Six coins of gold and silver of equal weights are melted and new coins are cast. The ratio of gold and silver in one of the coins is $2 : 1$, in another two coins $3 : 5$ and $7 : 5$ in the remaining coins. What will be the ratio between gold and silver respectively in the new coins? (R.R.B., 2006)

### Concept & Weighted Mixtures When combining items of equal total volumes but different internal ratios and differing quantities (weights), determine a unified weight scale using the LCM of the ratio sums, then aggregate the components based on the number of items in each group. ### Step-by-Step Solution * **Given:** - Total number of coins = $6$ (all equal weight). - Group 1: $1$ coin with Gold:Silver = $2 : 1$ (Sum of parts = $3$) - Group 2: $2$ coins with Gold:Silver = $3 : 5$ (Sum of parts = $8$) - Group 3: Remaining coins = $6 - 1 - 2 = 3$ coins with Gold:Silver = $7 : 5$ (Sum of parts = $12$) * **Calculation:** 1. Find the LCM of the ratio sums ($3, 8, 12$) to avoid fractions. The LCM is $24$. Let the weight of one coin be $24$ units. 2. Calculate the components for Group 1 ($1$ coin): - Gold = $\frac{2}{3} \times 24 = 16$ units - Silver = $\frac{1}{3} \times 24 = 8$ units - Total from Group 1 = $16$ Gold, $8$ Silver. 3. Calculate the components for Group 2 ($2$ coins): - Gold per coin = $\frac{3}{8} \times 24 = 9$ units. Total Gold for $2$ coins = $9 \times 2 = 18$ units. - Silver per coin = $\frac{5}{8} \times 24 = 15$ units. Total Silver for $2$ coins = $15 \times 2 = 30$ units. 4. Calculate the components for Group 3 ($3$ coins): - Gold per coin = $\frac{7}{12} \times 24 = 14$ units. Total Gold for $3$ coins = $14 \times 3 = 42$ units. - Silver per coin = $\frac{5}{12} \times 24 = 10$ units. Total Silver for $3$ coins = $10 \times 3 = 30$ units. 5. Aggregate the total amounts for the new cast coins: - Total Gold = $16 + 18 + 42 = 76$ units. - Total Silver = $8 + 30 + 30 = 68$ units. 6. The final ratio of Gold to Silver is $76 : 68$. 7. Simplifying the ratio by dividing by $4$ yields $19 : 17$. ### Exam Strategy & Shortcut Scale up each ratio directly using multipliers to reach the LCM of $24$, then multiply by the number of coins. Base scaling: $2:1 \rightarrow 16:8$. $3:5 \rightarrow 9:15$. $7:5 \rightarrow 14:10$. Multiply by coin counts: $1 \times (16:8) \rightarrow 16 : 8$ $2 \times (9:15) \rightarrow 18 : 30$ $3 \times (14:10) \rightarrow 42 : 30$ Sum: $(16+18+42) : (8+30+30) = 76 : 68 = 19 : 17$. ### Common Pitfall A frequent error is calculating the composition for a single coin of each type and just adding those together ($16+9+14 : 8+15+10$), failing to account for the fact that there are multiple coins of the second and third types. ### Final Answer Therefore, the correct answer is **$19 : 17$**.