Math Puzzle No. 1: Doubtful Doubling

Here's a puzzle that was first passed on to me by Al Lippert, who spent years providing the "Problem of the Day" at MathPath.  What goes after the last equals sign?

Hints: The answer we're looking for isn't 80.  And the equalities are all true!

Update: 11/6/20: read below for the solution.

There are undoubtedly many answers to the puzzle, but the solution we both came up with is: 130.

The hint with the puzzle said that all the equalities are true, and the answer was not 80.  How can this be?  The key is to write numbers in a different base!

We usually write numbers in base 10; starting from the right we have a 1's digit, then a 10's digit, then a 100's digit, and so on:

2345 = 2000 + 300 + 40 + 5 = 2 (1000) + 3 (100) + 4 (10) + 5(1).

In the puzzle above, the numbers on the left are written in base 10, but the numbers on the right are written in base 5, with a 1's digit, a 5's digit, a 25's digit, and so on.  Thus:

10 base 5 = 1(5) + 0(1) = 5 base 10

20 base 5 = 2(5) + 0(1) = 10 base 10

40 base 5 = 4(5) + 0(1) = 20 base 10

To answer the puzzle, we write 40 in base 5.  Because 40 = 25 + 15, the answer is 130:

130 base 5 = 1(25) + 3(5) + 0(1) = 40 base 10