Question

In a class in which the final course grade depends entirely on the average of four equally weighted 100-point tests, Frank has scored 82 , 77 , and 88 on the first three. What range of scores on the fourth test will give Frank a B for the semester (an average between 80 and 89 , inclusive)? Assume that all test scores have a non-negative value.

Answer 1

The equalli weighted average can be calculated using the following equation:

`A=(s_1+s_2+s_3+s_4)/(4)`

Since we know the values for the first 3, we can substitute them:

`\begin{gathered} A=(82+77+88+s_4)/(4) \\ A=(247+s_4)/(4) \end{gathered}`

The higher the last score, the highre will be the average.

So, to get the range that would get Frank a B (average between 80 and 89, inclusive), we can calculate the score for the minimum average and for the maximum average.

That is, the range of the score will be the from the score that will give him 80 to the score that will give him 89.

Mathmatically, this is:

`\begin{gathered} B\colon \\ 80\le A\le89 \\ A\ge80 \\ A\le89 \end{gathered}`

Now, we can substitute A into them and solve for s4:

`\begin{gathered} A\ge80 \\ (247+s_4)/(4)\ge80 \\ 247+s_4\ge80\cdot4 \\ 247+s_4\ge320 \\ s_4\ge320-247 \\ s_4\ge73 \end{gathered}`

And:

`\begin{gathered} A\le89 \\ (247+s_4)/(4)\le89 \\ 247+s_4\le89\cdot4 \\ 247+s_4\le356 \\ s_4\le356-247 \\ s_4\le109 \end{gathered}`

This would be between 73 and 109, inclusive, but the score has a maximum of 100 points, so the maximum can't be 109, it is 100.

Thus, the range of scores is from 73 to 100, inclusive, that is:

`\lbrack73,100\rbrack`