Question

A car averages 55 mph for the first 4 hours of a trip and averages 70 mph for each additional hour. The average speed for the entire trip was 60 mph. How many hours long is the trip?

• A. 6
• B. 8
• C. 11
• D. 12
• E. 14

Hint:

Remember that average speed is a weighted average of the individual speeds, weighted by the time spent at each speed. You cannot simply average 55 and 70. Problems involving average speed almost always require using the formula: Total Distance / Total Time.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem deals with average speed, which is not simply the average of the speeds. The average speed is defined as the total distance traveled divided by the total time taken. We need to set up an equation based on this definition.
Step 2: Key Formula or Approach:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \] We can also use the relationship: Distance = Speed \( \times \) Time.
Step 3: Detailed Explanation:
Part 1: Define variables and break down the trip
Let 't' be the number of additional hours the car traveled at 70 mph.
The trip has two parts:
Part 1:
- Speed\(_1\) = 55 mph
- Time\(_1\) = 4 hours
- Distance\(_1\) = Speed\(_1\) \( \times \) Time\(_1\) = \( 55 \times 4 = 220 \) miles.
Part 2:
- Speed\(_2\) = 70 mph
- Time\(_2\) = t hours
- Distance\(_2\) = Speed\(_2\) \( \times \) Time\(_2\) = \( 70 \times t = 70t \) miles.
Part 2: Set up the equation for the entire trip
- Total Time = Time\(_1\) + Time\(_2\) = \( 4 + t \) hours.
- Total Distance = Distance\(_1\) + Distance\(_2\) = \( 220 + 70t \) miles.
- Average Speed for the entire trip is given as 60 mph.
Using the average speed formula:
\[ 60 = \frac{220 + 70t}{4 + t} \] Part 3: Solve the equation for t
Multiply both sides by (4 + t) to eliminate the denominator.
\[ 60(4 + t) = 220 + 70t \] Distribute the 60 on the left side.
\[ 240 + 60t = 220 + 70t \] Now, isolate the variable 't'. Subtract 60t from both sides.
\[ 240 = 220 + 10t \] Subtract 220 from both sides.
\[ 20 = 10t \] Divide by 10.
\[ t = 2 \] So, the additional time traveled was 2 hours. Part 4: Answer the specific question
The question asks for the total length of the trip in hours.
\[ \text{Total Time} = 4 + t = 4 + 2 = 6 \text{ hours} \] Step 4: Final Answer
The entire trip was 6 hours long.