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Thoughts on the Cusp of a Presidential Election

What is the smallest percent of the popular vote needed to elect a president?

To simplify the problem, use these assumptions and givens:

  • Only two candidates are running
  • If a candidate loses a state, assume that the candidate received none of the popular vote in that state
  • The number of votes cast (n) in a state is proportional to the state's population...and thus also is proportional to the number of Representatives (r) for a state
  • The Electoral College has 538 members (2 Senators for each state + number of Representatives for each state + 3 for District of Columbia)
  • To win, a candidate must receive more than 269 electoral votes


Source: J. Witkowski's "Math Modeling and Presidential Elections" Mathematics Teacher Oct 1992

Hint: You will need an ordered list of the composition of the House of Representatives (e.g. 7 states have 1 Representative, 6 have 2, etc.).

If the "elected" candidate wins w states, then we have the sum of electoral votes for that candidate being: (r1+2) + (r2+2) +...+ (rw+2) > 269. You need to solve this equation...!


Solution Commentary: Performing some algebra on hint's equation, we have: r1 + r2 +...+ rw > 269 - 2w. Now, as you select states (with r Representatives) for this equation, you want to minimize r1 + r2 +...+ rw but also note that as the number of states w increases, the expression 269 - 2w decreases.

Using your ordered list, start with states with small populations first (e.g. with r = 1, w = 7+D.C. = 8, r1 + r2 +...+ rw = 8 and 269-2(8)=253. Continue until you find that for r = 11, w = 39+D.C., r1 + r2 +...+ rw = 188.

Thus, the ratio (Total votes in 39 states + D.C.)/(Total votes in nation) is proportional to 188/436. But, in each state, only a simple majority is needed (i.e. 0.5(188)+1 = 95). Thus, our final relationship is that (Total votes won in 39 states + D.C.)/(Total votes cast in nation) is proportional to 95/436 = 022%.

Under our assumptions, a candidate could win only 22% of the people's popular vote, win more than 269 electoral votes, and thus win the election!

Note: For a more complete analysis of the mathematics underlying this solution, see George Polya's "The Minimum Fraction of the Popular Vote That Can Elect the President of the United States," Mathematics Teacher, March 1961, pp. 130-133.