There is a two digit number with the unique property that if a decimal point is inserted between the two digits, the resulting number is the average of the digits. What is the number?

Let the two digit number be written as

*ab* in the decimal system. Hence,

*a* must be non zero.

By the problem:

*a.b* = Average (

*a, b*)

Or, (

*a.b*) = (

*a+b*)/2 ................(#)

Since

*a+b* is a whole number, it follows in terms of (#) that:

*b* is either 0 or 5.

If

*b* is 0, then:

*a*.0 =

*a*/2, giving:

*a* =0; which is a contradiction as a must be non zero.

Thus

*b*=5, giving:

2*(

*a*.5) =

*a*+ 5

Or, 2

*a*+1 =

*a*+5

Or,

*a *= 4

Thus, our desired two digit number is 45.

*Edited on ***May 4, 2007, 11:31 am**