2017 AMC 10A Problem 14

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\%

Let the ticket and soda costs be variables

Turn the two percent statements into a two-equation linear system

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{aligned} 5t &= A - \dfrac{A - t}{20} \\ 100t &= 20A - A + t \\ 99t &= 19A \\ t &= \dfrac{19A}{99}. \end{aligned}

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.

Problem 14 in other years