Rory left home and drove straight to the airport at an average speed of 45 miles per hour. He returned home along the same route, but traffic slowed him down and he only averaged 30 miles per hour on the return trip. If his total travel time was 2 hours and 30 minutes, how far is it, in miles, from Rory's house to the airport?

Answer

To solve the problem, we need to determine the distance from Rory's house to the airport. Let's break it down step by step.

### Step 1: Define the Variables
Let the distance from Rory's house to the airport be \( d \) miles.

### Step 2: Calculate Time for Each Trip
1. For the trip to the airport at an average speed of 45 miles per hour, the time taken \( t_1 \) can be calculated using the formula:
\[ t_1 = \frac{d}{45} \]

2. For the return trip home at an average speed of 30 miles per hour, the time taken \( t_2 \) can be calculated as:
\[ t_2 = \frac{d}{30} \]

### Step 3: Total Travel Time
The total travel time for both trips is given as 2 hours and 30 minutes. We convert this into hours:
\[ 2 \text{ hours} + 30 \text{ minutes} = 2.5 \text{ hours} = \frac{5}{2} \text{ hours} \]

### Step 4: Set Up the Equation
We can write an equation based on the total time:
\[ t_1 + t_2 = \frac{5}{2} \]
Substituting the expressions for \( t_1 \) and \( t_2 \):
\[ \frac{d}{45} + \frac{d}{30} = \frac{5}{2} \]

### Step 5: Find a Common Denominator
The least common multiple of 45 and 30 is 90. We can multiply the entire equation by 90 to eliminate the fractions:
\[ 90 \left(\frac{d}{45}\right) + 90 \left(\frac{d}{30}\right) = 90 \left(\frac{5}{2}\right) \]
This simplifies to:
\[ 2d + 3d = 225 \]

### Step 6: Combine Like Terms
Combining the terms on the left side gives:
\[ 5d = 225 \]

### Step 7: Solve for \( d \)
Now, divide both sides by 5 to find \( d \):
\[ d = \frac{225}{5} = 45 \]

### Final Answer
The distance from Rory's house to the airport is \( \boxed{45} \) miles.

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