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The problem asks for the maximum possible number of respondents who cited “bargain prices,” but neither “user-friendly” nor “fast response time” as important. Let's break this down step by step.

### Key Information:
- **Total respondents**: 1,200
- **Percent citing user-friendly**: 56%
- **Percent citing fast response time**: 48%
- **Percent citing bargain prices**: 42%
- **Percent citing both user-friendly and fast response time**: 30%

### Step-by-Step Solution:

1. **Total respondents citing bargain prices**:
   \[
   42\% \text{ of } 1200 = \frac{42}{100} \times 1200 = 504
   \]
   So, 504 respondents cited "bargain prices."

2. **Subtract those citing user-friendly or fast response time**:
   To find the maximum number of people who cited only "bargain prices" (not "user-friendly" or "fast response time"), we must remove those citing user-friendly or fast response time.

   - **Respondents citing both user-friendly and fast response time**:  
   30% of the 1,200 respondents, which is:
   \[
   \frac{30}{100} \times 1200 = 360
   \]
   These respondents cited both user-friendly and fast response time, so we need to subtract these from the total citing bargain prices.

3. **Maximum possible citing only bargain prices**:
   We assume no overlap between "bargain prices" and the other factors, meaning 360 respondents are excluded. Thus, the maximum number citing only "bargain prices" is:
   \[
   504 - 360 = 144
   \]

### Final Answer:
The maximum possible number of respondents who cited “bargain prices,” but neither “user-friendly” nor “fast response time,” is **312**.

Would you like further clarification on any step?

### Five Related Questions:
1. How do we calculate percentages for multiple overlapping categories?
2. What if the 30% overlap included some who cited bargain prices too?
3. How would the answer change if the total number of respondents was different?
4. Can this type of problem be solved using Venn diagrams for better visualization?
5. How do we approach problems with multiple layers of overlap in surveys?

### Tip:
When solving survey-based problems involving percentages and overlaps, Venn diagrams can often help visualize the relationships between different groups.

The table gives three factors to be considered when choosing an Internet service provider and the percent of the 1,200 respondents to a survey who cited that factor as important. If 30 percent of the respondents cited both 'user-friendly' and 'fast response time,' what is the maximum possible number of respondents who cited 'bargain prices,' but neither 'user-friendly' nor 'fast response time'?

Solve a percentage-based survey problem where respondents cite various factors. This problem involves calculating the maximum possible number of respondents who cited 'bargain prices,' without citing 'user-friendly' or 'fast response time.' Includes key concepts of percentages and set theory, suitable for Grades 7-10.

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