changing a cell value when the date changes to a new month

hi I've been stumped on this one for a while I am trying to set up an account reading system for water meters and it has a beginning amount (A1) and
end amount (B1). what I would like for it to do is when the new month comes around and I change the date on the spread sheet A1 will take the value of B1, but every time I try it with IF or Switch functions once I start to put a new amount in B1 it also changes A1 and I don't want it to do that.

Is there any way to accomplish this.

this is an example of one of the accounts and there will be a lot of these for the entire county. what i need is the
Beginning Read (column E) to replace its original value with the value of End Read (Column F) when i come in next month
to put in the new readings.

If the Minimum Value (Cell E5) -is going to get in the way of that i will add another column.