Exponential decay and time constants The time constant τ is the amount of time that
an exponentially decaying quantity takes to decay by a factor of 1/e. Because 1/e
is approximately 0.368, τ is the amount of time that the quantity takes to decay to
approximately 36.8% of its original amount.
(a) How are the time constant τ and the decay rate λ related?

I used the decay model:

$r = r_0e^{\lambda t} $

Where t = time 

r = amount of the particular exponentially decaying matter at time t

$-\lambda$ = decay rate

$e^{-\lambda t} = \frac{r}{r_0}$ 

Then I let $\frac{r}{r_0} = \frac{1}{e}$ and $t = τ$

$e^{-\lambda τ} = \frac{1}{e}$

Simplifying I got, $τ = \frac{1}{\lambda}$

Is this the correct way to show that the time constant is inversely proportional to the decay rate? 

(b) Express the time constant in terms of the half-life.

I'm not sure how to do this, do I just set $\frac{r}{r_0}  = 0.5$?
Half life=$ln 2$*$\tau$ and $\tau$=1/$\lambda$
In both cases, you are absolutely correct. You can prove this here