Question 231453: Amy drove the 200 miles to New Orleans at an average speed 10 miles per hour faster than her usual average speed. If she completed the trip in 1 hour less than usual, what is her usual driving speed in miles per hour.
I know the answer is 40 miles per hour, but how did they come up with this? I think there is numbers missing in this problem.
Thank you,
Katie
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! distance = 200 miles
Her usual speed is X miles per hour since we don't know it.
Her usual time is T hours because we don't know that either.
Problem states that she traveled 10 miles per hour faster than her usual speed.
This makes the speed she traveled at equal to X + 10.
Problem states that she took 1 hour less.
This makes the time she took equal to T - 1.
She normally travels at X miles per hours and takes T hours.
Since rate * time = distance, we get:
X * T = 200
This time she traveled 10 miles per hour faster and took 1 hour less.
This time, we get:
(X+10) * (T-1) = 200
Since they are both equal to 200, then they are both equal to each other, so we get:
X*T = (X+10)*(T-1)
We simplify this by multiplying out all terms to get:
X*T = X*T - X + 10T - 10
We have one equation in two unknowns, but we can substitute for one of the unknowns and solve for the other.
We'll take the equation X*T = 200 and solve for T to get T = 200/X
We'll then replace T in our equation of X*T = X*T - X + 10T - 10 to get:
X*200/X = X*200/X - X + 10*200/X - 10.
We'll simplify this equation to get:
200 = 200 - X + 2000/X - 10.
We'll subtract 200 and we'll add X and we'll subtract 2000/X and we'll add 10 to both sides of this equation to get:
X - 2000/X + 10 = 0
We'll multiply both sides of this equation by X to get:
X^2 + 10X - 2000 = 0
This factors out to:
(X-40) * (X+50) = 0
X can be either 40 or -50.
since X can't be negative, we are left with X = 40.
Since X*T = 200, We get T = 5
Our answer is:
X = 40
T = 5
40*5 = 200 which is the normal time at the normal rate.
50*4 = 200 which is 1 hour less than the normal time at 10 miles per hour more than the normal rate.
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