Question

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?

$9\%$

$19\%$

$22\%$

$23\%$

$25\%$

Solution:

Let $t$ be the cost of the ticket and $s$ be the cost of the soda. Then we get the following equations. $t = \dfrac{A - s}{5}$ $s = \dfrac{A - t}{20}$

Cross-multiplying the first equation gives us $5t = A - s.$ Substituting in the expression for $s$ yields $5t = A - \dfrac{A - t}{20}.$ Solving yields $\begin{align*} 5t &= A - \dfrac{A - t}{20} \\ 100t &= 20A - A + t \\ 99t &= 19A \\ t &= \dfrac{19A}{99}. \end{align*}$

This also gives us $s = \dfrac{A - \frac{19A}{99}}{20} = \dfrac{4A}{99}.$

Adding together the costs gives us $\dfrac{19A}{99} + \dfrac{4A}{99} = \dfrac{23A}{99} \approx 23 \%.$

Thus, D is the correct answer.

2017 AMC 10A Problem 14

Solution:

Problem 14 in Other Years