In a hotel, 60% had vegetarian lunch while 30% had non vegetarian lunch and 15% had both types of lunch. If 96 people were present, how many did not eat either type of lunch ?

Options

A. 20
B. 24
C. 26
D. 28

Answer

To solve the problem step by step, we will use the principle of set theory.

### Step 1: Define the percentages and total people
We know that:
- 60% had vegetarian lunch.
- 30% had non-vegetarian lunch.
- 15% had both types of lunch.
- Total number of people present = 96.

### Step 2: Calculate the number of people who had vegetarian lunch
The number of people who had vegetarian lunch can be calculated as:
\[ \text{Number of vegetarians} = 60\% \text{ of } 96 = \frac{60}{100} \times 96 = 57.6 \approx 58 \text{ (rounding to the nearest whole number)} \]

### Step 3: Calculate the number of people who had non-vegetarian lunch
The number of people who had non-vegetarian lunch can be calculated as:
\[ \text{Number of non-vegetarians} = 30\% \text{ of } 96 = \frac{30}{100} \times 96 = 28.8 \approx 29 \text{ (rounding to the nearest whole number)} \]

### Step 4: Calculate the number of people who had both types of lunch
The number of people who had both types of lunch is given as:
\[ \text{Number of both} = 15\% \text{ of } 96 = \frac{15}{100} \times 96 = 14.4 \approx 14 \text{ (rounding to the nearest whole number)} \]

### Step 5: Use the principle of inclusion-exclusion to find total lunch participants
Using the inclusion-exclusion principle:
\[ \text{Total who had lunch} = (\text{Vegetarians}) + (\text{Non-vegetarians}) - (\text{Both}) \]
Substituting the values:
\[ \text{Total who had lunch} = 58 + 29 - 14 = 73 \]

### Step 6: Calculate the number of people who did not eat either type of lunch
To find the number of people who did not eat either type of lunch:
\[ \text{People who did not eat lunch} = \text{Total people} - \text{Total who had lunch} \]
Substituting the values:
\[ \text{People who did not eat lunch} = 96 - 73 = 23 \]

### Conclusion
Thus, the number of people who did not eat either type of lunch is **23**.