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

1.	If a : (b + c) = 1 : 3 and c : (a + b) = 5 : 7, then b : (a + c) is equal to (a) 1 : 2 (b) 2 : 3 (c) 1 : 3 (d) 2 : 1
Answer» (a) 1 : 2 \(\frac{a}{b+c}=\frac13\) ⇒ a = \(\frac{b+c}{3}\) .......(i) \(\frac{c}{a+b}=\frac57\) ⇒ \(\frac{c}{\frac{b+c}{3}+b} = \frac57\) ⇒ \(\frac{3c}{b+c+3b}=\frac57\) ⇒ \(\frac{3c}{4b+c}=\frac57\) ⇒ 21c = 20b + 5c ⇒ 16c = 20b ⇒ b = \(\frac{4c}{5}\) Putting b = \(\frac{4c}{5}\) in (i) we get \(\frac{a}{\frac{4c}{5}+c} = \frac13\) ⇒ \(\frac{5a}{9c} = \frac13\) ⇒ a = \(\frac{3c}{5}\) \(\frac{b}{a+c}\) = \(\frac{\frac{4c}{5}}{\frac{3c}{5}+c}\) = \(\frac{4c}{8c} = \frac12\) = 1 : 2

If a : (b + c) = 1 : 3 and c : (a + b) = 5 : 7, then b : (a + c) is equal to (a) 1 : 2 (b) 2 : 3 (c) 1 : 3 (d) 2 : 1

Answer»

(a) 1 : 2

\(\frac{a}{b+c}=\frac13\) ⇒ a = \(\frac{b+c}{3}\) .......(i)

\(\frac{c}{a+b}=\frac57\) ⇒ \(\frac{c}{\frac{b+c}{3}+b} = \frac57\)

⇒ \(\frac{3c}{b+c+3b}=\frac57\) ⇒ \(\frac{3c}{4b+c}=\frac57\)

⇒ 21c = 20b + 5c ⇒ 16c = 20b ⇒ b = \(\frac{4c}{5}\)

Putting b = \(\frac{4c}{5}\) in (i) we get

\(\frac{a}{\frac{4c}{5}+c} = \frac13\) ⇒ \(\frac{5a}{9c} = \frac13\) ⇒ a = \(\frac{3c}{5}\)

\(\frac{b}{a+c}\) = \(\frac{\frac{4c}{5}}{\frac{3c}{5}+c}\) = \(\frac{4c}{8c} = \frac12\) = 1 : 2