When Mr. Smith returned from Europe in 1966, he found that he had in his possession 35 British sixpence coins, 55 French ten-centime pieces, and 77 Greek drachmas. Mr. Smith converted each of these coins to its value in American money (rounded off to the nearest cent) and found that the total was worth $5.86. How much was each coin worth in 1966 (to the nearest cent)?
I started by setting up the equation 35B+55F+77G=586, but I was not sure how to solve a three-variable diophantine equation.
This isn't generalized to other Diophantine equations but there is a pretty quick way to solve this: \begin{eqnarray} 35B+55F+77G&=&586\\ 8\cdot77=616&>&586 \rightarrow G < 8\\ 35B+55F+77G \rightarrow 2G&=&1\mod 5\rightarrow G=3\mod5 \\ 35B+55F+77\cdot 3&=&586 \rightarrow 35B+55F=355\\ 7B+11F&=&71\\ 4F&=&1 \mod7\rightarrow F=2\mod7\\ 11\cdot9&>&71\rightarrow F < 9\rightarrow F=2\\ 7B+22=71\rightarrow7B=49\rightarrow B&=&7\\ (B,F,G)&=&(7,2,3) \end{eqnarray}