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Question 285540: I am having problems with this problem, I would appreciate it if you can explain it to me.
At a university, 1/4 of the undergraduate students commute, and 1/3 of graduate commute. One-tenth of the undergraduates drive more than 40 miles daily, and 1/6 graduate drive more than 40 miles daily. If there are twice as many undergraduate students as there are graduate students, then what percent of the commuters drive more than 40 miles daily?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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
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