A survey asked 306 people what alternative transportation modes they use. The results are below. 148 walk 165 use the bus 136 ride a bicycle 79 walk and use the bus 59 walk and ride a bicycle 71 ride the bus and ride a bicyle 34 said they use all three modes of pransportation How many people only ride the bus? How many people don't use any alternate transportation? Question Help: Video 1 Video 2 Submit Question

To solve the problem, we can use the principle of inclusion-exclusion and a Venn diagram approach. Let's define the sets: - Let \( W \) be the set of people who walk. - Let \( B \) be the set of people who use the bus. - Let \( C \) be the set of people who ride a bicycle. From the survey, we have the following information: - \( |W| = 148 \) - \( |B| = 165 \) - \( |C| = 136 \) - \( |W \cap B| = 79 \) (people who walk and use the bus) - \( |W \cap C| = 59 \) (people who walk and ride a bicycle) - \( |B \cap C| = 71 \) (people who use the bus and ride a bicycle) - \( |W \cap B \cap C| = 34 \) (people who use all three modes) ### Step 1: Calculate the number of people in each individual category Using the inclusion-exclusion principle, we can find the number of people who use at least one mode of transportation: \[ |W \cup B \cup C| = |W| + |B| + |C| - |W \cap B| - |W \cap C| - |B \cap C| + |W \cap B \cap C| \] Substituting the values: \[ |W \cup B \cup C| = 148 + 165 + 136 - 79 - 59 - 71 + 34 \] Calculating this step-by-step: 1. \( 148 + 165 + 136 = 449 \) 2. \( 79 + 59 + 71 = 209 \) 3. \( 449 - 209 + 34 = 274 \) So, \( |W \cup B \cup C| = 274 \). This means 274 people use at least one mode of transportation. ### Step 2: Calculate the number of people who don't use any alternate transportation To find the number of people who don't use any alternate transportation, we subtract the number of people who use at least one mode from the total number of people surveyed: \[ \text{People who don't use any alternate transportation} = 306 - |W \cup B \cup C| = 306 - 274 = 32 \] ### Step 3: Calculate the number of people who only ride the bus To find the number of people who only ride the bus, we can use the following formula: \[ \text{Only bus} = |B| - (|W \cap B| + |B \cap C| - |W \cap B \cap C|) \] Substituting the values: \[ \text{Only bus} = 165 - (79 + 71 - 34) \] Calculating this step-by-step: 1. \( 79 + 71 = 150 \) 2. \( 150 - 34 = 116 \) 3. \( 165 - 116 = 49 \) So, the number of people who only ride the bus is 49. ### Summary of Results - The number of people who only ride the bus: **49** - The number of people who don't use any alternate transportation: **32**