Ashley can ride her bike to work in 2/3 of an hour. When she drives to work it takes 1/3 of an hour. If she drives 12 miles per hour faster than she rides her bike, how fast does she ride her bike?

To find out how fast Ashley rides her bike, we first need to determine the distance to her workplace.

Let's assume that the distance to her workplace is 'd' miles.

We know that the time it takes for Ashley to ride her bike to work is 2/3 of an hour, so the equation for the bike ride would be:

Time = Distance / Speed
2/3 = d / Bike Speed

Similarly, when she drives to work it takes 1/3 of an hour, so the equation for the car ride would be:

Time = Distance / Speed
1/3 = d / (Bike Speed + 12)

Since we want to find out Ashley's biking speed, we can set up a system of equations:

2/3 = d / Bike Speed
1/3 = d / (Bike Speed + 12)

To solve this system, we can use the method of substitution. Rearranging the first equation, we get:

d = (2/3) * Bike Speed

Now we substitute this value of 'd' into the second equation:

1/3 = [(2/3) * Bike Speed] / (Bike Speed + 12)

To simplify the equation, we can cross-multiply:

1 * (Bike Speed + 12) = (2/3) * Bike Speed

Expand the equation:

Bike Speed + 12 = (2/3) * Bike Speed

Multiply both sides of the equation by 3 to eliminate the fraction:

3 * (Bike Speed + 12) = 2 * Bike Speed

3 * Bike Speed + 36 = 2 * Bike Speed

Subtract 2 * Bike Speed from both sides:

Bike Speed + 36 = 0

Subtract 36 from both sides:

Bike Speed = -36

Since a negative speed doesn't make sense in this context, we can conclude that there is no valid solution to this problem. Please double-check the information provided or ensure that there are no errors in the question.