Answer:
Let's call Jamie's speed on her way to her parents' house "x" (in mph). Then, according to the problem, her speed on the way back was 11 mph slower than that, or "x - 11" (in mph).
We can use the formula:
time = distance / speed
to create two equations based on the information in the problem.
On the way to her parents' house, Jamie spent a certain amount of time driving (let's call it t1) and covered a distance of 360 miles. So we can write:
t1 = 360 / x
On the way back from her parents' house, Jamie spent a different amount of time driving (let's call it t2) and also covered a distance of 360 miles. But this time, her speed was x - 11. So we can write:
t2 = 360 / (x - 11)
We also know from the problem that Jamie spent a total of 12 hours driving, so we can write:
t1 + t2 = 12
Now we can substitute the expressions we have for t1 and t2 in terms of x into the last equation, and solve for x:
360 / x + 360 / (x - 11) = 12
Multiplying both sides by x(x-11) to eliminate the denominators, we get:
360(x-11) + 360x = 12x(x-11)
Simplifying and rearranging terms, we get a quadratic equation:
12x^2 - 492x + 3960 = 0
We can solve this quadratic equation using the quadratic formula:
x = [492 ± sqrt(492^2 - 4(12)(3960))] / (2*12)
x = [492 ± sqrt(12384)] / 24
x = [492 ± 111.27] / 24
We get two possible values for x:
x = 32.69 or x = 17.31
Since we know that Jamie's speed on the way back was 11 mph slower than her speed on the way there, we can rule out the solution x = 17.31 (which would give a negative speed on the way back). So the only valid solution is:
x = 32.69
This means Jamie's speed on the way to her parents' house was 32.69 mph, and her speed on the way back was 21.69 mph (which is 11 mph slower).
Therefore, the two rates (in mph) are 32.69 and 21.69, respectively.