Question 4 A survey asks 216 people "What beverage do you drink in the morning?", and offers choices: - Coffee only - Tea only - Both tea and coffee - Neither tea nor coffee Suppose 80 report tea only, 20 report coffee only, and 44 report both. How many people drink tea in the morning? How many people drink coffee in the morning? How many people drink neither tea nor coffee? Question Help: Video Submit Question

To solve this problem, we can use the principle of inclusion-exclusion. Let's denote the following: - \( T \) as the number of people who drink tea only - \( C \) as the number of people who drink coffee only - \( B \) as the number of people who drink both tea and coffee - \( N \) as the number of people who drink neither tea nor coffee We are given the following information: - Total number of people surveyed: 216 - Number of people who drink tea only: 80 - Number of people who drink coffee only: 20 - Number of people who drink both tea and coffee: 44 We can set up the following equations based on the principle of inclusion-exclusion: 1. Total number of people surveyed = Number of people who drink tea only + Number of people who drink coffee only + Number of people who drink both tea and coffee + Number of people who drink neither tea nor coffee 2. Number of people who drink tea only = Number of people who drink tea only + Number of people who drink both tea and coffee 3. Number of people who drink coffee only = Number of people who drink coffee only + Number of people who drink both tea and coffee 4. Number of people who drink neither tea nor coffee = Total number of people surveyed - (Number of people who drink tea only + Number of people who drink coffee only + Number of people who drink both tea and coffee) Let's solve these equations to find the number of people who drink tea, coffee, and neither in the morning. Solve the system of equations by following steps: - step0: Solution: \(\left\{ \begin{array}{l}80+20+44+N=216\\80+44=T\\20+44=C\\216-\left(80+20+44\right)=N\end{array}\right.\) - step1: Calculate: \(\left\{ \begin{array}{l}144+N=216\\124=T\\64=C\\72=N\end{array}\right.\) - step2: Calculate: \(\left\{ \begin{array}{l}N=72\\T=124\\C=64\\N=72\end{array}\right.\) - step3: Find the intersection: \(\left\{ \begin{array}{l}C=64\\N=72\\T=124\end{array}\right.\) - step4: Check the solution: \(\left\{ \begin{array}{l}C=64\\N=72\\T=124\end{array}\right.\) - step5: Rewrite: \(\left(C,N,T\right) = \left(64,72,124\right)\) The solution to the system of equations is: - Number of people who drink coffee only: \( C = 64 \) - Number of people who drink neither tea nor coffee: \( N = 72 \) - Number of people who drink tea in the morning: \( T = 124 \) Therefore, 124 people drink tea in the morning, 64 people drink coffee in the morning, and 72 people drink neither tea nor coffee.