In an academic term, a student submits 10 evaluations, which are qualified with grades between 0 and 5 (both inclusive). The average of the first 7 evaluations is 3.75. If the student's final grade (FG) is the average of the 10 grades, then:
-
\(2.625 \leq \text{FG} \leq 3.75 \)
-
\(0 \leq \text{FG} \leq 3.75 \)
-
\( 3.75 \leq \text{FG} \leq 4.125 \)
-
\( 2.625 \leq \text{FG} \leq 4.125 \)
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\( 4.125 \leq \text{FG} \leq 5 \)
Try to solve it before checking the answer.
\( 2.625 \leq \text{FG} \leq 4.125 \)
Solución
Let \(S_7\) be the sum of the first 7 evaluations. We have that:
\begin{aligned}
\frac{ S_7 }{ 7 } = 3.75
&\Rightarrow
S_7 = 3.75 \times 7
\\[.5em]
&\Rightarrow
S_7 = 26.25
\end{aligned}
\]
If \( S_{10} \) is the sum of the 10 evaluations, then \( \text{FG} = \frac{S_{10}}{ 10 } \).
The lowest grades that the student can obtain in the last three evaluations are 0, 0 and 0, while the highest are 5, 5 and 5. Hence:
\begin{aligned}
&S_7 + 0 + 0 + 0 \leq S_{10} \leq S_7 + 5 + 5 + 5
\\[.5em]
&\hspace{4em} \Rightarrow
S_7 \leq S_{10} \leq S_7 + 15
\\[.5em]
&\hspace{4em} \Rightarrow
\frac{S_7}{10} \leq \frac{S_{10}}{10} \leq \frac{ S_7 + 15 }{10}
\\[.5em]
&\hspace{4em} \Rightarrow
\frac{S_7}{10} \leq \text{FG} \leq \frac{S_7 + 15}{10}
\\[.5em]
&\hspace{4em} \Rightarrow
\frac{ 26.25 }{10} \leq \text{FG} \leq \frac{41.25}{10}
\\[.5em]
&\hspace{4em} \Rightarrow
\boldsymbol{ 2.625 \leq \text{FG} \leq 4.125 }
\end{aligned}
\]
\begin{aligned}
S_7 + 0 + 0 + 0 \leq S_{10} \leq S_7 + 5 + 5 + 5
&\Rightarrow
S_7 \leq S_{10} \leq S_7 + 15
\\[.5em]
&\Rightarrow
\frac{S_7}{10} \leq \frac{S_{10}}{10} \leq \frac{ S_7 + 15 }{10}
\\[.5em]
&\Rightarrow
\frac{S_7}{10} \leq \text{FG} \leq \frac{S_7 + 15}{10}
\\[.5em]
&\Rightarrow
\frac{ 26.25 }{10} \leq \text{FG} \leq \frac{41.25}{10}
\\[.5em]
&\Rightarrow
\boldsymbol{ 2.625 \leq \text{FG} \leq 4.125 }
\end{aligned}
\]