If $$(a + b) : (b + c) : (c + a) = 7 : 6 : 5$$ and $$a + b + c = 27$$, then what will be the value of $$\frac{1}{a} : \frac{1}{b} : \frac{1}{c}$$?

Let the (a + b), (b + c) and (c + a) be 7x, 6x and 5x respectively.
Now,
(a + b) + (b + c) + (c + a) = 7x + 6x + 5x
2(a + b + c) = 18x
Put the value of (a + b + c),
2 $$\times$$ 27 = 18x
x = 3
a + b = 7x = 3 $$\times$$ 7 = 21
c = (a + b + c) - a + b = 27 - 21 = 6
b + c = 6x = 3 $$\times$$ 6 = 18
b = 18 - 6 = 12
a + b + c = 27
a = 27 - 18 = 9
Now,
$$\frac{1}{a} : \frac{1}{b} : \frac{1}{c}$$ = $$\frac{1}{9} : \frac{1}{12} : \frac{1}{6}$$
= 4 : 3 : 6