1. Define the events

Let:

• $S$ = person is a smoker
• $\bar S$ = person is a non-smoker
• $C$ = person has lung cancer

Given:

• $P(S)=25\%=\dfrac{1}{4}$
• $P(\bar S)=75\%=\dfrac{3}{4}$

Also, a smoker has $27$ times the chance to develop lung cancer than a non-smoker. So if $P(C\mid \bar S)=x,$ then $P(C\mid S)=27x.$

2. Use Bayes' theorem

We need to find: $P(S\mid C).$

By Bayes' theorem, $P(S\mid C)=\frac{P(C\mid S)P(S)}{P(C\mid S)P(S)+P(C\mid \bar S)P(\bar S)}.$

Substitute the values: $P(S\mid C)=\frac{27x\cdot \frac14}{27x\cdot \frac14+x\cdot \frac34}.$

Factor out $x$: $P(S\mid C)=\frac{27\cdot \frac14}{27\cdot \frac14+\frac34}.$

Now simplify: P(S\mid C)=\frac{27/4}{27/4+3/4}=rac{27/4}{30/4}=rac{27}{30}=rac{9}{10}.

Thus, $P(S\mid C)=90\%.$

3. Match with the given form

The problem says the probability is $\dfrac{k}{10}\%$.

This is clearly intended to mean $k\times 10\%$ (since the probability found is $90\%$), so $90\%=k\times 10\% \Rightarrow k=9.$

4. Final answer

$\boxed{9}$

5. Comparison with stored answer

Stored correct answer = $9$.

Our derived answer also is $9$, so they agree.