1. Classify the 400 people into the given groups

Let:

• $S\cap N$ = smoker and non-vegetarian
• $S\cap V$ = smoker and vegetarian
• $\bar S\cap V$ = non-smoker and vegetarian

Given:

• $|S\cap N| = 160$
• $|S\cap V| = 100$
• remaining $140$ are $\bar S\cap V$

So the three groups are:

$160+100+140=400$

2. Use the given probabilities of chest disorder

The chances of getting the chest disorder are:

• for smoker and non-vegetarian: $35\% = 0.35$
• for smoker and vegetarian: $20\% = 0.20$
• for non-smoker and vegetarian: $10\% = 0.10$

3. Find the number of people with chest disorder from each group

From smoker and non-vegetarian:

$160\times 0.35 = 56$

From smoker and vegetarian:

$100\times 0.20 = 20$

From non-smoker and vegetarian:

$140\times 0.10 = 14$

4. Find total number of people suffering from the disorder

$56+20+14=90$

5. Apply conditional probability

We need:

$P(S\cap N\mid D)$

where $D$ = suffering from the chest disorder.

Thus,

$P(S\cap N\mid D)=\frac{\text{number of people who are smoker and non-vegetarian and have disorder}}{\text{total number of people who have disorder}}$

$=\frac{56}{90}=\frac{28}{45}$

6. Check options

$\frac{28}{45}$

So the correct option is D.