Duelling PercentagesDate: 06/11/2002 at 06:05:03 From: Pete Subject: Percentage question driving me mad! Hiya! Can you tell me why, for instance, when you add 15% to 100, you get 115, but if you subtract 15%, you get 86.95...instead of 85? Date: 06/11/2002 at 08:20:33 From: Doctor Ian Subject: Re: Percentage question driving me mad! Hi Pete, Are you sure about that? 100 + (15% of 100) = 100 + (0.15 * 100) = 100 + 15 = 115 100 - (15% of 100) = 100 - (0.15 * 100) = 100 - 15 = 85 I think what you're asking is why, when you reduce something by 15%, increasing the reduced figure by the same percentage doesn't get you back to where you started, e.g., 85 + (15% of 85) < 100 86.95 + (15% of 86.95) = 100 The clearest way to show why it works this way is to use symbols instead of concrete values. Suppose we start with some amount, A, and we reduce it by some percentage, P. Then the new value is A' = A(1 - P) Now if we increase the reduced amount, by the same percentage, we get A'' = A'(1 + P) = A(1 - P)(1 + P) = A(1 - P^2) which is smaller than what we started with! It can also be easier to grasp if you use extreme examples. If you start with $100, and I take 99% of it, that leaves you with $1. If I increase your dollar by 99 percent, will that get you anywhere near $100? And here is a third way to look at it. Suppose we start with some amount, +---+---+---+---+---+ | | | | | | +---+---+---+---+---+ | | | | | | +---+---+---+---+---+ | | | | | | +---+---+---+---+---+ | | | | | | +---+---+---+---+---+ | | | | | | +---+---+---+---+---+ and reduce it by some percentage (say, removing one item out of every 5): +---+---+---+---+---+ | | | | | - | +---+---+---+---+---+ | | | | | - | +---+---+---+---+---+ | | | | | - | +---+---+---+---+---+ | | | | | - | +---+---+---+---+---+ | | | | | - | +---+---+---+---+---+ Now we want to increase it by the same percentage (i.e., adding one item for every 5): +---+---+---+---+---+ | | | | | - | +---+---+---+---+---+ | | | | | - | +---+---+---+---+---+ | | | | | - | +---+---+---+---+---+ | | | | | - | +---+---+---+---+---+ | | | | | - | +---+---+---+---+---+ | + | + | + | + | +---+---+---+---+ The increase, being the SAME fraction of a SMALLER number, will be smaller than the reduction, so we end up replacing only SOME of the amount that was removed: +---+---+---+---+---+ | | | | |-/+| +---+---+---+---+---+ | | | | |-/+| +---+---+---+---+---+ | | | | |-/+| +---+---+---+---+---+ | | | | |-/+| +---+---+---+---+---+ | | | | | - | +---+---+---+---+---+ The difference between the increase and the reduction is the square of the percentage. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 06/12/2002 at 05:23:02 From: Pete Subject: Percentage question driving me mad! Hello Ian. Thanks very much for replying yesterday to my question. The problem lay in the fact that I was using a desktop calculator, and the -TAX button for some unknown reason does not give minus 10% of 100 as 90. Any ideas why this is so, as it happens on all models? Thanks.... Date: 06/12/2002 at 07:34:11 From: Doctor Ian Subject: Re: Percentage question driving me mad! Hi Pete, I've never seen a -TAX button on a calculator, but it sounds to me as though it might be assuming that you know the price WITH tax, and the tax rate, and you want to figure out the price WITHOUT tax. For example, if the price with tax is $100, and the tax is 10%, and the price without tax is P, then P(1 + 0.10) = $100 P = $100 / 1.10 = 90.91 In other words, a retail price of $90.91, with 10% tax added to it, comes to $100. Does that make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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