# Two formulas giving slightly different output

I am struggling to find a proper explanation why the two methods that I use for finding the difference in EXCEL are giving me two slightly different numbers.

The idea behind the task is that we have a post office and we have an X items that are coming in and Y items that are going out. The data is for the last two years and all of the items are categorized for each month of the year.

I was thinking that the difference between the average in "IN" and the average in "Out" must give the difference between the total "Old" and total "New". However, the results are by 0.09 off, which frankly is quite a big number if we take into consideration that instead of a post office we are working with some sort of a financial institutions.  I would be extremely grateful if you can share your opinion in this regard. Thank you in advance!

Yours sincerely, MathBeginner

Let $$a,b$$ be in/out for the previous year, and $$c,d$$ be the in/out for the current year. Your first and second calculations are $$p_1=\frac{c}{2a}+\frac{d}{2b}-1,\ \ p_2=\frac{d+c}{a+b}-1$$ And, these two quantities are not equal in general, which is why you see a difference. And, the difference is not guaranteed to be small. Letting one of $$a$$ or $$b$$ to be $$0$$ makes $$p_1\rightarrow\infty$$, while $$p_2$$ can still be between $$[0,1]$$.