SOLUTION: julie recently drove to visit her parents who live 240 miles away. on her way there her average speed was 11 miles per hour faster than on her way home (she ran into some bad weath

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Question 1135051: julie recently drove to visit her parents who live 240 miles away. on her way there her average speed was 11 miles per hour faster than on her way home (she ran into some bad weather). if julie spent a total of 8 hours driving, find the two rates

Found 3 solutions by Theo, ikleyn, MathTherapy:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
rate * time = distance.

on her way there, she drove 11 miles per hour faster than on her way home.

the total time spent driving was 8 hours.

let x = her rate of speed going.

let y = the time it took her to get there.

the formula for going to her parents home is x * y = 240

since she traveled 11 miles per hour faster going, then her rate of speed coming back has to be x - 11 because x miles per hour is 11 miles per hour faster than x - 11 miles per hour.

coming back her time was 8 - y.

the formula for coming back from her parent's home is (x - 11) * (8 - y) = 240

you have two equations that need to be solved simultaneously.

they are:

x * y = 240 and (x - 11) * (8 - y) = 240

since they are both equal to 240, you can set x * y equal to (x - 11) * (8 - y) to get:

x * y = (x - 11) * (8 - y)

multiply the factors in the right side of this equation to get:

xy = 8x - xy - 88 + 11y

in the first equation of xy = 240, solve for y to get y = 240 / x

replace y with 240 / x in the second equation to get:

x * 240/x = 8x - x * 240/x - 88 + 11 * 240 / x

simplify to get:

240 = 8x - 240 - 88 + 2640 / x

multiply both sides of this equation by x to get

240x = 8x^2 - 240x - 88x + 2640

subtract 240x from both sides of this equation to get:

0 = 8x^2 - 240x - 88x + 2640 - 240x

combine like terms to get:

0 = 8x^2 - 568x + 2640

divide both sides of this equation by 8 to get:

0 = x^2 - 71x + 330

factor this quadratic equation to get:

x = 5 or x = 66

in the first equation of x * y = 240, replace x with 5 and solve for y to get y = 240 / 5 = 48

still in the first equation of x * y = 240, replace x with 66 and solve for y to get y = 3.636

since the total time is 8 hours, y can't be 48, so y must be equal to 3.636363636 hours which forces x to be equal to 66.

x is the speed, so x is 66 miles per hour and it took 3.636363636 hours to get to her parent's house.

coming back she traveled 66 - 11 miles per hour and took 8 - 3.636363636 hours.

that means she traveled 55 miles per hour for 4.363636364 hours.

55 miles per hour for 4.363636364 hours equals 240 miles.

the solution looks good.

looks messy but it's right as far as i can tell.

your solution is that she traveled at 66 miles per hour to get to her parent's home and she traveled at 55 miles per hour coming back.

Answer by ikleyn(52802)   (Show Source):
You can put this solution on YOUR website!
.

Let x be the rate on the way to "there", in miles per hour. Then the rate on the way back was (x-11) mph, according to the condition. The time to travel "to there" was hours. The time to travel back was . The total time is the sum of these partial times, which gives you the "time" equation + = 8. To solve the equation, first divide both sides by 8; then multiply both sides by x*(x-11). You will get 30*(x-11) + 30*x = x*(x-11). Simplify it step by step and solve 30x - 330 + 30x = x^2 - 11x x^2 - 71x + 330 = 0 Factor left side {x+5)*(x-66) = 0 Only positive root x= 66 is meaningful, so the answer to the problem is THIS : The rate was 66 mph on the way to parents and 66-11 = 55 pmh on the way back. CHECK. Time to travel to "there" was = hours. Time to travel back was = hours. Total time was + = = 8 hours. ! Correct !

Solved.

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Using "time" equation is the STANDARD way to solve such problems.

The lesson to learn from my post is THREEFOLD:

a) Learn on how to write the "time" equation for such problems; b) also learn on how to solve it ! c) also learn on how to PRESENT the solution.

Answer by MathTherapy(10552)   (Show Source):
You can put this solution on YOUR website!

julie recently drove to visit her parents who live 240 miles away. on her way there her average speed was 11 miles per hour faster than on her way home (she ran into some bad weather). if julie spent a total of 8 hours driving, find the two rates

Let speed to destination be S
Then return speed = S - 11
We then get the following TIME equation:
240(S - 11) + 240S = 8S(S - 11) ------- Multiplying by LCD, S(S - 11)

(S - 66)(S - 5) = 0
OR