To determine the correct mathematical expression that represents the phrase "six more than the quotient of three and a number, decreased by eight," let's break down the phrase step-by-step:
1. Quotient of three and a number (n):
This describes a fraction where three is divided by [tex]\( n \)[/tex]. Mathematically, this is represented as:
[tex]\[
\frac{3}{n}
\][/tex]
2. Six more than the quotient:
Here we add 6 to the quotient obtained in the previous step. Therefore, we have:
[tex]\[
6 + \frac{3}{n}
\][/tex]
3. Decreased by eight:
Finally, we need to subtract 8 from the expression obtained in the previous step:
[tex]\[
6 + \frac{3}{n} - 8
\][/tex]
So, the expression that correctly represents the phrase is:
[tex]\[
6 + \frac{3}{n} - 8
\][/tex]
Among the given options, this matches the first option:
[tex]\[
6 + \frac{3}{n} - 8
\][/tex]
Therefore, the correct expression is:
[tex]\[
\boxed{6 + \frac{3}{n} - 8}
\][/tex]
