Luke and Slim have only one horse between them. Luke rides a certain time and then ties up the horse for Slim, who has been walking. Meanwhile Luke walks on ahead. They proceed in this way, alternately walking and riding. If they walk 4 miles per hour and ride 12 miles per hour, what part of the time is the horse resting?
Consider the 1st hour
Luke goes 12 miles while Luke goes 4
Luke then has to walk 2 more hours to get to the horse.
Now Luke is still 8 miles ahead, and during the next hour, Slim rides 12 miles while Luke walks another 4 miles, and they meet again.
By this time 4 hours have passed, and the horse has rested 2 hours.
so, ...
To find out the part of the time the horse is resting, we can analyze their walking and riding times.
Let's denote the time Luke rides as "x" hours. Since Luke walks on ahead while the horse is tied up for Slim, the time Slim rides will also be "x" hours.
Considering Luke walks at a speed of 4 miles per hour, the distance he covers while walking will be 4x miles. Similarly, Slim also covers a distance of 4x miles while walking.
The total distance covered by both of them during their riding time will be 12x + 12x = 24x miles since they cover 12 miles per hour while riding.
The entire distance covered during their travel is 4x + 4x + 24x = 32x miles.
The total time taken to cover this distance will be (4x / 4) + (4x / 4) + (24x / 12) + (24x / 12) = 2x + 2x + 2x + 2x = 8x hours.
Now, let's find out the resting time of the horse. The horse rests when neither Luke nor Slim is riding it. So, the resting time will be the total time taken to travel minus the riding time.
Resting time = Total time taken to travel - Riding time
Resting time = 8x - (x + x)
Resting time = 8x - 2x
Resting time = 6x
To find the part of the time the horse is resting, we need to divide the resting time by the total time taken to travel.
Part of the time the horse is resting = (Resting time / Total time taken) * 100
Part of the time the horse is resting = (6x / 8x) * 100
Part of the time the horse is resting = 6/8 * 100
Part of the time the horse is resting = 75%
Therefore, the horse is resting for 75% of the total time.