Question

Two athletes practice for a marathon by running back and forth on a 11-mile course. They start running simultaneously, one at a speed 2 miles faster than the other. How fast does each run if they meet 1 hr 6 minutes after starting? (The faster one is already returning at this point.) How far from the starting point do the runners meet?

Answer 1

The slower runner's speed is approximately 2.44 mph and the faster runner's speed is approximately 4.44 mph. The runners meet approximately 4.884 miles from the starting point.

Step-by-step explanation:

Let's assume that the slower runner's speed is x miles per hour. Since the faster runner is 2 miles per hour faster than the slower runner, their speed would be (x + 2) miles per hour.

In 1 hour and 6 minutes, the slower runner would have covered a distance of x miles, while the faster runner would have covered a distance of (x + 2) miles.

Since they meet after 1 hour and 6 minutes and the faster runner is already returning at this point, the distance between them would be double the distance covered by the slower runner:
2x = (x + 2) * (1 + 6/60)

Let's solve the equation to find the value of x:

2x = (x + 2) * (1.1)
2x = 1.1x + 2.2
0.9x = 2.2
x = 2.2/0.9
x = 2.44

So, the slower runner's speed is approximately 2.44 miles per hour, and the faster runner's speed is approximately 2 miles per hour faster than that, which is 4.44 miles per hour.

To find the distance from the starting point where the runners meet, we can substitute the value of x in the equation:
2x = (x + 2) * (1.1)
2 * 2.44 = (2.44 + 2) * (1.1)
4.88 = 4.44 * 1.1
4.88 = 4.884

So, the runners meet approximately 4.884 miles from the starting point.

Answer 2

Faster runner speed: 11 mph

Slower runner speed: 9 mph

Distance from starting point: 9.9 miles

Step-by-step explanation:

Let the slower runner speed be: x

Faster runner speed: x+2

1 hour 6 minutes = 1.1 hours

Slower: 1.1(x) = 1.1x miles

Faster: 1.1(x+2) = 1.1x+2.2 miles

This means that the faster runner had gone 2.2 miles more than the slower runner.

1.1x + 2.2 > 11 miles (This is true as we know the faster runner is coming back from the 11 mile run)

1.1x > 11 - 2.2 miles

1.1x > 8.8 miles

x > 8.8 / 2.2 miles

x > 8 mph

1.1x < 11 miles (we know this is true as the slower runner has to have gone less than 11 miles as the faster runner met him on his way back)

x < 11 / 1.1 miles

x < 10 miles

8 < x < 10

Although x could be any number including decimal between 8 and 10, the obvious first answer would be 9. However we need to check if this answer is correct.

Slower runner distance to the end:

1.1(9) = 9.9 miles

11 - 9.9 = 1.1 miles

Faster runners distance from the end (when coming back)

1.1(9+2) = 1.1(11) = 12.1 miles

12.1 - 11 = 1.1 miles

The speeds 9 mph and 11 mph are now proven to be the right speeds. Using our equation before, we can figure out the distance from the starting point that they meet at. This is just how far the slower runner has gotten. So, 1.1(9) = 9.9 miles from the starting point