To determine which algebraic expression represents the given word description, "The quotient of nine and the sum of a number and one," let's break down the statement step by step.
### Step-by-Step Solution:
1. Identify the key components in the word description:
- Quotient: This term means division.
- Nine: The numerator in our division.
- Sum of a number and one: We can represent the number as [tex]\( x \)[/tex], so the sum "a number and one" translates to [tex]\( x + 1 \)[/tex].
2. Build the algebraic expression:
- Since the word "quotient" indicates division, and we need to use nine as the numerator, with [tex]\( x + 1 \)[/tex] as the denominator, the algebraic expression will be:
[tex]\[
\frac{9}{x+1}
\][/tex]
3. Compare with the given choices:
- Choice A: [tex]\(\frac{9}{x} + 1\)[/tex] - This is not correct since it is the quotient of nine and [tex]\( x \)[/tex], then adding 1.
- Choice B: [tex]\(\frac{x}{9} + 1\)[/tex] - This is not correct since it represents the quotient of [tex]\( x \)[/tex] and nine, then adding 1.
- Choice C: [tex]\(\frac{x+1}{9}\)[/tex] - This is not correct since it represents the quotient of [tex]\( x + 1 \)[/tex] and nine.
- Choice D: [tex]\(\frac{9}{x + 1}\)[/tex] - This correctly represents the quotient of nine and the sum of [tex]\( x \)[/tex] and one.
### Conclusion:
The algebraic expression that represents the word description "The quotient of nine and the sum of a number and one" is:
[tex]\[
\boxed{\frac{9}{x+1}}
\][/tex]
Therefore, the correct choice is:
D. [tex]\(\frac{9}{x + 1}\)[/tex]
