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