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I have a base value of 100 and I apply a +10% and a + 7% increase to it. To get the calculated value, I do:

100 * 1.1 = 110
110 * 1.07 = 107

100 + 10 + 7 = 117

Now, to get back to 117, I was initially doing:

117 / 1.1 = 106.363636363636

106.363636363636 / 1.07 = 99.4052676295664 which is obviously not the original value of 100.

I then realized that I needed to apply the calculations in reverse order in which they were applied to get back to the original value, so:

117 / 1.07 = 110
110 / 1.1 = 100

Is there a way to do it so I get back to my original value (100 in this case) from 117 by going in the same order I used to get to 117 (that is, applying 10% first and then 7%

Here is a more detailed example which I calculated already and it appears to be correct, but I just want to make sure, so here is the setup:

My base value is 100 and I have a +10% and +7% compounded percentages and I have a +5% and +3% non-compounded perntages, so to first get my adjusted value, I do:

I have a variable adjustedValue which I will use to store the variables along the way, it is intially set to the total (100).

adjustedTotal = adjustedTotal * 1.1 = 110

adjustedTotal = adjustedTotal * 1.07 = 117.7

adjustedTotal = adjustedTotal + (value * 0.05) = 117.7 + 5 = 122.7

adjustedTotal = adjustedTotal + (value * 0.03) = 122.7 + 3 = 125.7

To get back to 100 from 125.7, I do:

adjustedTotal = adjustedTotal - (value * 0.03) = 125.7 - 3 = 122.7

adjustedTotal = adjustedTotal - (value * 0.05) = 122.7 + 5 = 117.7

adjustedTotal = adjustedTotal / 1.07 = 110

adjustedTotal = adjustedTotal / 1.1 = 100

Is the above the correct way to approach it?

The example below uses a base of 1000 and 3 rates (10%, 5%, and 3%).

If 10%, 5%, and 3% are compounded and they are all added to the base, I do:

1000 * 1.1 * 1.05 * 1.03 = 1189.65

To get back to 1000 from 1189.65, I do:

1189.65 / 1.1 / 1.05 / 1.03

If 10%, 5%, and 3% are non-compounded and they are all added to the base, I do:

1000 * (1 + 0.1 + 0.05 + 0.03) = 1180

To get back to 1000 from 1180, I do:

1180 / (1 + 0.1 + 0.05 + 0.03)

All of the above is well and works fine, the problem is when I need to mix and add rates, for example:

Assume I stick with my base of 1000, but my rates are all non-compounded and they are added and subtracted, hence(+10%, -5%, +3%). I know I can do this:

1000 * (1 + 0.1 - 0.05 + 0.03) = 1080 and to get back I do:

1080 / (1 + 0.1 - 0.05 + 0.03) = 1000

In the above, the confusing part to me is that no matter if I go up or down with the rates, will I always use division, basically, will I always do:

1080 / (1 + 0.1 - 0.05 + 0.03) and just change the signs in the parentheses. Here is an example:

1000 * (1 - 0.1 - 0.05 - 0.03) = 820

820 / (1 - 0.1 - 0.05 - 0.03) = 1000

In the above, will it always be 820 / ... when going backwards or will it ever be 820 * ...

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1  
110 * 1.07 = 107 : sure about this? –  Did Apr 18 '11 at 15:23
    
@Didier, the 10% and 7% are compounded, so that is why I am doing 1.07 –  Xaisoft Apr 18 '11 at 15:33
    
@Xaisoft, but 110*1.07=117.7 Also 117/1.07=109.3458 (approx) I don't understand what "get back to 117" means here. –  Ross Millikan Apr 18 '11 at 15:53
    
Your trouble has nothing to do with the order you use to perform your operations (hence 6312's answer is odd) but with the fact that if your initial stock of 100 increases by 10% and then the resulting stock increases by 7%, what you get is not a stock of 107 (your first try) nor a stock of 100+10+7=117 (your second try). But you get the same stock than if you increase your initial stock by 7% and then by 10%. –  Did Apr 18 '11 at 15:54
    
@Xaisoft: you've asked 4 questions on essentially the same topic. It is generally good form in this case to link to the previous ones and state what additional question you have. –  Willie Wong Apr 18 '11 at 20:25

1 Answer 1

up vote 2 down vote accepted

Suppose that there is a highway that goes from Los Angeles to San Francisco, then to Portland, then to Seattle. You have travelled on that highway from Los Angeles, through San Francisco and Portland. Schematically, LA to SF to P to S. To get back to LA, you need to visit your previous cities in reverse order, S to P to SF to LA. Roughly speaking that is why you have found that the "undoing" has to be done in reverse order.

If you have bumped into matrices, there is a corresponding phenomenon. In general let $U^{-1}$ be the inverse of the matrix $U$. Then then (if the specified inverses exist) $$(AB)^{-1}=B^{-1}A^{-1}$$ Note again the reversal of order.

But back to the problem that I have already seen posted a number of times. You have computed the result of applying $10$ percent interest to $100$, then applying $7$ percent interest, again to $100$, getting a total of $117$, and you want to recover certain numbers, presumably $110$, $107$, and $100$.

The simple minded way of doing this is to note that since there is no compounding, the combined rate is $17$ percent. So one recovers the initial sum by dividing by $1.17$, and then recomputes the $110$ and $107$. Why is this procedure, of which you are undoubtedly aware, not satisfactory? Any formula that recovers the $110$ first will be essentially equivalent to the procedure I suggest, it will have to use both interest rates. From $117$ you cannot recover the $110$ without using information about both interest rates.

If there were compounding, with the $10$ percent assessed first, then you would need to divide by $1.07$, then by $1.10$, or equivalently by $(1.07)(1.10)$. If there is compounding with the $7$ percent taken first, recovery involves in the same way dividing by $(1.10)(1.07)$. Note that the two results are the same. When there is compounding, the order in which the rates are assessed makes no difference, something that is perhaps not intuitively obvious.

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My math is not as good, so I understand your analogy, but after that my brain froze. Are you saying the only way to remove it is to go in reverse order. I will update my post with a better example. –  Xaisoft Apr 18 '11 at 15:35
    
So if I understand you correctly, if it is compound, I can do each one separately and order does not matter. If there is non-compounding values, I have to combine all the percentages first? Is that correct? –  Xaisoft Apr 18 '11 at 17:03
    
@User6312: Ok, got you so far. In the case of +3%, +6% compound, then 10% non-compound, then a 2% compound, and finally a 4% non-compound, can I take the non-compound values out and divide by 1.l4 and then process the compounded values in reverse (divide by 2%, then by 6%, then by 3%) –  Xaisoft Apr 18 '11 at 18:47
    
I updated my post with a base value of 1000 and 3 rates and 3 scenarios: 1 with all compounded rates, one with non-compounded rates, and the other with all non-compounded rates, but a mixuture of adding and subtracting the rates. –  Xaisoft Apr 18 '11 at 20:13
    
You have the right idea. In your example with $10$ percent, followed by $7$ percent "compounded" and $5$ percent, $3$ percent not compounded, let $V$ be initial value, and $T$ final value. Then $T=V[(1.10)(1.07)+0.05+0.03]$, so $V$ is $T$ divided by $[(1.10)(1.07)+0.05+0.03]$. The stuff you posted using negative numbers is OK. –  André Nicolas Apr 18 '11 at 21:22

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