two joggers used a 2 km circular road for their training. if they start together from the same place and jog in opposite directions, they meet after 15 minutes. However, if they jog in the same direction at the end of one hour and 15 minutes, the faster jogger would have gone 1 1/4 laps farther. How fast did each jog?
rate of faster --- x km/h
rate of slower --- y km/h
going in opposite direction:
distance covered by slower = (1/4)y
distance covered by faster = (1/4)x
(1/4)x + (1/4)y = 2
x + y = 8
going in same direction:
distance covered by faster = (5/4)x + 1/2 , (1/4 lap is 1/2 km)
distance covered by slower = (5/4)y
(5/4)x + 1/2 = (5/4)y
5x + 2 = 5y
5x - 5y = -2
multiply the first by 5
5x+5y = 40
5x - 5y = -2
10x = 38
x = 3.8
then y = 4.2
check:
opposite directions:
the faster went (1/4)(4.2) = 1.05 km
the slower went (1/4)(3.8) = .95
which adds up to 2 km, or 1 lap
same direction:
faster went (5/4)(4.2) = 5.25
slower went (5/4)(3.8) = 4.75
5.25 - 4.75 = .5 km, or 1/4 of a lap
all is good
To solve this problem, we can use the concept of relative speeds. Let's assume the speed of the slower jogger is "x" km/h and the speed of the faster jogger is "y" km/h.
When they jog in opposite directions, their speeds add up. Since they meet after 15 minutes (which is 1/4 hour), the total distance they cover is equal to the sum of their speeds multiplied by the time:
2 km = (x + y) * (1/4)
Next, let's consider the scenario where they jog in the same direction. In this case, their speeds will subtract because the faster jogger gains on the slower one. Given that they meet after 1 hour and 15 minutes (which is 5/4 hours), the distance covered by the faster jogger will be 1 1/4 laps or 5/4 laps more than the distance covered by the slower jogger:
2 km + (5/4) laps = (y - x) * (5/4)
Now, we have a system of equations:
1) 2 km = (x + y) * (1/4)
2) 2 km + (5/4) laps = (y - x) * (5/4)
To simplify the equations, we can multiply both sides of each equation by 4:
1) 8 km = (x + y)
2) 8 km + 5 laps = 5(y - x)
Now, let's solve the system of equations. We can rearrange equation 2 to isolate y - x:
5(y - x) = 8 km + 5 laps
5y - 5x = 8 km + 5 laps
5y - 5x = 8 km + 5(2 km)
5y - 5x = 18 km
y - x = 3.6 km
From equation 1, we know that x + y = 8 km. Solving these two equations simultaneously will give us the values of x and y:
x + y = 8
y - x = 3.6
Adding the two equations, we get:
2y = 11.6
y = 5.8 km/h
Substituting the value of y back into the equation x + y = 8, we can find x:
x + 5.8 = 8
x = 2.2 km/h
Therefore, the slower jogger is running at a speed of 2.2 km/h, and the faster jogger is running at a speed of 5.8 km/h.