Question

Brooke has scores of $84,72,90,95,$ and $87$ on her first five quizzes. After taking the sixth quiz, Brooke's mean score increased. Which could be Brooke's sixth quiz score? Select three options.

• A. $85$
• B. $90$
• C. $83$
• D. $86$
• E. $92$

Answer 1

Answer: (B), (D), (E)

$92$

Explanation of correct options:

Finding the sixth quiz score of Brooke's:

Given that, Brooke has scores of $84,72,90,95,$ and $87$ on her first five quizzes.

Initial mean,

$$\begin{gathered} M_i=\frac{84+72+90+95+87}{5} \\ \Rightarrow M_i=\frac{428}{5} \\ \Rightarrow M_i=85.6 \end{gathered}$$

Option (B): $90$

Final mean,

$$\begin{gathered} M_f=\frac{84+72+90+95+87+90}{6} \\ \Rightarrow M_f=\frac{518}{6} \\ \Rightarrow M_f=86.33 \\ \Rightarrow 86.33>85.6 \\ \Rightarrow M_f>M_i \end{gathered}$$

Here, the final mean is greater than initial mean.

Therefore, the option B is correct

Option (D): $86$

Final mean,

$$\begin{gathered} M_f=\frac{84+72+90+95+87+86}{6} \\ \Rightarrow M_f=\frac{514}{6} \\ \Rightarrow M_f=86.33 \\ \Rightarrow 85.666>85.6 \\ \Rightarrow M_f>M_i \end{gathered}$$

Here, the final mean is greater than initial mean.

Therefore, the option D is correct

Option (E): $92$

Final mean,

$$\begin{gathered} M_f=\frac{84+72+90+95+87+92}{6} \\ \Rightarrow M_f=\frac{520}{6} \\ \Rightarrow M_f=86.33 \\ \Rightarrow 86.666>85.6 \\ \Rightarrow M_f>M_i \end{gathered}$$

Here, the final mean is greater than initial mean.

Therefore, the option E is correct

Explanation of incorrect options:

For option A: $85$

Final mean,

$$\begin{gathered} M_f=\frac{84+72+90+95+87+85}{6} \\ \Rightarrow M_f=\frac{513}{6} \\ \Rightarrow M_f=85.5 \\ \Rightarrow 85.5<85.6 \\ \Rightarrow M_f<M_i \end{gathered}$$

Here, the final mean is less than initial mean.

Therefore, the option A is incorrect.

Option C: $83$

Final mean,

$$\begin{gathered} M_f=\frac{84+72+90+95+87+83}{6} \\ \Rightarrow M_f=\frac{511}{6} \\ \Rightarrow M_f=85.166 \\ \Rightarrow 85.166<85.6 \\ \Rightarrow M_f<M_i \end{gathered}$$

Here, the final mean is less than initial mean.

Therefore, the option C is incorrect

Hence, option (B), (D) and (E) is correct.