To solve the problem, let's denote the number of children as [tex]\(C\)[/tex] and the total number of people at the party as [tex]\(T\)[/tex].
According to the problem, [tex]\(\frac{1}{3}\)[/tex] of the people were children. Thus, we can write the following equation:
[tex]\[ C = \frac{1}{3} T \][/tex]
The problem also states that there were 50 more adults than children. Let's denote the number of adults as [tex]\(A\)[/tex]. Therefore,
[tex]\[ A = C + 50 \][/tex]
Since the total number of people is the sum of the number of children and adults, we have:
[tex]\[ T = C + A \][/tex]
Now substitute [tex]\(A = C + 50\)[/tex] into the equation for [tex]\(T\)[/tex]:
[tex]\[ T = C + (C + 50) \][/tex]
[tex]\[ T = 2C + 50 \][/tex]
We also know from the earlier equation that [tex]\(C = \frac{1}{3} T\)[/tex]. Substitute [tex]\(C = \frac{1}{3} T\)[/tex] into the equation:
[tex]\[ T = 2 \left(\frac{1}{3} T \right) + 50 \][/tex]
Simplify the equation:
[tex]\[ T = \frac{2}{3} T + 50 \][/tex]
To isolate [tex]\(T\)[/tex], subtract [tex]\(\frac{2}{3} T\)[/tex] from both sides of the equation:
[tex]\[ T - \frac{2}{3} T = 50 \][/tex]
This simplifies to:
[tex]\[ \frac{1}{3} T = 50 \][/tex]
To solve for [tex]\(T\)[/tex], multiply both sides by 3:
[tex]\[ T = 50 \times 3 \][/tex]
[tex]\[ T = 150 \][/tex]
So, the total number of people at the party is 150.
To confirm the solution, let's find the number of children [tex]\(C\)[/tex] and the number of adults [tex]\(A\)[/tex]:
[tex]\[ C = \frac{1}{3} T = \frac{1}{3} \times 150 = 50 \][/tex]
[tex]\[ A = C + 50 = 50 + 50 = 100 \][/tex]
Thus, there were 50 children and 100 adults, and the total number of people at the party is indeed 150.
