GMAT Practice Question: Of all the students in a certain dormitory, \frac{1}{2}...
Question
Of all the students in a certain dormitory, \frac{1}{2} are first-year students and the rest are second-year students. If \frac{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?
- \frac{1}{15}
- \frac{1}{5}
- \frac{4}{15}
- \frac{1}{3}
- \frac{2}{5}
Topics: word problems, fractions, translations
Solution
Step 1
We create variables for the numbers of first-year and second-year students out of a total of T students.
F + S = T
Step 2
We translate \textbf{"Of all the students in a certain dormitory, 1/2 are first-year students"} into an equation.
F = \frac{1}{2}\times T
Step 3
We translate \textbf{"the rest are second-year students"} into an equation.
S = T - F
Step 4
We translate \textbf{"4/5 of the first-year students have not declared a major"} into an equation for first-year students without a declared major.
F_{\text{not}} = \frac{4}{5}\times F
Step 5
We subtract to find the number of first-year students who have declared a major.
F_{\text{declared}} = F - F_{\text{not}}
Step 6
We translate \textbf{"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_{\text{declared}}}{S} = 3\times \frac{F_{\text{declared}}}{F} S_{\text{declared}} = 3\times \frac{F_{\text{declared}}}{F}\times S
Step 7
We subtract to find the number of second-year students who have not declared a major.
S_{\text{not}} = S - S_{\text{declared}}
Step 8
We substitute the previous results and simplify step by step.
F_{\text{declared}} = F - \frac{4}{5}\times F F_{\text{declared}} = \frac{1}{5}\times F S_{\text{declared}} = 3\times \frac{F_{\text{declared}}}{F}\times S S_{\text{declared}} = \frac{3}{5}\times S S = T - F S = T - \frac{1}{2}\times T S = \frac{1}{2}\times T S_{\text{not}} = S - S_{\text{declared}} S_{\text{not}} = S - \frac{3}{5}\times S S_{\text{not}} = \frac{2}{5}\times S S_{\text{not}} = \frac{2}{5}\times \frac{1}{2}\times T S_{\text{not}} = \frac{1}{5}\times T
Answer
\frac{1}{5}
