To determine which variable expressions correctly represent the phrase "the quotient of a number and 3," we need to understand that a quotient refers to the result of a division. So, we are looking for expressions that indicate a number is being divided by 3.
Let's evaluate each expression given:
1. [tex]\( n \div 3 \)[/tex]:
- This expression directly uses the division symbol ([tex]\(\div\)[/tex]). It means [tex]\(n\)[/tex] divided by 3, which correctly represents the quotient of a number [tex]\(n\)[/tex] and 3.
2. [tex]\( 3n \)[/tex]:
- This expression means [tex]\(3\)[/tex] multiplied by [tex]\(n\)[/tex]. Multiplication is not the same as division, so this does not correctly represent the given phrase.
3. [tex]\( \frac{1}{3n} \)[/tex]:
- This expression means 1 divided by the product of [tex]\(3\)[/tex] and [tex]\(n\)[/tex]. This is not the same as dividing [tex]\(n\)[/tex] by 3, so it does not correctly represent the given phrase.
4. [tex]\( \frac{n}{3} \)[/tex]:
- This expression means [tex]\(n\)[/tex] divided by 3, written as a fraction. This correctly represents the quotient of a number [tex]\(n\)[/tex] and 3.
Given these evaluations, the correct variable expressions Jean could have written to represent "the quotient of a number and 3" are:
- [tex]\( n \div 3 \)[/tex]
- [tex]\( \frac{n}{3} \)[/tex]
So, the correct answers are:
1. [tex]\( n \div 3 \)[/tex]
4. [tex]\( \frac{n}{3} \)[/tex]
Therefore, the indices of these correct expressions are:
1, 4
