How do I solve for a percentage of a total budget...
December 9, 2014 3:46 PM   Subscribe

This will be easy for someone, but math is my second language so I'm at a loss. How do I solve for the percentage of a TOTAL budget, when one line item is unknown (and is part of the total budget)?

Specifically, I have a budget with 20 line items. I know the value of all the line items, except for one, and that one unknown value has to equal 16.5% of the TOTAL budget. How do I figure this out? Without that line item to start with, I don't have the total budget - does this question make sense?

Bonus if you can tell me what formula to plug into Excel so that I never have to think about this again.

Thanks so much!
posted by coollibrarian to Work & Money (5 answers total) 3 users marked this as a favorite
Take the sum of all the values you have and divide that number by .835. That's the total budget. The total number times .165 is the missing value.
posted by Admiral Haddock at 3:50 PM on December 9, 2014 [5 favorites]

Or in Excel

Filling in a1:a19 with the relevant cells (assuming they're contiguous).
posted by Admiral Haddock at 3:56 PM on December 9, 2014 [2 favorites]

To explain why Admiral Haddock is correct: your nineteen items are 83.5% of the total budget.
posted by DanSachs at 4:46 PM on December 9, 2014 [1 favorite]

If you want to torture yourself see what a formal explanation of this looks like (for any number of values):

Let ci represent the cost of item i and express percentages as values between 0 and 1 (16.5 percent is equal to .165, for example -- just divide the percentage by 100 to get the appropriate value between 0 and 1). We have n total items (in your example, n=20). Let B represent the total budget. Let w be the weight (percentage of total budget) corresponding to whichever item k has the unknown cost. We know the following facts:

B = c1 + c2 + ... + cn
w * B = ck

Subtracting ck from both sides of the first equation gets us:

B - ck = c1 + c2 + ... + ck-1 + ck + ck+1 + ... + cn - ck

The ck values on the right-hand side cancel, and I can rewrite ck on the left as (w * B) due to the second equation above:

B - (w * B) = c2 + ... + ck-1 + ck+1 + ... + cn

We know all the values on the right-hand side, so let's call their sum T. This gets us (here I'm writing (1 * B) explicitly to make the next step clearer):

(1 * B) - (w * B) = T

Factor out the B on the left gives us:

(1 - w) * B = T

Divide both sides by (1 - w):

B = T / (1 - w)

Now everything on the right is a known quantity. Plug in the appropriate values and you can calculate B. Once you know B, you can plug in all the known values in the first equation at the top to get ck.
posted by axiom at 6:11 PM on December 9, 2014

Response by poster: Thank you, everyone!
Wow, I am not smart. I actually needed ALL of that explanation, lol.
posted by coollibrarian at 8:12 PM on December 9, 2014

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