Can you help me do this SAT math question?

Question

Esther drove to work in the morning at an average speed of 45 miles per hour. She returned home in the evening along the same route and averaged 30 miles per hour. If Esther spent a total of one hour communting to and from work, how many miles did Esther drive to work in the morning?

come on people, it can't be 45 miles.

45 miles PER HOUR t(to travel to work)but it takes Esther ONE HOUR to go and come back from work...

Answer 1

Let x be time taken in minutes from work to home (night).
Let y be time taken (in minutes) to get to work (morning)

First we convert the speed to miles per minutes...
she drove an average of 0.75mi/minute in the morning
she drove an average of 0.5mi/minute in the afternoon.

now we make equations:
y(.75)=x(.5)
x+y = 60

we know through subsitution that:
y = 60-x

therefore 60-x(.75)=.5x
therefore 45 -.75x = .5x
therefore 45 = 1.25x
x = 36... therefore Esther took 36 minutes to get home from work.
therefore, he took 60-36 = 24 minutes to get to work in the morning.
therefore he drove 24 x .75 = 18 miles to get to work in the morning.

Answer 2

To start out, let's say that the distance Esther drives to work is D, so a round trip from home to work and back to home again = 2D.

She spent a grand total of only one hour on the road, so it takes 1 hour to travel 2D. But because she didn't do it at the same speed the whole time, you have to break the problem up into two portions.

To Work: @ 45mph

For this part, look at 45mph hour. It is the distance of 45 miles divided by 1 hour. She traveled D miles at this speed, so how much time did it take her?

45m/1h = Dm/?h

Cross multiply to get Dmh = 45m?h

Divide by 45m to get Dh/45 = ?h

We don't yet know exactly how long she spent driving this leg of the route, but keep this equation for later.

To Home: @ 30mph

Treat this like the first part and you get 30m/1h = Dm/?h

Follow the pattern from before and you get Dh/30 = ?h

*** Please note, the unknown time measured in hours (?h) from this half does not equal the ?h from the first half.

Let the first ?h = H1 and the second ?h = H2

H1 + H2 = 1 hour (or simply h)

Plug in your equations from before to get Dh/45 + Dh/30 = h. Multiply each side by 1350 ( = 45 x 30) to get 30Dh + 45Dh = 1350h. Simplify this to 75Dh = 1350h, which can be further reduced to Dh = 18h. Divide both sides by h to see that D = 18

*** So, she drove 18 miles to work in the morning. ***

~ ~ ~ ~ ~
Check:

18 miles @ 45 mph takes 24 minutes.

18 miles @ 30 mph take 36 minutes.

Together, 24 minutes + 36 minutes = 1 hour.

~ ~ ~ ~ ~

OK, so I took the long route. It's been years since I did this simple stuff so I've forgotten my basic formulas. Now, if you want me to use matrices and find something to integrate or differentiate, I'll be ready to help ;)

Answer 3

45

Answer 4

Let X be the time taken by Esther to get to work in the morning.
Then distance to work from Home = 45 M/Hr * X

The time taken by Esther on her way back from work = (1 - X)
( Since it takes one hour to and fro from work)

So Distance covered on way back from work = 30* (1-X)

Since distance is same on the way to work and way back from work, we have...

45*x = 30*(1-x)
which gives,

45X = 30 - 30X
=> 75X = 30
=> X = 30/75 = 0.4 Hrs

Hence distance to work = 45 * .4 = 18 Miles....

18 Miles is the Answer.....

Answer 5

D=45*MornTime = 35 *EvenTime (velocity*time=Distance)
MornTime+EvenTime=1 (total time is 1 hr)

EvenTime=(45/30)*MornTime (from top)
EvenTime=1-MornTime ( from bottom)

1-MornTime=1.5*MornTime (45/30=1.5)
1=1.5*MornTime +MornTime
1/(1.5+1)=MornTime = .4 hours

D= 45 * MornTime = 45*.4= 18 miles

Check: D= 35(1-MornTime)= 30*.6= 18 miles :-)

Answer 6

45 miles.

Answer 7

Let total distance be = x
x/45 +x/30 = 1
or
(2x +3x)/90 =1
or
5x=90
or
x=18

Answer 8

that's what I thought. Average each way is 30 min. so is that 22.5 miles? I haven't done this in years, I shouldn't be helping you. good luck.