Of all the students in a certain dormitory, 1/2...

Question

Of all the students in a certain dormitory, 1/2 are first-year students and the rest are second-year students. If 4/5 of the first-year students have not declared a major and if the fraction of second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major, what fraction of all the students in the dormitory are second-year students who have not declared a major?

GMAT Panda · Accepted Answer

1/5 We create variables for the numbers of first-year and second-year students out of a total of T students. F + S = T We translate "Of all the students in a certain dormitory, 1/2 are first-year students" into an equation. F = 1/2 T We translate "the rest are second-year students" into an equation. S = T - F We translate "4/5 of the first-year students have not declared a major" into an equation for first-year students without a declared major. F_not = 4/5 F We subtract to find the number of first-year students who have declared a major. F_declared = F - F_not We translate "the fraction of second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major" into equations relating those groups. S_declaredS = 3 F_declaredF S_declared = 3 F_declaredF S We subtract to find the number of second-year students who have not declared a major. S_not = S - S_declared We substitute the previous results and simplify step by step. F_declared = F - 4/5 F F_declared = 1/5 F S_declared = 3 F_declaredF S S_declared = 3/5 S S = T - F S = T - 1/2 T S = 1/2 T S_not = S - S_declared S_not = S - 3/5 S S_not = 2/5 S S_not = 2/5 1/2 T S_not = 1/5 T

Explanation	Calculations	Help
$We create variables for the numbers of first-year and second-year students out of a total of T students.$	$F + S = T$
$We translate "Of all the students in a certain dormitory, 1/2 are first-year students" into an equation.$	$F = \frac{1}{2} \times T$
$We translate "the rest are second-year students" into an equation.$	$S = T - F$
$We translate "4/5 of the first-year students have not declared a major" into an equation for first-year students without a declared major.$	$F_{not} = \frac{4}{5} \times F$
$We subtract to find the number of first-year students who have declared a major.$	$F_{declared} = F - F_{not}$
$We translate "the fraction of second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major" into equations relating those groups.$	$\frac{S _{declared}}{S} = 3 \times \frac{F _{declared}}{F} S_{declared} = 3 \times \frac{F _{declared}}{F} \times S$
$We subtract to find the number of second-year students who have not declared a major.$	$S_{not} = S - S_{declared}$
$We substitute the previous results and simplify step by step.$	$F_{declared} = F - \frac{4}{5} \times F F_{declared} = \frac{1}{5} \times F S_{declared} = 3 \times \frac{F _{declared}}{F} \times S S_{declared} = \frac{3}{5} \times S S = T - F S = T - \frac{1}{2} \times T S = \frac{1}{2} \times T S_{not} = S - S_{declared} S_{not} = S - \frac{3}{5} \times S S_{not} = \frac{2}{5} \times S S_{not} = \frac{2}{5} \times \frac{1}{2} \times T S_{not} = \frac{1}{5} \times T$

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