Figuring a Tax
Date: 05/13/2003 at 12:07:08 From: Annette Subject: Percentage increase of an amount I work in insurance. We have a premium of $75,329. This premium needs to be increased by a tax of 3.75%. We have been doing the following: $75,329 / .9625 = $78,263.90 Why doesn't this work compared to the example below? If I follow what I read in the archives I should be using the following: (1.00 * Y) + (.0375 * Y) = X (1.00 + .0375) * Y = X 1.0375 * Y = X So 1.0375 * $75,329 = $78,153.84
Date: 05/13/2003 at 14:09:03 From: Doctor Ian Subject: Re: Percentage increase of an amount Hi Annette, If the premium before tax is P, and the tax is 3.75%, then the final charge should be P + 0.0375P = P(1 + 0.0375) = 1.0375P So if the premium is $75,329, the amount charged should be 1.0375 * $75,329 = 78,153.84 which is what you got at the end of your question. What are you doing when you divide by 0.9625, instead of multiplying by 1.0375? What amount are you ending up with? To make this easier to write, I'm going to use r = 0.0375 to represent the tax rate: something = P/0.9625 = P/(1 - r) something(1 - r) = P something - r*something = P something = P + r*something Now, since we can get 'something' by dividing P by a number that is less than 1, it has to be the case that something > P So what you're doing is computing the tax on an amount _greater_ than the premium, and adding it to the premium. Which is to say, you've been charging too much tax. Another way to look at it is this: Since something(1 - r) = P the 'something' that you're computing is the amount that you'd have to _reduce_ by 3.75% to end up with your original premium. An example with simpler numbers might make this clearer. Suppose the premium is $100, and the tax is 10%. The final amount should be $100 + (10% of $100) = $100 + $10 = $110 But suppose we ask instead: What number would we have to _reduce_ by 10%, in order to end up with $100? It's not $110. If you reduce $110 by 10%, you end up with $110 - (10% of $110) = $110 - $11 = $99 In fact, the amount A that you'd have to start with would be given by A(1 - 10/100) = 100 A = 100/(1 - 10/100) = 111.11 So what you're doing is kind of like this: premium with tax = $100 + (10% of 111.11) = $100 + (10% of (100 + 11.11)) = $100 + (10% of 100) + (10% of 11.11) ^ ^ ^ | | | premium tax overcharge Note that dividing by (1-rate) instead of multiplying by (1+rate) gives you the same answer only when P ------- = P(1 + r) (1 - r) 1 ------- = (1 + r) (1 - r) 1 = (1 + r)(1 - r) 1 = 1 - r^2 r^2 = 0 Which is to say, it works only when the rate is zero. For any other rate, you end up adding too much tax. How much too much? We can compute that this way: overcharge = P/(1-r) - P(1+r) = P[1/(1-r) - (1+r)] = P[r^2/(1 - r)] Let's check that on our $100 premium: overcharge = $100 * [(0.1)^2 / (1 - 0.1)] = $100 * [0.01 / 0.9] = $1.11 which agrees with what we got before. In the actual case, this works out to overcharge = $75,329 * [(0.0375)^2 / (1 - 0.0375)] = $100 * [0.00140625 / 0.9625] = $110.06 which is the difference between $78,263.90 and $78,153.84. I hope this helps. Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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