Maths Olympiad Prep

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Algebra Difficulty 2.9 Junior Find the answer

Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was AA dollars. The cost of his movie ticket was 20%20\% of the difference between AA and the cost of his soda, while the cost of his soda was 5%5\% of the difference between AA and the cost of his movie ticket. To the nearest whole percent, what fraction of AA did Roger pay for his movie ticket and soda?

Pick one

Solution

Let mm = cost of movie ticket
Let ss = cost of soda
We can create two equations:
m=15(A−s)m = \frac{1}{5}(A - s)
s=120(A−m)s = \frac{1}{20}(A - m)
Substituting we get:
m=15(A−120(A−m))m = \frac{1}{5}(A - \frac{1}{20}(A - m))
which yields:
m=1999Am = \frac{19}{99}A
Now we can find s and we get:
s=499As = \frac{4}{99}A
Since we want to find what fraction of AA did Roger pay for his movie ticket and soda, we add mm and ss to get:
1999A+499A  ⟹  (D) 23%\frac{19}{99}A + \frac{4}{99}A \implies \boxed{\textbf{(D)}\ 23\%}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.