Summing with minimum values

Question

I have the following practical question I guess it falls under basic arithmatic and it is not a homework. (Yes, I am kind of ashamed I don't know how to solve it myself after all these years....). Here it goes:

It is required to figure the total amount a person has kept in her account for 1 full year. Let's denot this by $R$.

She had $3$ accounts in $3$ different banks. The amount in each is shown in the picture below.

I approached this in 2 approaches hoping that the answer would be the same....:

Approach 1: for each bank, find the amount that remained 1 year in the account by calculating the minimum amoun in $2019,2020$. Repeat the process for each bank. At the end sum all these numbers to get the desired value $R3=1450$ as shown in the picuture below.

Approach 2: Sum the amounts in $2019$ in each bank, do the same for year $2020$, then calculate the minimum of the two sums. That is what is shown as $R2=1850$ in the picture below.

I thought that $R2$ will be equal to $R3$, but the values are different!!!

I understand that maybe there is no theory that says Sum (min. valuess)=Min(sum values), hwoever both approaches "make sense to me" :)

My question is which approach is in error and why?

Thank you.

Answer 1

The first method (which computes 1450) is the right one.

Here's what went wrong:

You were doing two operations.

One was "summing things up" The other was "taking the min of things"

You incorrectly assumed that whichever order you did them in would lead to the same result.

Mathematicians have a name for things where order doesn't matter: we talk about commutativity. Addition is commutative, for instance: if you take the number 10, and add 5, you get the same thing as if you took the number 5 and added 10. Division is NOT commutatitive: take the number 10 and divide by 5 and you get 2; take the number 5 and divide by 10, and you get $0.5$ --- very different results!

In your case, a more extreme example is this:

1000  0       0
  0  1000     0 
---------------
1000 1000    1000 OR 0

You can see that all the money got moved from account A to account B sometime mid-year, so clearly NO money remained in either account all year. That gives the two zeroes on the right, and the sum of these is the 0 at the bottom, which is clearly the right answer.

If, instead, you sum up the accounts, you get the two "1000"s at the bottom, and taking the min gets you 1000, which is not the right answer.

So "sum" and "min" don't commute: the min of the sums of some things is generally different than the sum of the mins of those things. Now it may happen that in some cases these two things are actually the same, but that's just a fluke.

As an example, look at 10 divided by 10. If you take these two (identical) numbers in the opposite order, you still get 10 divided by 10, and the two results are the same. But for division to be called "commutative", we would require that $a/b$ and $b/a$ always be the same, regardless of what numbers you use for $a$ and $b$.