To solve this problem, we need to determine the maximum number of questions Bill could have answered correctly given the different types of questions and their respective point values.

Step 1: Understand the problem.

Bill scored a total of 39 points, and the points are awarded based on the following question types:

• Knowledge: 2 points per question
• Skill: 5 points per question
• Application: 7 points per question

We need to maximize the number of questions, which means Bill should answer as many knowledge questions (worth 2 points) as possible, since they give the fewest points, allowing him to answer more questions overall.

Step 2: Approach

• First, divide the total points (39) by the number of points per knowledge question (2) to see how many knowledge questions he can answer if he only answered these:

  $\frac{39}{2} = 19.5$ This suggests a maximum of 19 questions could be answered if only knowledge questions were involved. However, because other types of questions are worth more points, this number will decrease when including skill and application questions.

Step 3: Use combinations of question types.

We need to determine if it's possible to replace some knowledge questions with skill or application questions, without exceeding the total of 39 points, but also maximizing the number of questions.

Let's try combinations where a few higher-point questions are mixed in:

• If Bill answered 1 application question (7 points):

  $39 - 7 = 32 \, \text{points left}$

  $\frac{32}{2} = 16 \, \text{knowledge questions}$

  In this case, Bill answered 1 application question and 16 knowledge questions, totaling:

  $1 + 16 = 17 \, \text{questions}$

This is a total of 17 questions, which is the maximum he could answer, as adding another skill or application question would reduce the total number of questions.

Step 4: Conclusion

The maximum number of questions Bill could have answered correctly is 17. Therefore, the correct answer is B (17).

Do you want more details on this solution or have any other questions?

Here are 5 related questions to explore:

1. What if Bill scored 50 points instead? How would the number of questions change?
2. Can Bill score exactly 39 points with a different mix of skill and knowledge questions?
3. What would happen if a new question type worth 10 points was added?
4. How can you determine the optimal number of questions if the point values changed?
5. How would the solution change if Bill had answered fewer than 39 points?

Tip: Always start with the smallest point value when maximizing the number of items within a constraint like total points.