In a student council election, five candidates contested for a single position. 75% of the students enrolled in the institution voted. One candidate secured 20% of the votes polled, while the votes obtained by the remaining four candidates were in the ratio $2:3:4:6$. The candidate with the highest votes got 120 votes more than the candidate with the second-highest votes. What is the total number of students enrolled in the institution?
Answer and solution
Correct answer: B
Solution
Let votes polled $= V$. The winning candidate gets $20\% \times V = 0.2V$. The remaining four share $0.8V$ in ratio 2:3:4:6 (sum = 15).
Highest of the four $= \dfrac{6}{15} \times 0.8V = 0.32V$. Second highest $= \dfrac{4}{15} \times 0.8V$.
Since $0.32V > 0.2V$, the highest vote-getter is the ratio-6 candidate (runner-up to winner by votes). Highest minus second-highest: $\left(\dfrac{6-4}{15}\right) \times 0.8V = \dfrac{2}{15} \times 0.8V = \dfrac{8V}{75}$ = 120 → $V = \dfrac{120 \times 75}{8} = 1125$.
Enrollment $= \dfrac{1125}{0.75} = 1500$.
Highest of the four $= \dfrac{6}{15} \times 0.8V = 0.32V$. Second highest $= \dfrac{4}{15} \times 0.8V$.
Since $0.32V > 0.2V$, the highest vote-getter is the ratio-6 candidate (runner-up to winner by votes). Highest minus second-highest: $\left(\dfrac{6-4}{15}\right) \times 0.8V = \dfrac{2}{15} \times 0.8V = \dfrac{8V}{75}$ = 120 → $V = \dfrac{120 \times 75}{8} = 1125$.
Enrollment $= \dfrac{1125}{0.75} = 1500$.