Commuting students. 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, then what fraction of the commuters drive more
than 40 miles daily?
number of grad students = x
number of undergrads = 2x
grads more than 40 miles = x/6
undergrads over 40 miles = 2x/10
total over 40 miles = 11x/30
undergrads commuting = 2x/4
grads commuting = x/3
total commuting = 5x/6
fraction of commuters over 40 miles = (11x/30)÷(5x/6) = 1/25
To find the fraction of commuters who drive more than 40 miles daily, we need to calculate the fraction of undergraduate and graduate students who commute and drive more than 40 miles daily, and then add them together.
Let's denote the number of graduate students as "G". Since there are twice as many undergraduate students as graduate students, the number of undergraduate students can be denoted as "2G".
Now, we can calculate the number of undergraduate students who commute and drive more than 40 miles daily. Given that 1/4 of undergraduate students commute, the number of undergraduate commuting students is (1/4) * (2G) = G/2. From this group, 1/10 drive more than 40 miles daily, so the number of undergraduate commuting students who drive more than 40 miles daily is (1/10) * (G/2) = G/20.
Similarly, we can calculate the number of graduate students who commute and drive more than 40 miles daily. Given that 1/3 of graduate students commute, the number of graduate commuting students is (1/3) * G. From this group, 1/6 drive more than 40 miles daily, so the number of graduate commuting students who drive more than 40 miles daily is (1/6) * (1/3) * G = G/18.
To find the total number of commuting students who drive more than 40 miles daily, we add the number of undergraduate commuting students who drive more than 40 miles daily and the number of graduate commuting students who drive more than 40 miles daily: G/20 + G/18.
Finally, we divide this sum by the total number of commuting students (undergraduate and graduate) to get the fraction: (G/20 + G/18) / ((1/4) * (2G) + (1/3) * G).
Now, we can simplify the fraction to get the final answer.