How many people ordered three different types of food?

Question

I have answered this question but I think my answer is inconsistent with the given information. Would you please help me? Thanks!

This restaurant offers Appetizers, Entrees, and Desserts. $167$ people come in tonight. Everyone orders at least one thing. $142$ people ordered exactly one thing. $100$ people ordered Appetizers, $50$ people Entrees, and $48$ Desserts. $13$ people ordered both Appetizers and Entrees, $14$ people both Entrees and Desserts, and $10$ people ordered Appetizers and Desserts. How many people ordered all three?

Here's my answer.

Let the number we seek be $T$. Then $$167=100+50+48-(13+14+10)+T,$$ then $$167=198-37+T,$$ then $T=6$.

I'm not using everything given in the problem, and I highly doubt this is correct but I don't know how to approach it differently. Thank you!

Answer 1

Your answer is correct.

The one piece of information you did not use was that $142$ people ordered exactly one thing. We can use that information to check your answer.

You found that $6$ people ordered all three courses. Since $13$ people ordered both an appetizer and an entree, $13 - 6 = 7$ people ordered only an appetizer and entree. Since $14$ people ordered both an entree and a dessert, $14 - 6 = 8$ people ordered only an entree and a dessert. Since $10$ people ordered both an appetizer and a dessert, $10 - 6 = 4$ people ordered only an appetizer and a dessert.

Hence, $100 - 6 - 7 - 4 = 83$ people ordered only an appetizer, $50 - 6 - 7 - 8 = 29$ people ordered only an entree, and $48 - 6 - 8 - 4 = 30$ people ordered only a dessert. Consequently, $83 + 29 + 30 = 142$ people ordered only one course.

Answer 2

You can solve the problem in the following way: If you denote the number of people that ordered only Appetizers, Entrees and Deserts by A, E and D, respectively, and the number of people that ordered all three by T, then the following is true:

A=100-13-10-T=77-T

E=50-13-14-T=23-T

D=48-14-10-T=24-T

This can be directly observed from a Venn diagram of this problem. But we know that the number of people that ordered only one thing is 142, so 142=A+E+D.

142=77+23+24-3T

3T=142-124=18

T=6 which confirms your answer.