If x + y : y + z : z + x = 6 : 7 : 8 and x + y + z = 14, find the value of z.

Question: If x + y : y + z : z + x = 6 : 7 : 8 and x + y + z = 14, find the value of z.

Solution:
Given,
x + y : y + z : z + x = 6 : 7 : 8
x + y + z = 14...........(i)

Let,
x + y = 6k, y + z = 7k, z + x = 8k

adding them we get,
2( x + y + z) = 21k
or, 2 × 14 = 21k
or, k = (2 × 14)/21
∴ k = 4/3

as, x + y = 6k
or, x + y = 6(4/3)
∴ x + y = 8

from (i) we get,
x + y + z = 14
8 + z = 14
z = 14 - 8
z = 6