If a : (b + c) = 1 : 3 and c : (a + b) = 5 : 7, then b : (a + c) is equal to.

Answer: Option A

$$\begin{array}{rl}& =\frac{a}{b+c}=\frac{1}{3}\\ & \Rightarrow a=\frac{b+c}{3}\\ & \Rightarrow \frac{c}{a+b}=\frac{5}{7}\\ & \Rightarrow 7c=5a+5b\\ & \Rightarrow 7c=\frac{5\left(b+c\right)}{3}+5b\\ & \Rightarrow 7c-\frac{5}{3}c=5b+\frac{5}{3}b\\ & \Rightarrow \frac{16c}{3}=\frac{20b}{3}\\ & \Rightarrow 16c=20b\\ & \Rightarrow b=\frac{4}{5}c.\\ & a=\frac{b+c}{3}=\frac{\frac{4}{5}c+c}{3}\\ & =\frac{9c}{5}\times \frac{1}{3}\\ & =\frac{3}{5}c.\\ & \therefore \frac{b}{a+c}=\frac{\left(\frac{4}{5}c\right)}{\left(\frac{3}{5}c+c\right)}\\ & =\frac{4c}{5}\times \frac{5}{8c}\\ & =\frac{1}{2}\\ & =1:2\end{array}$$