The formula d = rt is called the distance formula. In this formula, drepresents the distance traveled, r represents the rate of speed, and t represents the time traveled. If a person travels 90 miles in `2 1/2` hours, what is the value of r?

Options

A. 36 mph
B. 45 mph
C. 72 mph
D. 180 mph

Answer

To find the value of \( r \) (the rate of speed) using the distance formula \( d = rt \), we can follow these steps:

### Step 1: Identify the given values
We know from the problem:
- Distance (\( d \)) = 90 miles
- Time (\( t \)) = \( 2 \frac{1}{2} \) hours

### Step 2: Convert the time into an improper fraction
The time \( t = 2 \frac{1}{2} \) can be converted into an improper fraction:
\[ t = 2 \times 2 + 1 = 4 + 1 = 5 \quad \text{(the numerator)} \]
So,
\[ t = \frac{5}{2} \text{ hours} \]

### Step 3: Substitute the values into the distance formula
Using the distance formula \( d = rt \):
\[ 90 = r \times \frac{5}{2} \]

### Step 4: Solve for \( r \)
To isolate \( r \), we can rearrange the equation:
\[ r = \frac{90}{\frac{5}{2}} \]
This can be simplified by multiplying by the reciprocal:
\[ r = 90 \times \frac{2}{5} \]

### Step 5: Calculate the value of \( r \)
Now, we perform the multiplication:
\[ r = \frac{90 \times 2}{5} = \frac{180}{5} = 36 \]

### Step 6: State the final answer
Thus, the value of \( r \) is:
\[ r = 36 \text{ miles per hour} \]

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