Answer
See answers.
Work Step by Step
a. Assume that the ammeter has zero resistance and that the voltmeter has resistance $R_V$. Use Ohm’s law with the equivalent resistance. The voltmeter is reading the voltage across the battery.
$$V=IR_{eq}=I\frac{1}{\frac{1}{R}+\frac{1}{R_V}}$$
$$V(\frac{1}{R}+\frac{1}{R_V})=I$$
$$(\frac{1}{R}+\frac{1}{R_V})=\frac{I}{V}$$
$$\frac{1}{R}=\frac{I}{V}-\frac{1}{R_V}$$
b. Assume the voltmeter has infinite resistance and that the ammeter has resistance $R_A$. Use Ohm’s law with the equivalent resistance. The voltmeter is reading the voltage across the battery.
$$V=IR_{eq}=I(R+R_A)$$
$$ (R+R_A)=\frac{V}{I}$$
$$ R=\frac{V}{I}-R_A$$