To determine which expression represents "the quotient of nine times a number and seven," let's break down the phrase step by step:
1. Understanding the Terms:
- "Nine times a number": If we denote the unknown number as [tex]\( n \)[/tex], then "nine times a number" translates to [tex]\( 9n \)[/tex].
- "Quotient": The quotient refers to the result of division.
2. Phrase Interpretation:
- The complete phrase "the quotient of nine times a number and seven" means we need to divide [tex]\( 9n \)[/tex] by 7.
3. Translating to an Expression:
- The mathematical expression for dividing [tex]\( 9n \)[/tex] by 7 is [tex]\( \frac{9n}{7} \)[/tex].
4. Matching with Given Options:
- [tex]\( 9n - 7 \)[/tex]: This represents nine times a number minus seven, and is not a division.
- [tex]\( 9(n + 7) \)[/tex]: This represents nine times the sum of a number and seven, which isn't a division either.
- [tex]\( 9n \div 7 \)[/tex]: This properly represents nine times a number divided by seven.
Therefore, the correct expression corresponding to "the quotient of nine times a number and seven" is:
[tex]\[ \boxed{9n \div 7} \][/tex]
Thus, the correct answer is the third option.
