Middle School Fractions

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