WEEK'S PUZZLE 6

PUZZLE # 6 -from the 9th to the 15th of November (to be collected by the 16th of November)

FOUR DIGITS
This week your task is to try and find a four-digit number (abcd) so that when writing a decimal comma* between the second and the third digit (ab,cd) we get a number which is the mean or arithmetic average of the two digits which are left on both sides of the decimal comma.
*decimal comma or decimal point (ab.cd)


SOLUTION

Mentally:
The numbers are 49 and 50, because (49+50)/2 = 49,50

Mathematically:
If the number is abcd:

((10a + b) + (10c + d))/2 = 10a + b + c/10 + d/100

50 ( 10a + b) + 50 (10c + e) = 1000a + 100b + 10c + d

490c + 49d = 500a + 50b

49(10c + d) = 50(10a + b)

The only solution is 49 and 50.