Two alloys contain zinc and copper in the ratio of $2 : 1$ and $4 : 1$. In what ratio the two alloys should be added together to get a new alloy having zinc and copper in the ratio of $3 : 1$?

### Concept & Rule of Alligation The Rule of Alligation can be used to find the ratio in which two mixtures must be combined to yield a desired mixture. We apply alligation to the fractional concentration of a single component (e.g., zinc) across all mixtures. $$ \frac{\text{Quantity of 1st}}{\text{Quantity of 2nd}} = \frac{|\text{Cost of 2nd} - \text{Mean Price}|}{|\text{Cost of 1st} - \text{Mean Price}|} $$ ### Step-by-Step Solution * **Given:** - Zinc:Copper in Alloy 1 = $2 : 1 \Rightarrow$ Zinc fraction = $\frac{2}{3}$ - Zinc:Copper in Alloy 2 = $4 : 1 \Rightarrow$ Zinc fraction = $\frac{4}{5}$ - Zinc:Copper in Target Alloy = $3 : 1 \Rightarrow$ Zinc fraction = $\frac{3}{4}$ * **Calculation:** 1. We will use the fractional concentration of zinc for alligation. 2. Part 1 (Alloy 1): $\frac{2}{3}$ 3. Part 2 (Alloy 2): $\frac{4}{5}$ 4. Mean (Target): $\frac{3}{4}$ 5. Calculate the cross-differences: - Difference 1: $|\frac{4}{5} - \frac{3}{4}| = |\frac{16 - 15}{20}| = \frac{1}{20}$ - Difference 2: $|\frac{2}{3} - \frac{3}{4}| = |\frac{8 - 9}{12}| = \frac{1}{12}$ 6. The required ratio of Alloy 1 to Alloy 2 is $\frac{1}{20} : \frac{1}{12}$. 7. Multiply by the LCM ($60$) to clear fractions: $(\frac{1}{20} \times 60) : (\frac{1}{12} \times 60) = 3 : 5$. ### Exam Strategy & Shortcut To avoid fraction subtraction, assume a total volume that is the LCM of the denominators ($3, 5, 4$), which is $60$. Zinc in Alloy 1 = $\frac{2}{3}$ of $60 = 40$. Zinc in Alloy 2 = $\frac{4}{5}$ of $60 = 48$. Target Zinc = $\frac{3}{4}$ of $60 = 45$. Apply alligation on these integers: $|48 - 45| : |40 - 45| = 3 : 5$. This is much faster and less prone to errors. ### Common Pitfall Applying alligation directly to the ratio numbers (e.g., $2$ and $4$ to get $3$) instead of their fractional concentrations relative to the whole mixture. ### Final Answer Therefore, the correct answer is **$3 : 5$**.