Scarlett and Tristan are both running for student council president. At the start of the last day of voting, Tristan had half as many votes as Scarlett. Throughout the day, Scarlett gained 11 more votes, while Tristan gained 34 more votes. By the end of the day, Scarlett and Tristan had the same number of votes.

How many votes did Scarlett have at the start of the last day of voting?

1 answer

Let the number of votes Scarlett had at the start of the last day be denoted as \( S \). Since Tristan had half as many votes as Scarlett at the start, we can denote the number of votes Tristan had at the start as \( T = \frac{1}{2} S \).

Throughout the day, Scarlett gained 11 votes, while Tristan gained 34 votes. Therefore, by the end of the day, the number of votes for each candidate can be expressed as:

  • Votes for Scarlett at the end of the day: \[ S + 11 \]

  • Votes for Tristan at the end of the day: \[ T + 34 = \frac{1}{2} S + 34 \]

According to the problem, at the end of the day, both candidates had the same number of votes. Therefore, we can set the two expressions equal to each other: \[ S + 11 = \frac{1}{2} S + 34 \]

To eliminate the fraction, we'll multiply every term by 2: \[ 2(S + 11) = 2\left(\frac{1}{2} S + 34\right) \]

Expanding both sides gives: \[ 2S + 22 = S + 68 \]

Now, we can isolate \( S \) by subtracting \( S \) from both sides: \[ 2S - S + 22 = 68 \]

This simplifies to: \[ S + 22 = 68 \]

Subtracting 22 from both sides yields: \[ S = 46 \]

Thus, Scarlett had \( \boxed{46} \) votes at the start of the last day of voting.

To verify this, we can calculate Tristan's votes at the start: \[ T = \frac{1}{2} S = \frac{1}{2} \times 46 = 23 \]

Then, throughout the day:

  • Scarlett's final votes: \[ 46 + 11 = 57 \]
  • Tristan's final votes: \[ 23 + 34 = 57 \]

Both candidates indeed end up with the same number of votes, confirming that Scarlett had \( \boxed{46} \) votes at the start of the last day.