Problem 39 - Algebra 2

Question

Paul drove from his house to work at an average speed of 40 miles per hour. The drive took him 15 minutes. If the drive home took him 20 minutes and he used the same route in reverse, what was his average speed going home?

Step-by-Step Solution

Answer

Paul's average speed going home was 30 miles per hour.

Step 1: Convert Time to Hours

First, convert the time Paul took to drive back home from minutes to hours. Since he took 20 minutes, we divide 20 by 60 to convert it to hours: \( \frac{20}{60} = \frac{1}{3} \) hour.

Step 2: Determine Distance

Use the information from his drive to work to determine the distance. Since he drove to work at 40 miles per hour for 15 minutes, or \( \frac{15}{60} = \frac{1}{4} \) hour, the distance to work is computed using the formula \( \, \text{Distance} = \text{Speed} \times \text{Time} \) which gives: \[ \text{Distance} = 40 \times \frac{1}{4} = 10 \, \text{miles} \].

Step 3: Calculate Average Speed Going Home

To determine his average speed going home, use the speed formula \( \, \text{Speed} = \frac{\text{Distance}}{\text{Time}} \). We know the distance is 10 miles and the time back is \( \frac{1}{3} \) hour, so: \[ \text{Speed} = \frac{10}{\frac{1}{3}} = 10 \times 3 = 30 \, \text{miles per hour} \].

Key Concepts

Distance Calculation

Calculating distance is quite straightforward once you know the two essential components: speed and time. This relationship is expressed via the formula:

• Distance = Speed × Time

To find out how far Paul traveled to work, we need to consider his average speed and the time it took. From the original exercise, we know he drove at 40 miles per hour, and it took him 15 minutes to reach work.

However, before plugging these values into the formula, ensure that the time is in hours because the speed is in miles per hour. Convert 15 minutes to hours by dividing by 60, which equals \(\frac{1}{4}\) hour. Plug this into the formula:

• Distance = 40 × \(\frac{1}{4}\) = 10 miles.

This tells us that Paul's house and his workplace are 10 miles apart.

Time Conversion

Time conversion is a critical step when solving problems related to speed and distance. It ensures consistency in units, such as converting minutes to hours when dealing with speed in miles per hour. Knowing how to do these conversions is very useful.

To convert time from minutes to hours, use the relation:

• 1 hour = 60 minutes

Thus, for any given number of minutes, divide by 60 to convert to hours. In Paul's return journey scenario, he took 20 minutes. Converting 20 minutes to hours gives us:

• 20 minutes ÷ 60 = \( \frac{1}{3} \) hour.

Without this conversion, using the speed formula wouldn't provide the correct speed, as speed is measured over hours.

Speed Formula

The speed formula allows you to find out how fast something is moving by using the relationship between distance and time. It's expressed mathematically as:

• Speed = \( \frac{\text{Distance}}{\text{Time}} \)

For Paul's journey back home, we already know that the route was the same as going to work, so the distance is 10 miles. The time it took was converted to \(\frac{1}{3}\) hour.

Using the speed formula, substitute the known values:

• Speed = \(\frac{10}{\frac{1}{3}}\) = 10 × 3 = 30 miles per hour.

This result means going home, Paul's average speed was 30 miles per hour. Understanding how to manipulate the speed formula helps answer numerous questions related to moving speed, whether in travel, exercise, or other applications where speed, distance, and time are key considerations.