$\left(-\frac{6}{7}+3\right)+-\frac{3}{14}$

$-1 \frac{15}{16}+\left(\frac{1}{8}+\frac{1}{4}\right)$

$\left(1 \frac{5}{12}+3 \frac{2}{5}\right)+\left(-\frac{5}{8}\right)$

Mrs. Matthews is cooking ham for Sunday dinner. How many $\frac{3}{8}$-pound servings are there in a 9-pound ham?

Mr. Seeley has $\frac{3}{4}$ gallons of milk in a pitcher. If each glass holds $\frac{1}{16}$ gallon, how many glasses can he fill?

Lauren has 9 feet of wire. How many $\frac{2}{3}$-foot pieces can she cut? How much wire will be leftover?

Dean Karnazes achieved his goal by completing the 26.2-mile New York City marathon in $3 \frac{1}{2}$ hours. What was his average rate for this race?

$\boxed{23. \frac{27}{14}, 24. -\frac{25}{16}, 25. \frac{107}{30}, 26. 24, 27. 12, 28. 13 \text{ pieces, } \frac{1}{3} \text{ ft left}, 29. \frac{262}{35}}$

Explanation

Analysis and Methodology

The problems involve performing arithmetic operations with rational numbers, division of quantities, and rate calculations. We will process each part systematically using standard rules for fractions and mixed numbers.

Arithmetic with Rational Numbers (23-25)

Question 23: We solve $\left(-\frac{6}{7}+3\right) + \left(-\frac{3}{14}\right)$.

First, we find a common denominator for the terms inside the first parenthesis: $\left(-\frac{6}{7} + \frac{21}{7}\right) = \frac{15}{7}$. Then we add the remaining term: $\frac{15}{7} - \frac{3}{14} = \frac{30}{14} - \frac{3}{14} = \frac{27}{14}$.

Question 24: We solve $-1 \frac{15}{16} + \left(\frac{1}{8} + \frac{1}{4}\right)$.

Convert the mixed number to an improper fraction: $-1 \frac{15}{16} = -\frac{31}{16}$. Inside the parenthesis: $\frac{1}{8} + \frac{2}{8} = \frac{3}{8}$. Now perform the addition: $-\frac{31}{16} + \frac{6}{16} = -\frac{25}{16}$ (or $-1 \frac{9}{16}$).

Question 25: We solve $\left(1 \frac{5}{12} + 3 \frac{2}{5}\right) + \left(-\frac{5}{8}\right)$.

Convert mixed numbers: $\frac{17}{12} + \frac{17}{5} = \frac{85 + 204}{60} = \frac{289}{60}$. Now subtract $\frac{5}{8}$: $\frac{289}{60} - \frac{5}{8} = \frac{578 - 150}{120} = \frac{428}{120} = \frac{107}{30} = 3 \frac{17}{30}$.

Division, Rate, and Remainder Problems (26-29)

Question 26: To find the number of $\frac{3}{8}$-pound servings in 9 pounds of ham, divide: $9 \div \frac{3}{8} = 9 \times \frac{8}{3} = 3 \times 8 = 24$. There are 24 servings.
Question 27: To find how many $\frac{1}{16}$-gallon glasses fit in $\frac{3}{4}$ gallon: $\frac{3}{4} \div \frac{1}{16} = \frac{3}{4} \times 16 = 3 \times 4 = 12$. There are 12 glasses.
Question 28: For 9 feet of wire cut into $\frac{2}{3}$-foot pieces: $9 \div \frac{2}{3} = 9 \times \frac{3}{2} = 13.5$. She can cut 13 full pieces. The leftover wire is $9 - (13 \times \frac{2}{3}) = 9 - \frac{26}{3} = \frac{27 - 26}{3} = \frac{1}{3}$ foot.
Question 29: Average rate = Distance / Time. Distance = 26.2 miles, Time = $3 \frac{1}{2}$ hours = 3.5 hours. Rate = $26.2 \div 3.5 \approx 7.49$ mph. Expressed exactly: $26.2 \div \frac{7}{2} = 26.2 \times \frac{2}{7} = \frac{52.4}{7} = \frac{524}{70} = \frac{262}{35}$ mph.

Examples

Imagine you are helping a chef organize their kitchen. Using these calculations, you can determine exactly how many portions of a meal can be created from a large ingredient supply, or calculate the average cooking speed needed to finish a meal before a deadline. Understanding how to handle fractions and division ensures that ingredients are not wasted and time is used efficiently, showing that basic math is a powerful tool for daily productivity and planning.