3.In a four-digit number with distinct digits, the sum of
its middle digits equals the sum of its extreme digits.
The sum of its second and fourth digits equals five
times the sum of its other two digits. If the sum of its
digits is 18. what is the sum of all the possible
values of the hundreds digit?
(A) 21
(B) 24
(C) 27
(D) 30
Answers
Given : a four-digit number with distinct digits, the sum of its middle digits equals the sum of its extreme digits. The sum of its second and fourth digits equals five times the sum of its other two digits. the sum of its digits is 18.
To find : sum of all the possible values of the hundreds digit
Solution:
a four-digit number with distinct digits
ABCD
sum of its middle digits equals the sum of its extreme digits.
B + C = A + D
sum of its second and fourth digits equals five times the sum of its other two digits
B + D = 5 ( A + C)
sum of its digits is 18
=> A + B + C + D = 18
=> (A + D) + (B + C) = 18
=> A + D = B + C = 9
=> D= 9 - A
C = 9 - B
B + D = 5 ( A + C)
B + 9 - A = 5 ( A + 9 - B)
=> B - A + 9 = 5A + 45 - 5B
=> 6B = 6A + 36
=> B = A + 6
A can not be zero as its 1st digit
A = 1 => B = 7 Number is 1728
A = 2 => B = 8 Number is 2817
A = 3 => B = 9 Number is 3906
1728 , 2817 & 3906 are three possible number
Hundred digits are 7 , 8 & 9
Sum = 7 + 8 + 9 = 24
sum of all the possible values of the hundreds digit = 24
option B is correct
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