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: $\frac{9}{77}$

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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: $28 \frac{4}{7}$

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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: $-36$

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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: $11 \frac{11}{24}$

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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: $1 \frac{5}{12}$

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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

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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

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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

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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.