Dag rode down a hill

[sanju09]

For the first part of his bike trip, Dag rode down a hill at x miles per hour for t hours. For the rest of his trip, Dag rode up a hill at half that speed for twice as long. What was Dag's average speed, in miles per hour, for his entire trip?

A. ½ x
B. (2/3) x
C. ¾ x
D. (5/6) x
E. x

[truplayer256]

For the first part of his bike trip, Dag went a total of x*t miles and for the second part of his bike trip, Dag went x/2*2t or x*t miles. So, now we know are total distance: xt+xt=2xt. The total time that Dag took to finish his entire trip can be found by adding t and 2t, which comes out to be 3t.

2xt/3t=2x/3 B

[quocbao]

IMO: The answer is 2x / 3

[marcusking]

For the first part of his bike trip, Dag rode down a hill at x miles per hour for t hours. For the rest of his trip, Dag rode up a hill at half that speed for twice as long. What was Dag's average speed, in miles per hour, for his entire trip?

A. ½ x
B. (2/3) x
C. ¾ x
D. (5/6) x
E. x

Plug in easy number rode down the hill at 60mph for 1 hour
That means he rode up a hill at 30mph for 2 hours

He traveled 120 miles in 3 hours for an average speed of 40 mph.

40/60 = 2/3

B.

Please start posting the OA.

[sanju09]

For the first part of his trip, Dag rode for t hours at x miles per hour, so the distance he rode on this part of the trip was x t miles. For the rest of his trip, he rode at half the speed of the first part of the trip, and for twice as long. Therefore, he rode at a speed of x/2 miles per hour for 2 t hours, and thus he covered a distance of (x/2) (2 t) = x t miles in the rest of his trip. Dag's average speed, in miles per hour, for the entire trip is equal to the total distance of his trip divided by the total time of the trip. The total distance of his trip was x t + x t = 2 x t miles, and the total time of his trip was t + 2 t = 3 t hours. Therefore, Dag's average speed for the entire trip was [spoiler](2 x t)/(3 t) = (2/3) x[/spoiler] miles per hour, which is choice [spoiler](B)[/spoiler].