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# Percent increase then percent decrease results in no effect

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Percent increase then percent decrease results in no effect [#permalink]

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29 Jul 2012, 22:24
What conditions must be met for the percent increase and subsequent percent decrease to result in no change?
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Re: Percent increase then percent decrease results in no effect [#permalink]

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29 Jul 2012, 22:36
arnivorous wrote:
What conditions must be met for the percent increase and subsequent percent decrease to result in no change?

The multiplication factor of increase should be inverse of the multiplication factor of decrease.

e.g. Say you have a number 100.

You increase it by 25%. The multiplication factor is 5/4 i.e. when you multiply 100 by 5/4, you get 100*5/4 = 125. This is 25% more than 100.
Now you want to decrease it by a certain % such that you get 100 back.
Basically, 100*5/4 * x = 100
So x = 4/5 (inverse of 5/4)
Hence, you decrease by 20% (the multiplication factor of 20% is 4/5)

or
Use this formula: cumulative % change = a + b + ab/100
You want the cumulative change to be 0.

a + b + ab/100 = 0
If you know that you are increasing by 25% and want to find the % by which you should decrease to get the same number,
25 + b + 25b/100 = 0
5b/4 = -25
b = -20

So you need to decrease (hence you get the -ve sign) by 20%.

Check out these posts for details of multiplication factors and the formula:
http://www.veritasprep.com/blog/2011/02 ... rcentages/
http://www.veritasprep.com/blog/2011/02 ... e-changes/
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Re: Percent increase then percent decrease results in no effect [#permalink]

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29 Jul 2012, 22:43
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Expert's post
If we want to make two changes and end up back where we started, the two numbers we need to multiply need to be one another's inverses. For instance, 100 * 3/2 * 2/3 = 100.

This is a little harder to see with percents, because we might write the above like this: 100 increased by 50% then decreased by 33 1/3% equals 100. That's why it can be easier to figure out what number or fraction we're actually multiplying by (e.g. 50% increase = multiply by 1.5; 33 1/3% decrease = multiply by .6666). We can use these formulas:

Increase by x%: Multiply by (100+x)/100

Decrease by x%: Multiply by (100-x)/100

After that, we can often convert to fractions or decimals and find an inverse from there.

Here's an example:

x increased by 60% then decreased by y% equals x. What is y?

x increased by 60% = x * 160/100 = x * 16/10 = x * 8/5

So what do we need to do to get back to x? Multiply by the inverse of 8/5 . . . 5/8.

x * 8/5 * 5/8 = x

So we know that reducing by y is the same as multiplying by 5/8.

5/8 = .625 = 62.5% If we multiplied by 62.5%, y must be the remaining 37.5%.

We can translate the original sentence and plug this in, just to be see how it works:

x increased by 60% then decreased by y% equals x.
x * 160/100 * (100-y)/100 = x
x * 160/100 * (100-37.5)/100 = x
x * 160/100 * 62.5/100 = x
x * 8/5 * 5/8 = x
x = x

It works!
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Re: Percent increase then percent decrease results in no effect   [#permalink] 29 Jul 2012, 22:43
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# Percent increase then percent decrease results in no effect

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