$A$ and $B$ are two alloys of gold and copper prepared by mixing metals in the ratio $7 : 2$ and $7 : 11$ respectively. If equal quantities of the alloys are melted to form a third alloy $C$, then the ratio of gold and copper in alloy $C$ will be

### Concept & Mixture of Equal Quantities When mixing equal quantities of different mixtures, find the LCM of the sum of the ratio terms for each mixture. Assume this LCM as the total quantity for each mixture to easily determine the exact amounts of each component without dealing with complex fractions. ### Step-by-Step Solution * **Given:** - Ratio of Gold:Copper in Alloy A = $7 : 2$ (Total parts = $9$) - Ratio of Gold:Copper in Alloy B = $7 : 11$ (Total parts = $18$) - Equal quantities of A and B are mixed. * **Calculation:** 1. The sum of ratio parts for A is $7 + 2 = 9$. 2. The sum of ratio parts for B is $7 + 11 = 18$. 3. Let's assume the quantity of each alloy taken is the LCM of $9$ and $18$, which is $18$ units. 4. In $18$ units of Alloy A: - Gold = $\frac{7}{9} \times 18 = 14$ units - Copper = $\frac{2}{9} \times 18 = 4$ units 5. In $18$ units of Alloy B: - Gold = $\frac{7}{18} \times 18 = 7$ units - Copper = $\frac{11}{18} \times 18 = 11$ units 6. In the new Alloy C (total $36$ units): - Total Gold = $14 + 7 = 21$ units - Total Copper = $4 + 11 = 15$ units 7. The ratio of Gold to Copper in C is $21 : 15$. 8. Simplifying the ratio by dividing by $3$: $7 : 5$. ### Exam Strategy & Shortcut Instead of fractions like $(\frac{7}{9} + \frac{7}{18}) : (\frac{2}{9} + \frac{11}{18})$, equate the total parts directly. Alloy A is $7:2$ (sum $9$). Alloy B is $7:11$ (sum $18$). Multiply Alloy A's ratio by $2$ to make its total parts $18$. Alloy A becomes $14:4$. Now just add them: $(14+7) : (4+11) = 21:15 = 7:5$. ### Common Pitfall A very common mistake is simply adding the numerators and denominators of the ratios (i.e., $7+7 : 2+11 = 14:13$). Ratios can only be added if they represent components of the same total quantity. ### Final Answer Therefore, the correct answer is **$7 : 5$**.