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.