A survey of 1800 commuters in New York City showed that 1130 take the subway, 660 take the bus, and 170 do not take either the bus or the subway.
(a) How many commuters take both the bus and the subway?
commuters

(b) How many commuters take only the subway?
commuters

To solve these problems, we can use set theory and the principle of inclusion and exclusion.

Let:
- \( S \) represent the set of commuters who take the subway.
- \( B \) represent the set of commuters who take the bus.
- \( S \cap B \) represent the set of commuters who take both the bus and the subway.

### Given dataTo solve these problems, we can use set theory and the principle of inclusion and exclusion.

Let:
- \( S \) represent the set of commuters who take the subway.
- \( B \) represent the set of commuters who take the bus.
- \( S \cap B \) represent the set of commuters who take both the bus and the subway.

### Given 
- Total number of commuters surveyed: 1800
- Number of commuters who take the subway: \( |S| = 1130 \)
- Number of commuters who take the bus: \( |B| = 660 \)
- Number of commuters who do not take either the bus or the subway: 170

### Step 1: Find the number of commuters who take either the bus or the subway (or both).
If 170 commuters do not take either the bus or the subway, then the number of commuters who take either the bus or the subway (or both) is:
\[
|S \cup B| = 1800 - 170 = 1630
\]

### Step 2: Use the principle of inclusion and exclusion to find the number of commuters who take both the bus and the subway.
The principle of inclusion and exclusion states:
\[
|S \cup B| = |S| + |B| - |S \cap B|
\]
Plugging in the known values:
\[
1630 = 1130 + 660 - |S \cap B|
\]
Solving for \( |S \cap B| \):
\[
|S \cap B| = 1130 + 660 - 1630 = 160
\]
So, **160 commuters take both the bus and the subway.**

### Step 3: Find the number of commuters who take only the subway.
The number of commuters who take only the subway is:
\[
\text{Only Subway} = |S| - |S \cap B| = 1130 - 160 = 970
\]
So, **970 commuters take only the subway.**

### Final Answers:
(a) **160 commuters take both the bus and the subway.**

(b) **970 commuters take only the subway.**

Would you like more details or have any questions?

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Here are 5 related questions:

1. How many commuters take only the bus?
2. What percentage of commuters take both the bus and the subway?
3. If the survey was conducted with 2500 commuters instead of 1800, how would this affect the number of commuters taking both the bus and the subway?
4. How many commuters take the bus or the subway, but not both?
5. If 5% more commuters start taking the subway, how many additional people would that be?

**Tip:** When solving problems involving sets and totals, always remember to check if using a Venn diagram or inclusion-exclusion principle simplifies the calculation.

A survey of 1800 commuters in New York City showed that 1130 take the subway, 660 take the bus, and 170 do not take either the bus or the subway. (a) How many commuters take both the bus and the subway? (b) How many commuters take only the subway?

Explore how to solve a problem involving 1800 commuters in New York City, where 1130 take the subway, 660 take the bus, and 170 do not take either. Learn how to use the inclusion-exclusion principle to determine how many take both the bus and subway, and how many take only the subway. Suitable for Grades 9-12.

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