A survey asked 275 people what alternative transportation modes they use. The results are
below.
148 walk
154 use the bus
150 ride a bicycle
89 walk and use the bus
69 walk and ride a bicycle
85 ride the bus and ride a bicyle
50 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

UpStudy AI · Accepted Answer

**Step 1. Define the sets and given values**

Let  
$$ W $$ = set of people who walk,  
$$ B $$ = set of people who use the bus,  
$$ R $$ = set of people who ride a bicycle.

The survey gives:  
- $$ |W| = 148 $$  
- $$ |B| = 154 $$  
- $$ |R| = 150 $$  
- $$ |W \cap B| = 89 $$  
- $$ |W \cap R| = 69 $$  
- $$ |B \cap R| = 85 $$  
- $$ |W \cap B \cap R| = 50 $$

**Step 2. Find the number of people who only ride the bus**

The number of people who only ride the bus (and do not walk or ride a bicycle) is given by  
$$
\text{Only } B = |B| - |W \cap B| - |B \cap R| + |W \cap B \cap R|
$$

Substitute the given numbers:  
$$
\text{Only } B = 154 - 89 - 85 + 50
$$

Calculate:  
$$
154 - 89 = 65
$$
$$
65 - 85 = -20
$$
$$
-20 + 50 = 30
$$

Thus, the number of people who only ride the bus is $$30$$.

**Step 3. Find the number of people who don't use any alternate transportation**

To find this, first we calculate the number of people using at least one mode, i.e., the number of people in the union $$ W \cup B \cup R $$ using the inclusion-exclusion principle:

$$
|W \cup B \cup R| = |W| + |B| + |R| - |W \cap B| - |W \cap R| - |B \cap R| + |W \cap B \cap R|
$$

Substitute the values:  
$$
|W \cup B \cup R| = 148 + 154 + 150 - 89 - 69 - 85 + 50
$$

Now, compute step by step:  
$$
148 + 154 + 150 = 452
$$

$$
89 + 69 + 85 = 243
$$

Thus,  
$$
|W \cup B \cup R| = 452 - 243 + 50 = 259
$$

There were 275 people surveyed, so the number of people who don't use any alternate transportation is  
$$
275 - 259 = 16
$$

**Final Answers:**  
- The number of people who only ride the bus is $$30$$.  
- The number of people who don't use any alternate transportation is $$16$$.
- 30 people only ride the bus. - 16 people don't use any alternate transportation.

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