Question

The length of an aluminum wire is quadrupled and the radius is doubled. By which factor does the resistance change?

Answer 1

The resistance of a conductive wire is given by:
$R= \frac{\rho L}{A}$
where
$\rho$ is the material resistivity
$L$ is the wire length
$A$ is the cross-sectional area of the wire

The length of the wire is quadrupled, so if we call L the original length and L' the new length, we can write 
$L'=4 L$

Similarly, the radius of the wire is doubled (r'=2r), so the new area is
$A'= \pi (r')^2 = \pi (2r)^2 = 4 \pi r^2 = 4A$

And if we substitute into the equation, we find that the new resistance of the wire is
$R'= \frac{\rho L'}{A'}= \frac{\rho (4L)}{4 A'} = \frac{\rho L}{A}=R$
Therefore, R=R': this means that the resistance of the wire did not change.