AMC 10 2017a Problem 14

Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20\%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5\%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda? $\textbf{(A) } 9\% \textbf{(B) } 19\% \textbf{(C) } 22\% \textbf{(D) } 23\% \textbf{(E) } 25\%$