2017 AMC 10A Problems/Problem 14
Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was dollars. The cost of his movie ticket was of the difference between and the cost of his soda, while the cost of his soda was of the difference between and the cost of his movie ticket. To the nearest whole percent, what fraction of did Roger pay for his movie ticket and soda?
Let = cost of movie ticket
Let = cost of soda
We can create two equations:
Substituting we get:
Now we can find s and we get:
Since we want to find what fraction of did Roger pay for his movie ticket and soda, we add and to get:
We have two equations from the problem: and If we replace with we get a system of equations, and the sum of the values of and is the percentage of . Solving, we get and . Adding, we get , which is closest to which is .
WLOG let Let be the price of the movie ticket. Let be the price of the soda. Thus, Simplifying, we have
Multiplying the first equation by and adding them, we have
Finding and is straightforward from there.
Let be the price of a movie ticket and be the price of a soda.
and Then, we can turn this into
Subtracting and getting rid of A, we have . Assume WLOG that , , thus making a solution for this equation. Substituting this into the 1st equation, we get . Hence,
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