Julie drives 100 mi to Grandmother's house. On the way to Grandmother's, Julie drives half the distance at 40 mph and half the distance at 60 mph. On her return trip, she drives half the time at 40 mph and half the time at 60 mph. What is Julie's average speed on the way to Grandmother's house?

Verified step by step guidance

Step 1: Understand the concept of average speed. Average speed is defined as the total distance traveled divided by the total time taken. For the trip to Grandmother's house, Julie drives half the distance at 40 mph and the other half at 60 mph. We need to calculate the total time taken for the trip and divide the total distance (100 mi) by this time.

Step 2: Divide the total distance into two equal parts since Julie drives half the distance at each speed. Each part is 50 mi (100 mi ÷ 2).

Step 3: Calculate the time taken to travel each segment. Use the formula for time: \( t = \frac{d}{v} \), where \( d \) is distance and \( v \) is speed. For the first segment (50 mi at 40 mph), \( t_1 = \frac{50}{40} \). For the second segment (50 mi at 60 mph), \( t_2 = \frac{50}{60} \).

Step 4: Add the times for both segments to find the total time taken for the trip to Grandmother's house: \( t_{\text{total}} = t_1 + t_2 \).

Step 5: Calculate the average speed using the formula \( v_{\text{avg}} = \frac{d_{\text{total}}}{t_{\text{total}}} \), where \( d_{\text{total}} \) is the total distance (100 mi) and \( t_{\text{total}} \) is the total time calculated in Step 4.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Average Speed

Average speed is defined as the total distance traveled divided by the total time taken. It provides a measure of how fast an object moves over a given distance, regardless of variations in speed during the journey. In this scenario, calculating average speed requires determining the time taken for each segment of the trip to find the overall average.

Distance and Time Relationship

The relationship between distance, speed, and time is fundamental in physics, expressed by the formula: distance = speed × time. This relationship allows us to calculate one variable if the other two are known. In Julie's case, understanding how to break down her trip into segments based on distance and speed is crucial for finding the total time taken.

Segmented Travel

Segmented travel refers to a journey divided into parts, each with potentially different speeds or conditions. In Julie's trip, she travels half the distance at one speed and the other half at another speed. This concept is important for calculating average speed, as it requires analyzing each segment separately to determine the total time and distance.