During a hurricane evacuation from the east coast of Georgia, a family traveled 450 miles west. For the first part of the trip, they averaged 25 mph less than the second part, where their average speed was 62mph. If the total time of travel was 9 hours, how many miles did they drive at the reduced speed?

I have tried every which way to figure this out. I thought that I needed to figure out how fast they were going, 49.5 at reduced speed and 74.5 at full speed. Totally lost.

the problem gives the speeds in both parts

... 62 mph in the 2nd part
... and 25 mph less in the 1st part

let t be the time of reduced speed
... 37 t + 62(9 - t) = 450

find the time, then multiply by speed to find the distance

once I find the time, which would be in a xx.xx format, I would calculate the distance at d=rt , correct?

To solve this problem, let's break it down into smaller steps:

Step 1: Assign variables
Let's assign the following variables to the unknown quantities in the problem:
- Let x be the speed at the first part of the trip (at the reduced speed).
- Let y be the speed at the second part of the trip (62 mph).
- Let d be the distance traveled at the reduced speed.

Step 2: Set up equations
We can use the formula: Speed = Distance / Time, where Time = Distance / Speed.

For the first part of the trip:
Time spent = Distance / Speed
So, Time1 = d / x

For the second part of the trip:
Time spent = Distance / Speed
So, Time2 = (450 - d) / y

The total time spent is given as 9 hours, so:
Time1 + Time2 = 9 hours

Step 3: Solve the equations
Now, substitute the given values and equations into each other:

Time1 = d / x
Time2 = (450 - d) / y
Time1 + Time2 = 9 hours

Substitute Time1 and Time2 into the equation for total time:

d / x + (450 - d) / y = 9

Step 4: Simplify the equation
To make the equation easier to work with, let's eliminate the fractions by multiplying every term by xy:

xy * (d / x) + xy * ((450 - d) / y) = 9 * xy
d * y + x * (450 - d) = 9 * xy

Step 5: Expand and rearrange the equation
Now, simplify the equation further:

d * y + x * 450 - x * d = 9 * xy
d * y + 450x - xd = 9xy

Rearrange the terms:

d * y - 9xy = xd - 450x
d * y - 9xy - xd = -450x

Step 6: Factor out the common variables
Factor out the common variables on the left-hand side:

d * (y - 9x) - x * (y - 9x) = -450x
(d - x) * (y - 9x) = -450x

Step 7: Solve for d
Divide both sides of the equation by (y - 9x):

d - x = (-450x) / (y - 9x)
d = (-450x) / (y - 9x) + x

Step 8: Substitute the values and solve for d
Now, substitute the given values back into the equation:

d = (-450 * x) / (y - 9x) + x

Given x = 62 - 25 = 37 mph (25 mph less than the second part), and y = 62 mph, we can calculate d:

d = (-450 * 37) / (62 - (9 * 37)) + 37
d = -16650 / (-235) + 37
d = 70.64 + 37
d ≈ 107.64 miles

Therefore, the family traveled approximately 107.64 miles at the reduced speed.

Note: The value obtained for d is an approximation since we rounded off the speeds in the calculation.