When Amy drives her car to work, the trip takes 1/2 hour. When she rides the bus, it takes 3/4 hour. The average speed of the bus is 12 mph less than her speed driving. Find the distance she travels to work.

Let D be the distance to work.

D=Drivespeed*1/2

D=Busspeed*.75

so Bus*.75=Drivespeed*1/2
but bus=drivspeed-12

drivspeed*1/2=(.75)(drivspeed-12)
solve for drivespeed, then
distance=drivspeed*1/2

To find the distance Amy travels to work, we need to use the formula:

Distance = Speed × Time

Let's start by finding the speed at which Amy drives. We are told that the trip takes her 1/2 hour, so the formula becomes:

Distance = Speed × (1/2)

Next, let's find the speed of the bus. We are told that when Amy takes the bus, it takes her 3/4 hour. We are also told that the average speed of the bus is 12 mph less than her speed driving. So, the formula becomes:

Distance = (Speed - 12) × (3/4)

Since we want to find the distance Amy travels to work, we can set these two formulas equal to each other:

Speed × (1/2) = (Speed - 12) × (3/4)

Now let's solve for the speed at which Amy drives:

(1/2)Speed = (3/4)(Speed - 12)

To simplify this equation, let's multiply both sides by 4 to get rid of the fractions:

2Speed = 3(Speed - 12)

Next, we distribute the 3 on the right side:

2Speed = 3Speed - 36

Now let's isolate the variable Speed by subtracting 3Speed from both sides:

- Speed = - 36

Finally, divide both sides by -1 to get the value of Speed:

Speed = 36 mph

Now that we know that Amy drives at a speed of 36 mph, we can find the distance she travels to work by substituting this speed back into either of the distance formulas. Let's use the first formula with the driving time of 1/2 hour:

Distance = 36 mph × (1/2)

Distance = 18 miles

Therefore, Amy travels a distance of 18 miles to work.