You want a seat on the board of directors of Four Keys, Inc. The company has 240,000 shares of stock outstanding and the stock sells for \( \$ 57 \) per share. There are currently 3 seats up for election. If the company uses cumulative voting, how many shares do you need to guarantee that you will be elected to the board? a. 54,001 shares b. 120,001 shares c. 80,000 shares d. 60,001 shares e. 90,001 shares

We first note that in cumulative voting a shareholder can cast one vote per share for each board seat, and the key idea is that to guarantee a seat on a board with \( r \) seats, you must ensure that no other candidate can accumulate votes exceeding yours. The standard formula to guarantee election is to have at least \[ \left\lfloor \frac{N}{r+1} \right\rfloor + 1 \] shares. Here, \( N = 240,\!000 \) (the total number of shares) and \( r = 3 \) (the number of board seats up for election). 1. Calculate the critical fraction: \[ \frac{240,\!000}{3+1} = \frac{240,\!000}{4} = 60,\!000 \] 2. To guarantee election, you need one more share than this value: \[ 60,\!000 + 1 = 60,\!001 \] Thus, you need \( \boxed{60,\!001 \text{ shares}} \) to guarantee a seat on the board. The correct answer is (d) \( 60,\!001 \) shares.