Six coins of equal weight made of gold and silver are melted together to form new coins. The ratio of gold to silver in these coins is as follows: 2:1 in one coin, 3:5 in two coins, and 7:5 in the remaining three coins. Determine the ratio of gold to silver in the newly formed coins.

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Accepted Answer

To find the ratio of gold to silver in the newly formed coins:

1. **Gold Calculation**:  
   The amount of gold in the new coins is calculated by summing the contributions from each type of coin:  
   - For the first coin (ratio 2:1): $$ \frac{2}{3} $$  
   - For the next two coins (ratio 3:5): $$ 2 	imes \frac{3}{8} $$  
   - For the last three coins (ratio 7:5): $$ 3 	imes \frac{7}{12} $$

Adding these values gives:  
   $$
   	ext{Gold} = \left( \frac{2}{3} + 2 	imes \frac{3}{8} + 3 	imes \frac{7}{12} ight) = \left( \frac{2}{3} + \frac{6}{8} + \frac{21}{12} ight) = \frac{19}{6}
   $$

2. **Silver Calculation**:  
   Similarly, calculate the amount of silver from each type of coin:  
   - For the first coin (ratio 2:1): $$ \frac{1}{3} $$  
   - For the next two coins (ratio 3:5): $$ 2 	imes \frac{5}{8} $$  
   - For the last three coins (ratio 7:5): $$ 3 	imes \frac{5}{12} $$

Adding these values gives:  
   $$
   	ext{Silver} = \left( \frac{1}{3} + 2 	imes \frac{5}{8} + 3 	imes \frac{5}{12} ight) = \left( \frac{1}{3} + \frac{10}{8} + \frac{15}{12} ight) = \frac{17}{6}
   $$

3. **Final Ratio**:  
   The ratio of gold to silver is therefore:
   $$
   (	ext{Gold}) : (	ext{Silver}) = \frac{19}{6} : \frac{17}{6} = 19 : 17
   $$

Thus, the final ratio is **19:17**.

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