2017 AMC 8 Problems/Problem 14

Problem

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?

$\textbf{(A) }89\qquad\textbf{(B) }92\qquad\textbf{(C) }93\qquad\textbf{(D) }96\qquad\textbf{(E) }98$

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.

Solution 3

In the problem, we can see that Chloe solved 80% of the problems she solved alone, but 88% of her answers are correct. If 80 and another number's average is 88, the other number must be 96. Then, Zoe solved 90% of the problems she did alone, but 96% of her answers were correct. Then, the average of 90 and 96 is $\boxed{\textbf{(C) } 93}$ .

Solution 4

(Slightly different Solution) Suppose we said that there were $100$ problems in their assignment. Then, Chloe had $40$ correct and $10$ incorrect on her portion, and $48$ correct and $2$ incorrect on the portion she and Zoe solved. Zoe has $45$ correct and $5$ incorrect on her portion, and $48$ correct and $2$ incorrect on the portion that she and Chloe solved. Then, Zoe has $48 + 45 = 93$ correct answers out of $100$ , so the answer is $\boxed{\textbf{(C) } 93}$ .

-HW73

Solution 5

This a hard question and I cannot do it so don't use this solution.

-fn106068

Never Mind

https://www.youtube.com/watch?v=dQw4w9WgXcQ ~Its a me a mario

Video Solution

https://youtu.be/WgoAEitW5D4

https://youtu.be/1VWcwRNNJoI

~savannahsolver

2025 AMC 8 (Problems • Answer Key • Resources)
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