Why do significant figure rules contradict each other?
Bear in mind the significant digit rules are just  simplifications of the rules of absolute (+, -) and relative (*,/) variance propagation.

The rule of significant digit numbers for summing is for 2 statistically independent variables. Their  correlation coefficient = 0 and the error is $$  s_\mathrm{C}= \sqrt{(s_\mathrm{A})^2+(s_\mathrm{B})^2}$$
and therefore about at the same order as for less precisely known variable.

Multiplying a variable by number means summing the same variable $N$ times, with the correlation coefficient 1 and the error is growing linearly by $N$. As here applies the multiplicative  rule for propagation of relative variance.

$$ s_\mathrm{C,r}= \sqrt{(s_\mathrm{A,r})^2+(s_\mathrm{B,r})^2}\\=\sqrt{(s_\mathrm{A,r})^2+(s_\mathrm{3})^2}=s_\mathrm{A,r}$$

I agree with the comment to follow the rule of significant digits on the final result only, to avoid cummulative rounding errors. 

On the other side, it can be hard to track the digits but for simple formulas. Applying the variance propagation rules may be easier way to go.