Question

If a man runs 4 miles per hour (mph) for 3 hours but tires and so continues by walking at 2.5 mph, how long will it take him to cover 27 miles? A. 7 hours 45 minutes B. 8 hours C. 9 hours D. 9 hours 15 minutes E. 9 hours and 45 minutes

Short answer

C. 9 hours

Step-by-step solution

Calculate the distance covered while running

The man runs at a speed of 4 mph for 3 hours. Use the formula for distance: \[ \text{Distance} = \text{Speed} \times \text{Time} \]So, the distance covered while running is \[ 4 \text{ mph} \times 3 \text{ hours} = 12 \text{ miles} \]

Determine the remaining distance to be covered

The total distance to be covered is 27 miles. Subtract the distance covered while running from the total distance: \[ 27 \text{ miles} - 12 \text{ miles} = 15 \text{ miles} \]

Calculate the time to cover the remaining distance by walking

The remaining distance of 15 miles is covered by walking at a speed of 2.5 mph. Use the formula for time: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]So, the time to cover the remaining distance is \[ \frac{15 \text{ miles}}{2.5 \text{ mph}} = 6 \text{ hours} \]

Add the total time taken

Add the time spent running and walking: \[3 \text{ hours} + 6 \text{ hours} = 9 \text{ hours} \]Therefore, the total time taken to cover 27 miles is 9 hours.

Key concepts

Distance-Speed-Time Relationship

To solve this problem, we use the relationship between distance, speed, and time. This is a key concept for many GMAT word problems and is expressed with the formulas:
\[ \text{Distance} = \text{Speed} \times \text{Time} \] and \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
In our exercise, the man runs first and then walks. We calculate the distance covered during the running phase by multiplying his running speed (4 mph) by the time he spends running (3 hours). This gives us 12 miles.

With this calculation, we determine that he has covered 12 miles out of the 27-mile distance. The remaining distance (27 miles - 12 miles) is 15 miles, which he covers by walking, using the formula \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
This methodology allows us to effectively break down and understand any distance-speed-time problem.

Multiple-Step Problem-Solving

This problem requires solving in multiple steps. First, we calculate the distance covered while running. Then, determine the remaining distance and finally, compute the time needed to walk the rest. Breaking down a problem into smaller, manageable steps ensures a clear and organized approach.

Here is a step-by-step breakdown:

• Compute distance covered by running using \[ \text{Distance} = \text{Speed} \times \text{Time} \] (4 mph \times 3 hours = 12 miles).
• Subtract this distance from the total (27 miles - 12 miles = 15 miles).
• Calculate time required to walk the remaining distance using \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] (15 miles / 2.5 mph = 6 hours).
• Add the running time (3 hours) and walking time (6 hours) to get the total (9 hours).

Following these logical steps can help tackle even more complex word problems.

Mathematical Reasoning

Mathematical reasoning is essential for solving GMAT word problems. It involves understanding the problem, identifying the mathematical concepts required, and applying them in a structured way.

In this exercise, we demonstrate mathematical reasoning by:

• Recognizing that we need to use the distance-speed-time relationships.
• Understanding that the problem can be divided into two parts: running and walking.
• Using basic arithmetic to subtract distances and divide to find the time.
• Adding times together to find the total duration.

This approach ensures a thorough and clear understanding of the problem, making it easier to find the correct solution. Logical thinking, combined with these key mathematical principles, guides us toward solving such problems effectively.