$$ \begin{array}{l} ext { Brooke has scores of } 84,72,90,95 ext {, and } 87 ext { on her first } \ ext { Which could be Brooke's sixth quiz score? Select three } \ ext { five quizzes. After taking the sixth quiz, Brooke's mean } \ ext { score increased. } \ ext { options. } \ \square 85 \ \square 90 \ \square 83 \ \square 86 \ \square 92\end{array} $$

Question

$$ \begin{array}{l}	ext { Brooke has scores of } 84,72,90,95 	ext {, and } 87 	ext { on her first } \ 	ext { Which could be Brooke's sixth quiz score? Select three } \ 	ext { five quizzes. After taking the sixth quiz, Brooke's mean } \ 	ext { score increased. } \ 	ext { options. } \ \square 85 \ \square 90 \ \square 83 \ \square 86 \ \square 92\end{array} $$

UpStudy AI · Accepted Answer

To determine which of the given options could be Brooke's sixth quiz score such that her mean score increases after taking the sixth quiz, we will follow these steps:

1. **Calculate the current mean score of the first five quizzes.**
2. **Determine the new mean score after adding the sixth quiz score.**
3. **Identify which of the options would result in an increased mean score.**

### Step 1: Calculate the current mean score

Brooke's scores for the first five quizzes are: $$84, 72, 90, 95, 87$$.

First, we calculate the total score of these quizzes:

$$
\text{Total score} = 84 + 72 + 90 + 95 + 87
$$

Calculating this:

$$
\text{Total score} = 84 + 72 = 156
$$
$$
156 + 90 = 246
$$
$$
246 + 95 = 341
$$
$$
341 + 87 = 428
$$

Now, we find the mean score of the first five quizzes:

$$
\text{Mean} = \frac{\text{Total score}}{\text{Number of quizzes}} = \frac{428}{5}
$$

Calculating this:

$$
\text{Mean} = 85.6
$$

### Step 2: Determine the new mean score after adding the sixth quiz score

Let $$x$$ be the score of the sixth quiz. The new total score after adding the sixth quiz will be:

$$
\text{New total score} = 428 + x
$$

The new mean score after six quizzes will be:

$$
\text{New mean} = \frac{428 + x}{6}
$$

### Step 3: Identify which options would result in an increased mean score

For the mean score to increase, the new mean must be greater than the old mean:

$$
\frac{428 + x}{6} > 85.6
$$

Multiplying both sides by 6 to eliminate the fraction:

$$
428 + x > 513.6
$$

Subtracting 428 from both sides:

$$
x > 85.6
$$

### Step 4: Evaluate the options

Now we will evaluate the options to see which scores are greater than $$85.6$$:

- $$85$$ (not greater than $$85.6$$)
- $$90$$ (greater than $$85.6$$)
- $$83$$ (not greater than $$85.6$$)
- $$86$$ (greater than $$85.6$$)
- $$92$$ (greater than $$85.6$$)

### Conclusion

The scores that could be Brooke's sixth quiz score, which would result in an increased mean score, are:

- $$90$$
- $$86$$
- $$92$$

Thus, the three options are $$90$$, $$86$$, and $$92$$.
Brooke's sixth quiz score could be 90, 86, or 92 to increase her mean score.

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