Problem (Algebra):
Let $T=67$. A group of children and adults go to a rodeo. A child's admission ticket costs $\$ 5$, and an adult's admission ticket costs more than $\$ 5$. The total admission cost for the group is $\$ 10 \cdot T$. If the number of adults in the group were to increase by $20 \%$, then the total cost would increase by $10 \%$. Compute the number of children in the group.

Solution:
Suppose there are $x$ children and $y$ adults in the group and each adult's admission ticket costs $\$ a$. The given information implies that $5 x+a y=10 T$ and $5 x+1.2 a y=11 T$. Subtracting the first equation from the second yields $0.2 a y=T \rightarrow a y=5 T$, so from the first equation, $5 x=5 T \rightarrow x=T$. With $T=67$, the answer is 67 .

Final answer: 67