64. Commuting Students. At a well known university, ¼ of the undergraduate students commute, and 1/3 of 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, then what fraction of the commuters drive more than 40 miles daily?

let the number of graduate students be x

let the number of undergrads be 2x

undergrads who drive = 2x/4 = x/2
grads who drive = x/3

total commuters = x/2 + x/3 = 5x/6

undergrads more than 40 miles = 2x/10 = x/5
grads more than 40 miles = x/6
total driving more than 40 miles = x/5 + x/6 = 11x/30

fraction asked for = (11x/30) ÷ ( 5x/6)
= 11/25

You can verify this by picking some arbitrary numbers, say
graduates = 600
undergrads = 1200

so grads who drive = 200
undergrads who drive = 300
total driving = 500

More than 40 miles:
undergrads = 120
grads = 100
total over 40 miles = 220

fraction = 220/500 = 22/50 = 11/25
same as above

thank you very much I tried this many times and was not able to solve it.

To find the fraction of commuters who drive more than 40 miles daily, we need to calculate the fraction of undergraduates and graduates who commute and drive more than 40 miles daily separately, and then combine them.

Let's assume there are "x" graduate students in total. Since there are twice as many undergraduate students as graduate students, the number of undergraduate students would be 2x.

It is given that 1/4 of the undergraduate students commute. So the number of commuting undergraduate students would be (1/4) * 2x = x/2.

Similarly, 1/3 of the graduate students commute. So the number of commuting graduate students would be (1/3) * x = x/3.

Now, it is stated that one-tenth of the undergraduate students drive more than 40 miles daily. So the number of undergraduate students driving more than 40 miles daily would be (1/10) * (x/2) = x/20.

Also, 1/6 of the graduate students drive more than 40 miles daily. So the number of graduate students driving more than 40 miles daily would be (1/6) * (x/3) = x/18.

Now, let's calculate the total number of students who drive more than 40 miles daily. This would be the sum of undergraduate students and graduate students driving more than 40 miles daily.

Total number of students driving more than 40 miles daily = (x/20) + (x/18).

To find the fraction of commuters who drive more than 40 miles daily, we divide the total number of students driving more than 40 miles daily by the total number of commuters.

Fraction of commuters driving more than 40 miles daily = [(x/20) + (x/18)] / [(x/2) + (x/3)].

Simplifying the expression:
Fraction of commuters driving more than 40 miles daily = [(9x + 10x) / (360)] * [(6x) / (3x)] = (19/18) * (2/3).

So the fraction of commuters who drive more than 40 miles daily is (19/18) * (2/3), which can be simplified further if necessary.