a, b and c are whole numbers. If 1/(a+(1/b+1/c))=6/23 find the value of a+b+c
As a hint, lets start from Subhotosh Khan's hint but with a different problem
a + (1/b + 1/c) = 3.5 = 3 + 1/2
Since a, b, and c, are whole numbers 1/b+1/c must add to 2 or less.
(1) They can add to 2 only if both b and c are one but they must add to include the fractional part so they can't both be 1.
(2) If one of them were 1 a whole number then the other would have to be 1/2 and we have
a = 2, b = 1, c = 2 or a=2, b=2, c=1
(3) If both b and c are greater than 1, 1/b + 1/c is less than 1 and a must be 3 leaving 1/b+1/c=1/2 or 2 (b + c) = b c. So either b or c must be even. If we choose b (or c) even, b = 2 b', we have
2 b' + c =2 b' c
and we see that c (b) is also even, c = 2 c'. Thus
b' + c' = b' c'
or
c' = b' / (b'-1)
The only b' which will make c' a whole number is b'=2 which implies
a=3, b=4, c=4