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

1 : 2

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2 : 3

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1 : 3

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2 : 1

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Solution

The correct option is A $1 : 2$
It is given that $a : (b + c) = 1 : 3$ and $c : (a + b) = 5 : 7$ and we solve these expressions as follows:

$\frac{a}{b + c} = \frac{1}{3} \Rightarrow 3 a = b + c \Rightarrow 3 a - b = c . . . . (1)$

$\frac{c}{a + b} = \frac{5}{7} \Rightarrow 7 c = 5 (a + b) \Rightarrow 7 c = 5 a + 5 b . . . . (2)$

Multiplying the first equation by $7$ we get:

$7 (3 a - b) = 7 c \Rightarrow 7 c = 21 a - 7 b . . . . (3)$

Now, subtracting equation 2 from equation 3, we have:

$7 c - 7 c = (21 a - 7 b) - (5 a + 5 b) \Rightarrow 0 = 21 a - 7 b - 5 a - 5 b \Rightarrow 16 a = 12 b \Rightarrow b = \frac{16 a}{12} \Rightarrow b = \frac{4 a}{3}$

Substituting the value of $b$ in equation 1:

$3 a - \frac{4 a}{3} = c \Rightarrow c = \frac{9 a}{3} - \frac{4 a}{3} \Rightarrow c = \frac{9 a - 4 a}{3} \Rightarrow c = \frac{5 a}{3}$

Now, lets find the value of $b : (a + c)$ as shown below:

$\frac{b}{a + c} = \frac{\frac{4 a}{3}}{a + \frac{5 a}{3}} = \frac{\frac{4 a}{3}}{\frac{3 a}{3} + \frac{5 a}{3}} = \frac{\frac{4 a}{3}}{\frac{8 a}{3}} = \frac{4 a}{3} \times \frac{3}{8 a} = \frac{4 a}{8 a} = \frac{1}{2} = 1 : 2$

hence, $b : (a + c) = 1 : 2$ .

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