Let's solve the problem step-by-step to determine how many [tex]\(\frac{3}{8}\)[/tex]-pint servings of vinegar Ms. Nellies can provide to the students.
1. Convert the Vinegar Quantities into Improper Fractions:
- Ms. Nellies has [tex]\(2 \frac{1}{2}\)[/tex] pints of vinegar. We convert this mixed number into an improper fraction:
[tex]\[
2 \frac{1}{2} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}
\][/tex]
2. Determine the Serving Size:
- Each group of students needs [tex]\(\frac{3}{8}\)[/tex] pint of vinegar.
3. Set Up the Division:
- To find out how many [tex]\(\frac{3}{8}\)[/tex]-pint servings are in [tex]\(\frac{5}{2}\)[/tex] pints of vinegar, we divide [tex]\(\frac{5}{2}\)[/tex] by [tex]\(\frac{3}{8}\)[/tex].
[tex]\[
\frac{5}{2} \div \frac{3}{8}
\][/tex]
4. Perform the Division:
- When dividing by a fraction, we multiply by its reciprocal. So, we take the reciprocal of [tex]\(\frac{3}{8}\)[/tex], which is [tex]\(\frac{8}{3}\)[/tex], and multiply:
[tex]\[
\frac{5}{2} \times \frac{8}{3} = \frac{5 \times 8}{2 \times 3} = \frac{40}{6} = \frac{20}{3} \approx 6.6667
\][/tex]
5. Interpret the Answer:
- The improper fraction [tex]\(\frac{20}{3}\)[/tex] simplifies to approximately [tex]\(6.67\)[/tex], which means Ms. Nellies can provide around 6.67 [tex]\(\frac{3}{8}\)[/tex]-pint servings of vinegar with the amount she has.
Next, let's address Trey's number line model and equation:
- The given volume is [tex]\(\frac{5}{2}\)[/tex] pints, and the serving size is [tex]\(\frac{3}{8}\)[/tex] pint, making the correct equation [tex]\(2 \frac{1}{2} \div \frac{3}{8}\)[/tex].
Finally, summarizing the conclusions for the question:
- Trey's number line model is correct because it represents the expression [tex]\(2 \frac{1}{2} \div \frac{3}{8}\)[/tex].
- Ms. Nellies has enough vinegar for approximately [tex]\(6.67\)[/tex] groups of students.
Therefore, the final answers for the blanks are:
Trey's number line model is correct because it represents the expression [tex]\(2 \frac{1}{2} \div \frac{3}{8}\)[/tex]. The correct equation is [tex]\(2 \frac{1}{2} \div \frac{3}{8}\)[/tex]. Ms. Nellies has enough vinegar for approximately [tex]\(6.67\)[/tex] groups of students.
