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Question

Of all the students in a certain dormitory, are first-year students and the rest are second-year students. If 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?

Answer Choices

A. $\frac{1}{15}$
B. $\frac{1}{5}$
C. $\frac{4}{15}$
D. $\frac{1}{3}$
E. $\frac{2}{5}$

Steps

Explanation	Calculations	Help
$We create variables for the numbers of first-year and second-year students out of a total of T students.$	$F + S = T$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We translate "Of all the students in a certain dormitory, 1/2 are first-year students" into an equation.$	$F = \frac{1}{2} \times T$	Theory & Tactics Method Card TRANS2-B Click to view full details
$We translate "the rest are second-year students" into an equation.$	$S = T - F$	Theory & Tactics Method Card TRANS1-D Click to view full details
$We translate "4/5 of the first-year students have not declared a major" into an equation for first-year students without a declared major.$	$F_{not} = \frac{4}{5} \times F$	Theory & Tactics Method Card TRANS2-B Click to view full details
$We subtract to find the number of first-year students who have declared a major.$	$F_{declared} = F - F_{not}$	Theory & Tactics Method Card TRANS1-D Click to view full details
$We translate "the fraction of second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major" into equations relating those groups.$	$\frac{S _{declared}}{S} = 3 \times \frac{F _{declared}}{F} S_{declared} = 3 \times \frac{F _{declared}}{F} \times S$	Theory & Tactics Method Card TRANS1-E Click to view full details
$We subtract to find the number of second-year students who have not declared a major.$	$S_{not} = S - S_{declared}$	Theory & Tactics Method Card TRANS1-D Click to view full details
$We substitute the previous results and simplify step by step.$	$F_{declared} = F - \frac{4}{5} \times F F_{declared} = \frac{1}{5} \times F S_{declared} = 3 \times \frac{F _{declared}}{F} \times S S_{declared} = \frac{3}{5} \times S S = T - F S = T - \frac{1}{2} \times T S = \frac{1}{2} \times T S_{not} = S - S_{declared} S_{not} = S - \frac{3}{5} \times S S_{not} = \frac{2}{5} \times S S_{not} = \frac{2}{5} \times \frac{1}{2} \times T S_{not} = \frac{1}{5} \times T$	Theory & Tactics Method Card FDPR2-A Click to view full details

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Final Answer

\frac{1}{5}

Question

Of all the students in a certain dormitory, are first-year students and the rest are second-year students. If 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?

Answer Choices

A. $\frac{1}{15}$
B. $\frac{1}{5}$
C. $\frac{4}{15}$
D. $\frac{1}{3}$
E. $\frac{2}{5}$

Steps

Explanation	Calculations	Help
$We create variables for the numbers of first-year and second-year students out of a total of T students.$	$F + S = T$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We translate "Of all the students in a certain dormitory, 1/2 are first-year students" into an equation.$	$F = \frac{1}{2} \times T$	Theory & Tactics Method Card TRANS2-B Click to view full details
$We translate "the rest are second-year students" into an equation.$	$S = T - F$	Theory & Tactics Method Card TRANS1-D Click to view full details
$We translate "4/5 of the first-year students have not declared a major" into an equation for first-year students without a declared major.$	$F_{not} = \frac{4}{5} \times F$	Theory & Tactics Method Card TRANS2-B Click to view full details
$We subtract to find the number of first-year students who have declared a major.$	$F_{declared} = F - F_{not}$	Theory & Tactics Method Card TRANS1-D Click to view full details
$We translate "the fraction of second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major" into equations relating those groups.$	$\frac{S _{declared}}{S} = 3 \times \frac{F _{declared}}{F} S_{declared} = 3 \times \frac{F _{declared}}{F} \times S$	Theory & Tactics Method Card TRANS1-E Click to view full details
$We subtract to find the number of second-year students who have not declared a major.$	$S_{not} = S - S_{declared}$	Theory & Tactics Method Card TRANS1-D Click to view full details
$We substitute the previous results and simplify step by step.$	$F_{declared} = F - \frac{4}{5} \times F F_{declared} = \frac{1}{5} \times F S_{declared} = 3 \times \frac{F _{declared}}{F} \times S S_{declared} = \frac{3}{5} \times S S = T - F S = T - \frac{1}{2} \times T S = \frac{1}{2} \times T S_{not} = S - S_{declared} S_{not} = S - \frac{3}{5} \times S S_{not} = \frac{2}{5} \times S S_{not} = \frac{2}{5} \times \frac{1}{2} \times T S_{not} = \frac{1}{5} \times T$	Theory & Tactics Method Card FDPR2-A Click to view full details

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Final Answer

\frac{1}{5}