SOLUTION: We are learning rational, radical complex expression word problems. I do NOT understand this at all PLEASE help!!! :( (1) At a well known university, 1/4 of the undergraduat

Question 419056: We are learning rational, radical complex expression word problems.
I do NOT understand this at all PLEASE help!!! :(
(1) At a well known university, 1/4 of the undergraduate students commute and 1/3 of the graduate students commute. One-tenth of the undergraduate students drive more than 40 miles daily, and 1/6 of the graduate students drive more than 40 miles daily. If there are twice as many undergraduate students as there are graduate students, than what fraction of the commuters drive more than 40 miles daily?
I'm COMPLETELY lost.....please help!!!!
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
you are given:

1/4 of the undergraduate students commute.
1/3 of the graduate students commute.
1/10 of the undergraduate students drive more than 40 miles daily.
1/6 of the graduate students drive more than 40 miles daily.

if there are twice as many undergraduate students as there are graduate students, then what fraction of the commuters drive more than 40 miles daily.

let u = number of undergraduate students.
let g = number of graduate students.
let c = number of students that commute.
let d = number of students that drive more than 40 miles daily.

we know that the number of students is equal to the number of undergraduates plus the number graduates. this would be an implied assumption of the problem statement since we are not given any information to indicate there are other classes of students other than these.

we are given that c = (1/4 * u) + (1/3 * g)

this means that the number of students that commute is equal to 1/4 times the number of undergraduate students plus 1/3 times the number of graduate students.

we are given that d = (1/10 * u) + (1/6 * g)

this means that the number of students that drive more than 40 miles per day is equal to 1/10 times the number of undergraduate students plus 1/6 times the number of graduate students.

we are asked what fraction of the students that commute drive more than 40 miles daily, assuming that there are twice as many undergraduate students as graduate students.

this fraction should be equal to d / c because d is equal to the number of students that commute more than 40 miles per day and c is equal to the number of students that commute.

we are given to assume that u = 2*g

this means that there are twice as many undergraduate students as graduate students.

in the equations:

c = (1/4 * u) + (1/3 * g)

d = (1/10 * u) + (1/6 * g)

we can substitute 2 * g for u to get:

c = (1/4 * 2 * g) + (1/3 * g)

d = (1/10 * 2 * g) + (1/6 * g)

we simplify these to get:

c = (1/2 * g) + (1/3 * g)

d = (1/5 * g) + (1/6 * g)

we combine like terms to get:

c = (3/6 * g) + (2/6 * g) = 5/6 * g

d = (6/30 * g) + (5/30 * g) = 11/30 * g

we divide d by c to get (11/30 * g) / (5/6) * g)

the g's cancel out and we are left with d / c = (11/30) / (5/6)

this is equivalent to d / c = (11/30) * (6/5)

we simplify to get d / c = 11/25 which is equivalent to d / c = .44

our answer should be that .44 of the commuters drive more than 40 miles each day.

that's .44 of the commuters, not the whole student body.

since that was the question, you can stop here.

we can test this out as follows:

assume that the number of students is equal to 4*3*10*6 = 720

we have twice as many undergraduate students as graduate students.

this makes the number of graduate students equal to 240 and the number of undergraduate students equal to 480.

1/4 of the undergraduates = 480/4 = 120
1/3 of the graduates = 240/3 = 80

total students that commute equals 120 + 80 equals 200

1/10 of the undergraduates drive more than 40 miles.
1/6 of the graduates drive more than 40 miles.

1/10 of the undergraduates = 480/10 = 48.
1/6 of the graduates = 240/6 = 40.

total students that drive more than 40 miles is equal to 48 + 40 equals 88.

fraction of students that drive more than 40 miles is equal to 88 / 200 = .44

numbers check out.

I took the number of students as 4 * 3 * 10 * 6 = 720 in order so that I wouldn't end up with partial students after all the multiplying and dividing was done.

I basically used the denominators of 1/4 and 1/3 and 1/10 and 1/6 in order to do that.

I wasn't totally sure if it would work since this is the first time I tried something like that in order to confirm my answer was correct.

It did, at least in this case.

This is not to say that the total number of students was 720, but it was probably either that or some multiple of that or some other number that would yield no partial students after working the problem through.

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