SOLUTION: Edward takes 15 minutes longer than Kate does to make the 112-mile drive between two cities. Kate drives 8 miles an hour faster. How fast do Edward and Kate drive?

Algebra ->  Customizable Word Problem Solvers  -> Travel -> SOLUTION: Edward takes 15 minutes longer than Kate does to make the 112-mile drive between two cities. Kate drives 8 miles an hour faster. How fast do Edward and Kate drive?      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 1162900: Edward takes 15 minutes longer than Kate does to make the 112-mile drive between two cities. Kate drives 8 miles an hour faster. How fast do Edward and Kate drive?
Found 5 solutions by jim_thompson5910, amarjeeth123, ikleyn, josgarithmetic, MathTherapy:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

x = time in hours it takes Kate to make the drive
x+0.25 = time in hours it takes Edward to make the drive
15 minutes = 1/4 hour = 0.25 hours

y = Edward's speed in miles per hour (mph)
y+8 = Kate's speed in mph

For Edward,
distance = rate*time
112 = y*(x+0.25)
which solves to
y = 112/(x+0.25)

For Kate,
distance = rate*time
112 = (y+8)*x

Plug in y = 112/(x+0.25) and solve for x

112 = (y+8)*x
112 = ( 112/(x+0.25) +8)*x
112 = (112x)/(x+0.25) + 8x
112-8x = (112x)/(x+0.25)
(112-8x)(x+0.25) = 112x
112x+28-8x^2-2x = 112x
112x+28-8x^2-2x-112x = 0
-8x^2-2x+28 = 0
-2(4x^2+x-14) = 0
4x^2+x-14 = 0

Use the quadratic formula to solve from here
We have a = 4, b = 1, c = -14.
x+=+%28-b%2Bsqrt%28b%5E2-4ac%29%29%2F%282a%29 or x+=+%28-b-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-1%2Bsqrt%28%281%29%5E2-4%284%29%28-14%29%29%29%2F%282%284%29%29 or x+=+%28-1-sqrt%28%281%29%5E2-4%284%29%28-14%29%29%29%2F%282%284%29%29

x+=+%28-1%2Bsqrt%28225%29%29%2F%288%29 or x+=+%28-1-sqrt%28225%29%29%2F%288%29

x+=+%28-1%2B15%29%2F%288%29 or x+=+%28-1-15%29%2F%288%29

x+=+%2814%29%2F%288%29 or x+=+%28-16%29%2F%288%29

x+=+1.75 or x+=+-2

Ignore the negative x value as a negative time value makes no sense.

x = 1.75 means it takes Kate 1.75 hours to make the drive
1.75 hours = 1 hr + 0.75 hr = 1 hr + 45 min = 1 hr, 45 min

Add 15 minutes to this value to get 2 hours, which is the time it takes Edward to make the drive.

Once we know the time values, we can compute the speed values

speed = distance/time = 112/1.75 = 64 mph
is Kate's speed

while,
speed = distance/time = 112/2 = 56 mph
is Edwards speed, exactly 8 mph less than Kate's speed.


=============================================
Answers:

Kate's speed = 64 mph
Edward's speed = 56 mph

Answer by amarjeeth123(569) About Me  (Show Source):
You can put this solution on YOUR website!
Let Edward's speed be s.
Let Kate's speed be s+8.
Distance=112 miles
Time=Distance/speed
Edward's time=112/s
Kate's time=112/(s+8)
(112/s)=(1/4)+(112/(s+8))
112(32)=s(s+8)
s^2+8s-3584=0
Solving we get s=56
Edward's speed is 56 mph and Kate's speed is 64 mph respectively.

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

Let x be the Edward's rate driving, in miles per hour.


Then the Kate's rate driving is  (x+8) mph.


Write the "time" equation


    112%2Fx - 112%2F%28x%2B8%29 = 1%2F4   of an hour.


To solve it, multiply both sides by 4x*(x+8).  You will get


    112*4*(x+8) - 112*4*x = x*(x+8)

    112*4(x+8) - 112*4x = x*(x+8)

    3584 = x*(x+8)      (*)


From this point, there are two ways to complete the solution.


First way is to solve this quadratic equation formally.
In this way, you will get the ANSWER 56 mph for Edwards and 64 mph for Kate.


The other way is to introduce new variable y = x+4.
Then x+8 = y-4,  and your equation (*)  takes the form

    3584 = (y-4)*(y+4)

    3584 = y^2 - 16

    3584 + 16 = y^2  

    3600 = y^2

    y  =  60


and you get the same answer practically MENTALLY.

Solved.




Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
              SPEED      TIME      DISTANCE

Edward        r         112/r      112

Kate         r+8        112/(r+8)  112

Difference              1/4

112%2Fr-112%2F%28r%2B8%29=1%2F4---------simplify and solve...

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
Edward takes 15 minutes longer than Kate does to make the 112-mile drive between two cities. Kate drives 8 miles an hour faster. How fast do Edward and Kate drive?
Let Edward's speed be S

Then Kate's is: S + 8

We then get the following TIME equation: 

112(4)(S + 8) = 112(4)(S) + S(S + 8) ------ Multiplying by LCD, 4S(S + 8)

 

 

matrix%281%2C3%2C+0%2C+%22=%22%2C+S%5E2+%2B+8S+-+112%2832%29%29 

matrix%281%2C3%2C+0%2C+%22=%22%2C+S%5E2+%2B+8S+-+%223%2C584%22%29%29 

(S - 56)(S + 64) = 0

Edward's speed or highlight_green%28matrix%281%2C4%2C+S%2C+%22=%22%2C+56%2C+mph%29%29        OR            S = - 64 (ignore)

Can you now find Kate's speed?