2017 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 14 Medium

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? (2017 AMC 10A Problem, Question#14)

  • A.

    9%

  • B.

    19%

  • C.

    22%

  • D.

    23%

  • E.

    25%

Answer:D

Let m= cost of movie ticket,

Let s=cost of soda,

We can create two equations:

m= \dfrac{1}{5}(A-s).

s= \dfrac{1}{20}(A-m).

Substituting we get:

m= \dfrac{1}{5}(A- \dfrac{1}{20}(A-m)).

which yields:

m= \dfrac{19}{99}A.

Now we can find s and we get:

s= \dfrac{4}{99}A.

Since we want to find what fraction of A did Roger pay for his movie ticket and soda, we

add m and s to get:

\dfrac{19}{99}A+ \dfrac{4}{99}A \Rightarrow (\rm D)23 %.

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