2017 AMC 8

Complete problem set with solutions and individual problem pages

Problem 14 Medium

Chloe and Zoe are both students in Ms. Demeanor's math class. Last night, they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only 80\% of the problems she solved alone, but overall 88\% of her answers were correct. Zoe had correct answers to 90\% of the problems she solved alone. What was Zoe's overall percentage of correct answers?

  • A.

    89

  • B.

    92

  • C.

    93

  • D.

    96

  • E.

    98

Answer:C

Solution 1

Let the number of questions that they solved alone be x. Let the percentage of problems they correctly solve together be a%. As given,

\frac{80x}{100} + \frac{ax}{100} = \frac{2 \cdot 88x}{100}.

Hence, a = 96.

Zoe got \frac{90x}{100} + \frac{ax}{100} = \frac{186x}{100} problems right out of 2x. Therefore, Zoe got \frac{\frac{186x}{100}}{2x} = \frac{93}{100} = \boxed{\textbf{(C) } 93} percent of the problems correct.

 

Solution 2

Assume the total amount of problems is 100 per half homework assignment since we are dealing with percentages, not values. Then, we know that Chloe got 80 problems correct by herself and got 176 problems correct overall. We also know that Zoe had 90 problems she did correctly alone. We can see that the total amount of correct problems Chloe and Zoe did together was 176-80=96. Therefore, Zoe did 96+90=186 problems out of 200 problems correctly. This is \boxed{\textbf{(C) } 93} percent.

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