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Answer 9 through 16 with simplified fractions

Let's go through questions 9 to 16 one by one, simplifying fractions where necessary.

### 9. 
\[
-\frac{3}{7} + \frac{6}{11}
\]
To add these, we need a common denominator. The least common denominator (LCD) between 7 and 11 is 77.

\[
-\frac{3}{7} = -\frac{33}{77}, \quad \frac{6}{11} = \frac{42}{77}
\]

Now, adding these:

\[
-\frac{33}{77} + \frac{42}{77} = \frac{9}{77}
\]

**Answer: \(\frac{9}{77}\)**

---

### 10.
\[
7 \frac{1}{7} \times 4
\]
First, convert the mixed number to an improper fraction:

\[
7 \frac{1}{7} = \frac{50}{7}
\]

Now, multiply:

\[
\frac{50}{7} \times 4 = \frac{200}{7}
\]

As a mixed number, this is:

\[
28 \frac{4}{7}
\]

**Answer: \(28 \frac{4}{7}\)**

---

### 11.
\[
-20 \div \frac{5}{9}
\]
Dividing by a fraction is the same as multiplying by its reciprocal:

\[
-20 \times \frac{9}{5} = -\frac{180}{5} = -36
\]

**Answer: \(-36\)**

---

### 12.
\[
10 \frac{5}{6} + \frac{5}{8}
\]
Convert the mixed number to an improper fraction:

\[
10 \frac{5}{6} = \frac{65}{6}
\]

Now, find a common denominator between 6 and 8, which is 24.

\[
\frac{65}{6} = \frac{260}{24}, \quad \frac{5}{8} = \frac{15}{24}
\]

Now, add the fractions:

\[
\frac{260}{24} + \frac{15}{24} = \frac{275}{24} = 11 \frac{11}{24}
\]

**Answer: \(11 \frac{11}{24}\)**

---

### 13.
Bill's ladder is \(8 \frac{1}{2}\) feet tall, and there are 6 equal spaces between the rungs.

Convert the mixed number to an improper fraction:

\[
8 \frac{1}{2} = \frac{17}{2}
\]

To find the size of each space, divide by 6:

\[
\frac{17}{2} \div 6 = \frac{17}{2} \times \frac{1}{6} = \frac{17}{12}
\]

As a mixed number, this is:

\[
1 \frac{5}{12}
\]

**Answer: \(1 \frac{5}{12}\)**

---

### 14.
Mrs. Jones has \(3 \frac{3}{5}\) pounds of fruit and splits it into portions of \( \frac{2}{5} \) pounds each.

Convert the mixed number to an improper fraction:

\[
3 \frac{3}{5} = \frac{18}{5}
\]

Now, divide:

\[
\frac{18}{5} \div \frac{2}{5} = \frac{18}{5} \times \frac{5}{2} = \frac{90}{10} = 9
\]

**Answer: 9 portions**

---

### 15.
A relay race covers \(3 \frac{1}{2}\) miles, and each runner runs \( \frac{1}{4} \) mile.

Convert the mixed number to an improper fraction:

\[
3 \frac{1}{2} = \frac{7}{2}
\]

Now, divide:

\[
\frac{7}{2} \div \frac{1}{4} = \frac{7}{2} \times 4 = 14
\]

**Answer: 14 runners**

---

### 16.
The concert lasts 27 minutes and 30 seconds, and there are 15 songs of equal length.

First, convert the total time to seconds:

\[
27 \, \text{minutes} = 27 \times 60 = 1620 \, \text{seconds}
\]
\[
27 \, \text{minutes} \, 30 \, \text{seconds} = 1620 + 30 = 1650 \, \text{seconds}
\]

Now, divide the total time by 15:

\[
\frac{1650}{15} = 110 \, \text{seconds}
\]

Convert 110 seconds back to minutes:

\[
110 \, \text{seconds} = 1 \, \text{minute} \, 50 \, \text{seconds}
\]

**Answer: 1 minute and 50 seconds per song**

---

Let me know if you'd like more details or explanations on any of the steps! Here are five related questions for practice:

1. How would you simplify \(\frac{7}{9} + \frac{2}{3}\)?
2. What is \(-5 \div \frac{2}{7}\)?
3. If a ladder is 12 feet tall and has 8 equal spaces, how large is each space?
4. How do you convert 15 \(\frac{3}{4}\) into an improper fraction?
5. A concert has 10 songs and lasts 40 minutes. How long is each song?

**Tip:** When dividing by a fraction, always multiply by its reciprocal.

Learn how to solve math problems involving fraction and mixed number operations, including addition, subtraction, division, and multiplication. Problems 9-16 cover key concepts such as simplifying fractions and working with mixed numbers. Suitable for Grades 6-8.

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