18. A cyclist bikes x distance at 10 miles per hour and returns over the same path at 8 miles per hour. What is the cyclist's average rate for the round trip in miles per hour?

time for first trip = x/10

time for second trip = x/8
total time = x/10 + x/8

avg rate = total distance/total time
= 2x/(x/10 + x/8)

simplify, your x's should cancel out.

To find the cyclist's average rate for the round trip, we can use the formula for average speed:

Average Speed = Total Distance / Total Time

The total distance for the round trip is twice the distance that the cyclist bikes one way, since they travel the same path twice. So the total distance is 2x.

To find the total time, we need to calculate the time it takes for the cyclist to go one way and then return. Time is equal to distance divided by rate.

Time for one way = x miles / 10 mph = x/10 hours
Time for return trip = x miles / 8 mph = x/8 hours

The total time for the round trip is the sum of the time for one way and the time for the return trip:

Total Time = (x/10 hours) + (x/8 hours)

Now we have the total distance and the total time, so we can calculate the average speed:

Average Speed = Total Distance / Total Time

Average Speed = 2x miles / [(x/10 hours) + (x/8 hours)]

To simplify the expression, we can find a common denominator:

Average Speed = 2x miles / [(8x/80 hours) + (10x/80 hours)]

Average Speed = 2x miles / [(18x/80 hours)]

Average Speed = (2x * 80 hours) / (18x)

Simplifying further:

Average Speed = (160x) / (18x)

Average Speed = 8.89 mph (rounded to two decimal places)

So, the cyclist's average rate for the round trip is 8.89 miles per hour.