I am thinking of a three digit number. If you take away the first digit the remaining number is 1 fifth of the original three digit number. If you take away the next digit, the remaining number is 1 fifth of the two digit number.

What was the original three digit number?

Digits: a,b,c
Number: 100a + 10b + c

[1] 5(10b + c) = 100a + 10b + c

[2] 5c = 10b + c

Do a couple headstands with [1] and [2]
to end up with a unique 125:

Combine [1] and [2], eliminate c to get b = 2a; so:
a b
1 2
2 4
3 6
4 8

Use b = 2a in [2] to get c = 5a; so:
a b c
1 2 5
2 4 10 : nope
3 6 15 : nope
4 8 20 : nope

Cute li'l problem!

Nicely explained, Denis!