The problem presents a survey of 800 students. 245 students take the train. 305 students take the bus. 335 students take neither the train nor the bus. The task is to find: a. The number of students who take both the train and the bus. b. The number of students who take only the train.

1. Problem Description

The problem presents a survey of 800 students.

245 students take the train.

305 students take the bus.

335 students take neither the train nor the bus.

The task is to find:

a. The number of students who take both the train and the bus.

b. The number of students who take only the train.

2. Solution Steps

Let $T$ be the set of students who take the train, and $B$ be the set of students who take the bus.

We are given:

Total number of students = 800

Number of students who take the train, $|T| = 245$

Number of students who take the bus, $|B| = 305$

Number of students who take neither, $|(T \cup B)^c| = 335$

The number of students who take either the train or the bus or both is:

$|T \cup B| = 800 - 335 = 465$

We know that:

$|T \cup B| = |T| + |B| - |T \cap B|$

$465 = 245 + 305 - |T \cap B|$

$465 = 550 - |T \cap B|$

$|T \cap B| = 550 - 465 = 85$

So, the number of students who take both the train and the bus is

8

5.

To find the number of students who take only the train, we subtract the number of students who take both from the number of students who take the train:

$|T \text{ only}| = |T| - |T \cap B|$

$|T \text{ only}| = 245 - 85 = 160$

3. Final Answer

a. The number of commuters who take both the train and the bus is:

$85$

b. The number of commuters who take only the train is:

$160$