SOLUTION: A family drove to a beach resort at an average speed of 55 MPH and later returned over a road at an average speed of 65 MPH. Find the distance to the resort if the total driving ti

Question 71389: A family drove to a beach resort at an average speed of 55 MPH and later returned over a road at an average speed of 65 MPH. Find the distance to the resort if the total driving time was 12 hours.
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
We'll use the equation that says Distance equals Rate times Time and in equation form this
is:
.
D = R*T
.
On their way to the beach the family drives at a rate of 55 mph. So the equation becomes
.
D = 55*T
.
On the way back to the starting point the family drives at a rate of 65 mph. So the equation
for the return trip is:
.
D = 65*t
.
I used a small t for the time it takes to get home. Driving faster means that it should
take less time on the return trip. And this shows that the time on the return trip
is different from the time spent going to the beach.
.
The trip back to the start is the same mileage as the trip going, so the two distances
are equal. Since the two distances are equal, the right sides of these two equations
must also be equal. Set them equal and you get:
.
55*T = 65*t
.
Now we can make use of the fact that the total driving time for the trip is 12 hours.
So the time going plus the time coming home is 12 hours. In equation form this is:
.
T + t = 12
.
We can now subtract t from both sides of this equation to get that:
.
T = 12 - t
.
So we can substitute 12 - t for T in the equation. When we do it becomes:
.
55*(12 - t) = 65*t
.
Multiply out the left side to get:
.
660 - 55t = 65t
.
Add 55t to both sides of the equation and you get:
.
660 = 120t
.
Solve by dividing both sides by 120 and you find that the time t for the return trip is
5.5 hours (5 hours and 30 minutes).
.
Recall that we said Distance = Rate*Time. The rate on the way home was 65 mph and the
time driven was 5.5 hours. Multiply these two and you find that the distance from the
beach to the point of starting out was 357.5 miles. (Note that the trip to the beach was
the same distance, so its distance also has to be 357.5 miles.)
.
Since the total driving time on the round trip was 12 hours and 5.5 hours was the time
it took to get back, we know that the trip to the beach had to take 6.5 hours. Since it
was driven at 55 mph, the distance going to the beach equals the rate times the time T or:
.
55 mph * 6.5 hours = 357.5 miles.
.
This checks and verifies that the distance between the starting point and the beach was
357.5 miles.
.
Hope this helps you to see your way through the problem and gives you some insight
into doing time, rate, and distance problems.
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