To find out what score Anna needs on her fifth quiz so that her mean quiz score is 85, we proceed as follows:
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Calculate the sum of the first four quiz scores: \[ 78 + 86 + 90 + 82 = 336 \]
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Let \( x \) be the score Anna needs on her fifth quiz. Then the sum of all five quiz scores will be: \[ 336 + x \]
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The mean of the five quiz scores is given by the formula: \[ \text{Mean} = \frac{\text{Total Sum}}{\text{Number of Quizzes}} = \frac{336 + x}{5} \]
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We want this mean to be 85: \[ \frac{336 + x}{5} = 85 \]
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Multiply both sides by 5 to eliminate the fraction: \[ 336 + x = 425 \]
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Solving for \( x \) gives: \[ x = 425 - 336 = 89 \]
So, Anna should get a score of 89 on her fifth quiz for her mean quiz score to be exactly 85. Thus, the answer is:
89