Question

a college food court surveyed students to gather information about the types of drinks they like. here are the results. 603 students were surveyed in all. 297 of the students drink tea. 29 of the students drink both tea and milk, but dont drink soda. 62 of the students drink both tea and soda, but dont drink milk. 47 of the students drink tea, but drink neither soda nor milk. 56 of the students drink soda, but drink neither tea nor milk. 117 of the students drink milk, but drink neither tea nor soda. 572 of the students drink at least one of these three types of drinks (tea, soda, milk). construct a venn - diagram illustrating these results. then answer the question. how many of the students dont drink milk?

Explanation:

Step1: Analyze the given data

Let \(T\) be the set of tea - drinkers, \(S\) be the set of soda - drinkers, and \(M\) be the set of milk - drinkers. We know \(n(T\cup S\cup M)=572\), \(n(T\cap\overline{S}\cap\overline{M}) = 47\), \(n(T\cap S\cap\overline{M})=62\), \(n(T\cap S\cap M)=29\), \(n(\overline{T}\cap S\cap\overline{M}) = 56\), \(n(\overline{T}\cap\overline{S}\cap M)=117\), \(n(\overline{T}\cap S\cap M)=297\), and the total number of students surveyed \(N = 603\).

Step2: Calculate the number of students who drink milk

We can use the principle of inclusion - exclusion. First, we know that \(n(T\cup S\cup M)=n(T\cap\overline{S}\cap\overline{M})+n(T\cap S\cap\overline{M})+n(T\cap S\cap M)+n(\overline{T}\cap S\cap\overline{M})+n(\overline{T}\cap\overline{S}\cap M)+n(\overline{T}\cap S\cap M)+n(\overline{T}\cap\overline{S}\cap\overline{M})\).
The number of students who drink milk \(n(M)=n(T\cap S\cap M)+n(\overline{T}\cap S\cap M)+n(\overline{T}\cap\overline{S}\cap M)\)
\(n(M)=29 + 297+117=443\).

Step3: Calculate the number of students who don't drink milk

The number of students who don't drink milk is the total number of students minus the number of students who drink milk.
Let \(x\) be the number of students who don't drink milk. Then \(x = 603-443 = 160\).

Answer:

160