If $a : (b + c) = 1 : 3$ and $c : (a + b) = 5 : 7$, then $b : (a + c)$ is equal to (S.S.C., 2006)

### Concept & Equating Total Parts When given multiple ratios involving the same set of variables partitioned differently (like $a$ vs $b+c$ and $c$ vs $a+b$), the sum of all variables ($a+b+c$) must be constant. By making the total number of parts in both ratios equal, you can directly compare and find the individual values of the variables. ### Step-by-Step Solution - **Given Ratios:** 1. $a : (b + c) = 1 : 3$ 2. $c : (a + b) = 5 : 7$ - **Calculate Total Parts:** - For ratio 1, total parts = $1 + 3 = 4$. - For ratio 2, total parts = $5 + 7 = 12$. - **Equalize Total Parts:** - The LCM of $4$ and $12$ is $12$. - Multiply the first ratio by $3$ to make its total parts $12$: $a : (b + c) = (1 \times 3) : (3 \times 3) = 3 : 9$. - **Identify Individual Values:** - Now, total parts = $12$. - From the modified first ratio, $a = 3$. - From the second ratio, $c = 5$. - **Find the Missing Variable:** - We know $a + b + c = 12$. - $3 + b + 5 = 12 \implies b + 8 = 12 \implies b = 4$. - **Calculate Target Ratio:** - We need $b : (a + c)$. - Substitute the values: $4 : (3 + 5) = 4 : 8 = 1 : 2$. ### Exam Strategy & Shortcut Whenever a question gives part-to-rest ratios for the same total pool, immediately look at the sums ($1+3=4$ and $5+7=12$). Scale the ratios so the sums match. Once matched, pluck the independent variables directly ($a=3, c=5$), subtract from the total to get the last one ($b=4$), and form your final ratio. ### Common Pitfall A common mistake is trying to set up complex simultaneous algebraic equations (e.g., $3a = b+c$ and $7c = 5a+5b$), which is highly prone to calculation errors and wastes precious time compared to simply scaling the ratio parts. ### Final Answer Therefore, the correct answer is **$1 : 2$**.