Last month’s Ponder This challenge was as follows:
In an election, Charles came in last and Bob received 24.8% of the votes. After counting two additional votes, he overtook Bob with 25.1% of the votes. Assuming there were no ties and all the results are rounded to the nearest promille (one-tenth of a percent), how many votes did Alice get?
And here‘s the solution. It was the easiest Ponder This ever.
Let be the number of votes gotten by Alice, Bob, and Charles, respectively. If Charles came in last, then and . Since Bob received 24.8% of the votes and he was not the last one, then we deduce that Alice was the first with more than 50% of the votes. Therefore, we have .
In the first counting, Bob got 24.8% of the votes. Since the results are rounded off, we have
where is the round-off error. Hence, we obtain
Multiplying both sides by , and letting , we get
Note that, since , we have that . After counting two additional votes, Charles overtook Bob with 25.1% of the votes. Let be the new number of votes that Charles got. Hence,
where is the round-off error. Thus, we have
and, multiplying both sides by , and letting , we get
where . Since there were no ties and these two additional votes were enough for Charles to overtake Bob, then we have . As are natural numbers, we conclude that .
Hence, we have a linear system of equations in
where . Note that must be natural numbers, which means that there should exist such that the linear system of equations has a solution in . Let us relax the system of equations by making and allowing the unknowns to take values in rather than . Then, we obtain the new linear system of equations
whose solution is . Taking the floor of each component, we obtain a solution in
Note that , and , meaning that and , which are admissible values. Note also that . Hence, all the conditions are satisfied. We thus conclude that Alice got 84 votes.