To determine which algebraic expression represents the given phrase, let's break down the phrase step by step and construct the correct mathematical expression.
The phrase is:
"five times the sum of a number and eleven, divided by three times the sum of the number and eight."
1. Understanding the phrase:
- "A number" is typically represented by a variable, usually [tex]\( x \)[/tex].
- "The sum of a number and eleven" means [tex]\(x + 11\)[/tex].
- "Five times the sum of a number and eleven" translates to [tex]\( 5 \times (x + 11) \)[/tex].
2. Next part of the phrase:
- "The sum of the number and eight" means [tex]\( x + 8 \)[/tex].
- "Three times the sum of the number and eight" translates to [tex]\( 3 \times (x + 8) \)[/tex].
3. Putting it all together:
- The entire phrase "five times the sum of a number and eleven, divided by three times the sum of the number and eight" translates to:
[tex]\[
\frac{5 \times (x + 11)}{3 \times (x + 8)}
\][/tex]
Now, let's compare this with the given choices:
1. [tex]\( 5(x+11) + 3(x+8) \)[/tex]:
- This expression represents adding [tex]\(5(x + 11)\)[/tex] and [tex]\(3(x + 8)\)[/tex], which does not involve division.
2. [tex]\( \frac{5 x + 11}{3 x + 8} \)[/tex]:
- This expression represents dividing [tex]\(5x + 11\)[/tex] by [tex]\(3x + 8\)[/tex], which does not capture the multiplication inside the sums as per the phrase.
3. [tex]\( \frac{5(x+11)}{3(x+8)} \)[/tex]:
- This expression accurately represents "five times the sum of a number and eleven, divided by three times the sum of the number and eight."
4. [tex]\( 5x + 11 + 3x + 8 \)[/tex]:
- This expression represents adding the individual terms [tex]\(5x + 11 + 3x + 8\)[/tex], which is not the correct interpretation either.
Thus, the correct algebraic expression that represents the given phrase is:
[tex]\[
\boxed{\frac{5(x+11)}{3(x+8)}}
\][/tex]
