FORTNIGHT'S PUZZLE 6

PUZZLE 6: 14th-20th December (to be e-mailed by the 21st December)
POSTED BY IES CAMPANAR

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 = 49,50Mathematically:If the number is abcd:((10a + b) + (10c + d))/2 = 10a + b + c/10 + d/10050 ( 10a + b) + 50 (10c + e) = 1000a + 100b + 10c + d490c + 49d = 500a + 50b49(10c + d) = 50(10a + b)The only solution is 49 and 50.