A cyclist and a jogger start from a town at the same time and head for a destination 6 mi away. The rate of the cyclist is twice the rate of the jogger. The cyclist arrives 3 hr ahead of the jogger. find the rate of the cyclist.

This is what i got so far but its not adding up.
r t d
cyclist 2x*3=6
jogger x*-3=6

It always helps to define the variables used, with the units, and cite the equations you use.

Distance = speed * time, or
time = distance/speed
Let
x=speed of cyclist, m.p.h.
x/2=speed of jogger, m.p.h.
D=distance, 6 mi.

"The cyclist arrives 3 hr ahead of the jogger. find the rate of the cyclist. "
means that the difference of the time taken by each is 3 hours.
So we write the equation:
(D/(x/2) - D/x = 3 hours
On simplifying,
D/x = 3 hours
x=D/3=2 mph (cyclist)
x/2=2/1=1 mph (jogger)

Check:
Time for cyclist: 6/2= 3 hours
Time for jogger: 6/1=6 hour.
OK

To find the rate of the cyclist, we can create two equations based on the given information.

Let x be the rate of the jogger (in mi/hr).
The rate of the cyclist is twice that, so the rate of the cyclist is 2x (in mi/hr).

For the cyclist:
Distance = Rate × Time
6 = (2x) × (3 + t) (since the cyclist arrives 3 hours ahead of the jogger)

For the jogger:
Distance = Rate × Time
6 = x × t

Now we can solve these equations to find the value of x, which is the rate of the jogger.

To solve this problem, let's start by setting up the equation for the cyclist and the jogger:

Let "x" represent the rate of the jogger.
Since the rate of the cyclist is twice the rate of the jogger, the rate of the cyclist would be 2x.

Now, let's calculate the time it takes for both the cyclist and the jogger to reach the destination:

For the cyclist:
Distance = Rate * Time
6 mi = (2x) * (t - 3) -> The cyclist arrives 3 hours earlier than the jogger, so we subtract 3 from the time (t - 3).

For the jogger:
Distance = Rate * Time
6 mi = x * t

Now we have two equations:
1) 6 = (2x) * (t - 3)
2) 6 = x * t

We can solve this system of equations to find the values of x and t.

From equation 2), we can isolate t:
t = 6 / x

Substitute this value of t in equation 1):
6 = (2x) * (6 / x - 3)

To simplify further, we can cancel out the x term:
6 = 2(6 - 3x)

Simplify the equation:
6 = 12 - 6x

Now isolate the x term by subtracting 12 from both sides:
-6 = -6x

Divide both sides by -6:
1 = x

So, the rate of the jogger (x) is 1.

To find the rate of the cyclist, substitute the value of x into the equation t = 6 / x:
t = 6 / 1
t = 6

The time it takes for the jogger to reach the destination is 6 hours. Since the cyclist arrives 3 hours earlier, the time it takes for the cyclist is (6 - 3) = 3 hours.

Therefore, the rate of the cyclist is:
Rate = Distance / Time = 6 mi / 3 hr = 2 mi/hr

So, the rate of the cyclist is 2 miles per hour.