Question

For the first 60 miles of a 120-mile trip, a person drive 50 mph. What speed would he have to travel for the last half of the trip so that the average speed for the entire trip would be 61 mph. The answer is supposed to be 78 8/39 miles. How can this be?

Answer 1

Here's my logic:

61 = 50*x
2

Cross multiply:

61*2 = 50*x

122 = 50 x – subtract 50 from 122 and I get 72 = x

but this isn't correct.

Answer 2

I would not use logic. I would use the definition of average speed
total distance/total time = average speed

to use that equation we need some numbers.
time spent on the first half of the trip:

rate * time = distance
50*x = 60
x= 60/50= 1.2 hours

he spend 1.2 hours going 60 miles. next find the time to go the next 60 miles, using the first equation
\[ \frac{ total \ distance}{total \ time}= average \ speed\\
\frac{120}{x+1.2}= 61
\]

Answer 3

solve for x to get the time it takes to travel the last 60 miles
61x+ 73.2= 120
61x= 46.8
x= 46.8/61
that is the time it takes to go the second 60 miles
it will have a speed of 60 miles/x
\[ speed = 60 \cdot \frac{61}{46.8}= \frac{3660}{46.8}= 78 \ \frac{8}{39}\ mph \]

Answer 4

@phi Thank you! I didn't think of it that way.