Coin

Let the weight in the coin and price of copper be $w_c$ and $p_c$ and those of gold are $w_g$ and $p_g$ respectively. Then the value of metal in the coin:

$\begin{aligned} v & = \blue{w_c p_c} + w_g p_g & \small \blue{\text{Given }w_c p_c = \$0.10} \\ & = \blue{0.10} + 50\red{w_g} & \small \red{\text{also }\frac {w_g}{w_c} = \frac 3{97}} \\ & = \blue{0.10} + 50 \times \red{\frac {3 \blue{w_c}}{97}} & \small \blue{\text{and }w_c = \frac {0.10}{p_c}} \\ & = \blue{0.10} + 50 \times \frac 3{97} \times \blue{\frac {0.10}{0.05}} \\ & \approx \boxed{3.19} & \small \blue{\text{to 3 significant figures}} \end{aligned}$