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?
It seems to me that you need the average of 10 and 8.
18/2 = 9 mph
No, it's not that simple
Time for first trip = x/10
time for second trip =x/8
distance of total trip = 2x
rate = total distance/total time
=2x/(x/10 +x/8)
=..
=..
=8.88 mph
To find the average rate for the round trip, we can use the formula:
Average Speed = Total Distance / Total Time.
Let's assume the distance of the round trip is 'D' miles.
The time it takes to travel one way at a speed of 10 miles per hour is given by:
Time = Distance / Speed = D / 10.
The time it takes to travel back the same distance at a speed of 8 miles per hour is given by:
Time = Distance / Speed = D / 8.
The total time taken for the round trip is the sum of the time taken to go and the time taken to return:
Total Time = D/10 + D/8.
Now, we can substitute this in the formula for average speed:
Average Speed = Total Distance / Total Time = D / (D/10 + D/8).
Simplifying, we can find the LCD (Least Common Denominator) of 10 and 8, which is 40:
Average Speed = D / ((8D + 10D) / 40) = D / (18D / 40).
Now, we can simplify further by multiplying the numerator by the reciprocal of the denominator:
Average Speed = D * (40 / 18D) = 40 / 18 miles per hour.
Simplifying this fraction, we get:
Average Speed = 20 / 9 miles per hour.
Therefore, the cyclist's average rate for the round trip is approximately 2.22 miles per hour.
To find the cyclist's average rate for the round trip, we can use the formula:
Average speed = Total distance / Total time
In this case, the cyclist bikes a certain distance at 10 miles per hour and returns over the same path at 8 miles per hour. Since the distance is the same for both directions, we can say the total distance is 2 times the distance of one leg of the trip.
Let's assume the distance of one leg of the trip is 'd' miles.
For the first leg of the trip:
Time = Distance / Speed = d / 10
For the return leg of the trip:
Time = Distance / Speed = d / 8
The total time for the round trip is the sum of these two times:
Total Time = d / 10 + d / 8
Now, let's substitute the value of the total distance (2d) and total time into the formula for average speed:
Average Speed = Total distance / Total time
= 2d / (d/10 + d/8)
To simplify this expression, we can find a common denominator for the fractions on the right side of the equation:
Average Speed = 2d / ((8d + 10d) / (8 * 10))
= 2d / (18d / 80)
= (2d * 80) / 18d
= 160d / 18d
Now, we can cancel out 'd' from the numerator and denominator:
Average Speed = 160 / 18
= 8.89 miles per hour (rounded to two decimal places)
Therefore, the cyclist's average rate for the round trip is approximately 8.89 miles per hour.