Question

One hour after Yolanda started walking from X to Y, a distance of 45 miles, Bob started walking along the same road from Y to X. If Yolanda's walking rate was 3 miles per hour and Bob's was 4 miles per hour, how many miles had Bob walked when they met?

• A. 24
• B. 23
• C. 22
• D. 21
• E. 19.5

Hint:

In relative motion problems, carefully note the starting times and directions. A simple approach is to calculate the distance covered by the person who started earlier, subtract it from the total distance, and then solve a simpler problem where they start simultaneously. Here, in the first hour, Yolanda covers 3 miles. The remaining distance is 42 miles. Their combined speed is 3 + 4 = 7 mph. Time to meet = 42/7 = 6 hours. In these 6 hours, Bob walks 6 \(\times\) 4 = 24 miles.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a relative speed problem where two objects are moving towards each other. The key is to account for the head start that one person has. We need to find the point in time and distance where they meet.
Step 2: Key Formula or Approach:
The fundamental formula is Distance = Rate \(\times\) Time.
When two objects move towards each other, their relative speed is the sum of their individual speeds. However, due to the one-hour head start, it's easier to set up an equation where the sum of the distances they travel equals the total distance.
Let \(t\) be the time in hours that Bob has been walking.
Since Yolanda started one hour earlier, she has been walking for \(t + 1\) hours.
Step 3: Detailed Explanation:
Distance covered by Yolanda = Yolanda's Rate \(\times\) Yolanda's Time
\[ D_{Yolanda} = 3 \times (t + 1) \] Distance covered by Bob = Bob's Rate \(\times\) Bob's Time
\[ D_{Bob} = 4 \times t \] They meet when the sum of the distances they have walked equals the total distance between X and Y, which is 45 miles.
\[ D_{Yolanda} + D_{Bob} = 45 \] Substituting the expressions for their distances:
\[ 3(t + 1) + 4t = 45 \] Now, we solve for \(t\):
\[ 3t + 3 + 4t = 45 \] \[ 7t + 3 = 45 \] \[ 7t = 45 - 3 \] \[ 7t = 42 \] \[ t = \frac{42}{7} = 6 \text{ hours} \] This means Bob walked for 6 hours before they met.
The question asks for the number of miles Bob had walked.
\[ D_{Bob} = 4 \times t = 4 \times 6 = 24 \text{ miles} \] Step 4: Final Answer:
Bob had walked 24 miles when they met. This corresponds to option (A).