Let's analyze the problem step by step.

The survey says:

• Total commuters: 1,700
• People who take the subway: 1,150
• People who take the bus: 640
• People who do not take either: 130

We are tasked with finding:

1. How many commuters take both the bus and the subway?
2. How many commuters take only the subway?

We can solve this using the principle of set theory, particularly with two overlapping sets (those who take the bus and those who take the subway). Let:

• $S$ represent the commuters who take the subway.
• $B$ represent the commuters who take the bus.
• $x$ represent the commuters who take both the bus and the subway.

Step 1: Set up the total equation

The total number of commuters is: $S + B - x + \text{commuters who take neither} = 1,700$ Substituting the known values: $1,150 + 640 - x + 130 = 1,700$ Simplifying this equation: $1,920 - x + 130 = 1,700$ $2,050 - x = 1,700$ $x = 2,050 - 1,700 = 350$

Thus, 350 commuters take both the bus and the subway.

Step 2: Find how many commuters take only the subway

To find how many take only the subway, subtract the commuters who take both from those who take the subway: $1,150 - 350 = 800$

Thus, 800 commuters take only the subway.

Final answers:

(a) 350 commuters take both the bus and the subway. (b) 800 commuters take only the subway.

Would you like more details or have any questions?

Related questions:

1. How many commuters take only the bus?
2. How would the solution change if the number of commuters taking neither increased?
3. Can this problem be solved using a Venn diagram?
4. How do you calculate the number of commuters who take either the bus or the subway but not both?
5. What if the total number of commuters was different, say 2,000?

Tip:

Always verify your answers by substituting them back into the original problem to ensure the total counts are consistent.