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?

Answer 1

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

Answer 2

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