Let \( S \) be the number of votes Scarlett had at the start of the last day of voting, and let \( T \) be the number of votes Tristan had at that time. According to the problem, we have the following information:
-
At the start of the last day, Tristan had half as many votes as Scarlett: \[ T = \frac{1}{2} S \]
-
Throughout the day, Scarlett gained 11 votes, and Tristan gained 34 votes. By the end of the day, they had the same number of votes, which gives us the equation: \[ S + 11 = T + 34 \]
Now we can substitute the first equation into the second equation. Since \( T = \frac{1}{2} S \), we can substitute that into the second equation: \[ S + 11 = \frac{1}{2} S + 34 \]
Next, we need to eliminate the fraction by multiplying everything by 2: \[ 2(S + 11) = 2\left(\frac{1}{2} S + 34\right) \] This simplifies to: \[ 2S + 22 = S + 68 \]
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, we find: \[ S = 68 - 22 \] \[ S = 46 \]
Thus, Scarlett had \(\boxed{46}\) votes at the start of the last day of voting.
To ensure our solution is correct, let's calculate the initial number of votes for Tristan: \[ T = \frac{1}{2} S = \frac{1}{2} \times 46 = 23 \]
Now, let's consider how many votes each candidate has by the end of the day:
- At the end of the day, Scarlett has: \[ 46 + 11 = 57 \]
- At the end of the day, Tristan has: \[ 23 + 34 = 57 \]
Since they both end up with the same number of votes (57 votes), our solution is verified: Scarlett had \(\boxed{46}\) votes at the start.
1 answer