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, 2a+1 = a+5
Or, a = 4

Thus, our desired two digit number is 45. 

Edited on May 4, 2007, 11:31 am

Posted by K Sengupta on 2007-05-04 11:26:06