A car goes uphill for 1 mile at 20mph and then downhill for 1 mile at 30mph. What is the average speed of the car? (It is NOT 25mph!)

Question

Remember that average speed is total distance traveled divided by total time. The total distance is clearly 2 miles. So you need to figure out the total time.

Answer 1

The average speed is total distance divided by total time. The total distance is 2 miles.

The uphill time is distance/speed = 1/20 hour.

The downhill time is distance/speed= 1/30 hour

So the total time is 1/20 + 1/30 = 50/600 hour

So the average speed is 2/(50/600)= (2/1)/(50/600) = 1200/50 = 24mph

Thinking that the average speed is (20 + 30)/2 = 25mph is a very common error. Focusing on the definition of the average speed avoids the error.

Note that if the car goes uphill at 20mph for one HOUR, and then downhill at 30mph for one HOUR, then the average speed IS 25mph, and now you can probably do the calculation which shows this result.

Answer 2

v_avg = distance_total/time_total

distance_total = 1mi + 1mi = 2mi

v_1 = d_1/t_1

t_1 = d_1/v_1 = 1mi/20mph = (1/20)hr

Same logic for t_2

t_2 = d_2/v_2 = 1mi/30mph = (1/30)hr

t_t = t_1 + t_2 = (1/20 + 1/30)hr = (5/60)hr = (1/12)hr

v_avg = (2mi)/((1/12)hr) = 24(mi/hr) = 24mph

It makes sense that it should be less than 25mph. If it spent an equal amount of time at 20mph and 30mph, then the average would be 25mph. But it went an equal distance at each speed. That means it spent more time at the slower speed. Since it spent more time at the slower speed, the average speed should be less than the average of the two values.