Question 285540
1/4 undergraduates commute.
1/3 graduates commute.
1/10 undergraduates commute more than 40 miles.
1/6 graduates commute more than 40 miles.
number of undergraduates = 2 * number of graduates.


let u = number of undergraduates.
let g = number of graduates.


u = 2*g


since 1/4 of undergraduates commute, and 1/10 of those drive more than 40 miles, then 1/10 * 1/4 * u commute more than 40 miles.


since 1/3 of graduates commute, and 1/6 of those drive more than 40 miles, then 1/6 * 1/3 * g commute more than 40 miles.


The total number of commuters = 1/4*u + 1/3*g.


The total number of commuters that travel more than 40 miles = 1/10*1/4*u + 1/6*1/3*g


Since u = 2*g, we can replace u with 2*g in the equations.


They become:


The total number of commuters = 1/4*2*g + 1/3*g = 3/12*2*g + 4/12*g = 6/12*g + 4/12*g = 10/12*g = .83333333*g


The total number of commuters that travel more than 40 miles = 1/10*1/4*2*g + 1/6*1/3*g = 1/20*g + 1/18*g = .105555556*g


The proportion of commuters that travel more than 40 miles = .105555556*g / .83333333*g = .12666666667


The percentage of commuters that travel more than 40 miles = 12.67%


To see how this works, we'll use numbers.


Assume that the number of undergraduates is 7200.


Assume that the number of graduates is 3600.


The number of undergraduates is 2 * the number of graduates.


1/4 of the undergraduates commute = 1/4 * 7200 = 1800.


1/3 of the graduates commute = 1/3 * 3600 = 1200.


1/10 of the undergraduates drive more than 40 miles = 1800/10 = 180.


1/6 of the graduates drive more than 40 miles = 1200/6 = 200.


Total of students that drive more than 40 miles = 180 + 200 = 380.


Total of students that commute = 1800 + 1200 = 3000.


Proportion of commuting students that drive more than 40 miles = 380/3000 = .12666666667.


Percentage of commuting students that drive more than 40 miles = 12.67%.


The numbers confirm the letters so the answer is good.


number of graduate students is 3600.
number of undergraduate students is 7200 = 2*3600.
1/4 of 7200 = 1800
1/10 of 1800 = 180
1/3 of 3600 = 1200
1/6 of 1200 = 200
total commuters = 1800 + 1200 = 3000
total drivers over 40 miles = 180 + 200 = 380
380/3000 = .12666666667