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I'll solve a few of the problems here, starting from the top. Let's go through the initial ones together.

### Problem 1
> The ratio of good guys to bad guys was 15 to 4. If there were 75 good guys, how many bad guys were there?

1. **Set up the proportion**:
   \[
   \frac{\text{good guys}}{\text{bad guys}} = \frac{15}{4}
   \]
   
2. **Use 75 as the number of good guys**:
   \[
   \frac{75}{\text{bad guys}} = \frac{15}{4}
   \]

3. **Solve for the number of bad guys**:
   Cross-multiply:
   \[
   75 \times 4 = 15 \times \text{bad guys}
   \]
   \[
   300 = 15 \times \text{bad guys}
   \]
   \[
   \text{bad guys} = \frac{300}{15} = 20
   \]

**Answer**: There were 20 bad guys.

---

### Problem 2
> The ratio of blue marbles to green marbles was 17 to 5. If there were 110 marbles in the bag, how many were blue marbles?

1. **Let \( x \) be the number of blue marbles** and \( y \) be the number of green marbles.

2. Set up the system using the total and the ratio:
   \[
   x + y = 110
   \]
   \[
   \frac{x}{y} = \frac{17}{5}
   \]

3. **Express \( y \) in terms of \( x \)**:
   \[
   y = \frac{5}{17}x
   \]

4. Substitute \( y = \frac{5}{17}x \) into \( x + y = 110 \):
   \[
   x + \frac{5}{17}x = 110
   \]
   \[
   \frac{22x}{17} = 110
   \]
   \[
   x = \frac{110 \times 17}{22} = 85
   \]

**Answer**: There were 85 blue marbles.

---

### Problem 3
> Bob and Nigel found that 4 times the sum of a number and -5 equals 24. What is the number?

1. **Set up the equation**:
   \[
   4(x + (-5)) = 24
   \]
   Simplify:
   \[
   4(x - 5) = 24
   \]

2. **Solve for \( x \)**:
   Divide both sides by 4:
   \[
   x - 5 = 6
   \]
   Add 5 to both sides:
   \[
   x = 11
   \]

**Answer**: The number is 11.

---

### Problem 4
> If the product of a number and -5 is reduced by 2, the result is 26 less than twice the opposite of the number. What is the number?

1. **Set up the equation**:
   Let the number be \( x \). Then, the equation is:
   \[
   -5x - 2 = 2(-x) - 26
   \]

2. **Simplify and solve for \( x \)**:
   Expand and simplify:
   \[
   -5x - 2 = -2x - 26
   \]
   Add \( 5x \) to both sides:
   \[
   -2 = 3x - 26
   \]
   Add 26 to both sides:
   \[
   24 = 3x
   \]
   Divide by 3:
   \[
   x = 8
   \]

**Answer**: The number is 8.

---

Would you like more details on any of these solutions, or help with the next set of problems?

---

Here are 5 related questions to expand on these types of problems:

1. How do you set up a proportion when given two parts and the whole?
2. What steps can be taken to solve ratios with multiple unknowns?
3. How can you create an equation from a word problem involving "the product of a number"?
4. Why is it necessary to isolate variables when solving equations?
5. What other methods can be used to solve proportions besides cross-multiplying?

**Tip**: When translating word problems to equations, identify keywords (like "product," "sum," and "difference") to understand the operations involved.

A variety of algebraic problems, including ratios, inequalities, functions, simplifying expressions, and solving equations.

This problem set includes solutions for a variety of algebra topics, such as ratios, inequalities, functions, simplification of expressions, and solving equations. Perfect for students in Grades 8-10, covering essential algebraic concepts and step-by-step solutions.

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