Math Problem Statement

Björn took a poll for an upcoming local election and found that 27% of voters were likely to vote for the incumbent. His poll has a 4% margin of error.

Which of the following equations represents the likely minimum and maximum percentages of voters, v , that will vote for the incumbent?

Choose TWO correct answers A.|27−4|=v

B.|4−v|=27

C.|4−27|=v

D.|v−27|=4

E.|v−4|=27

F.|27−v|=4

Solution

In this problem, the margin of error tells us that the actual percentage of voters likely to vote for the incumbent could be as much as 4% higher or lower than the reported 27%. We need to find two equations that represent this situation.

Let's break it down:

  1. Maximum likely percentage:
    The highest percentage vv could be is 27+4=3127 + 4 = 31%.

  2. Minimum likely percentage:
    The lowest percentage vv could be is 274=2327 - 4 = 23%.

The problem is asking for equations that express the relationship between vv and the given numbers, taking into account the margin of error.

Now, let's check the options:

  • Option A: 274=v|27 - 4| = v
    This simplifies to 23=v|23| = v, or v=23v = 23, which matches the minimum percentage. This is a correct option.

  • Option B: 4v=27|4 - v| = 27
    This equation does not logically represent the situation, as it suggests that the difference between 4 and vv is 27, which doesn't relate to the problem. This is not correct.

  • Option C: 427=v|4 - 27| = v
    This simplifies to 23=v|−23| = v, or v=23v = 23, which matches the minimum percentage, but it is redundant with Option A and does not cover the full range. This is not correct.

  • Option D: v27=4v - 27 = 4
    This represents the maximum percentage, since v=27+4=31v = 27 + 4 = 31. This is a correct option.

  • Option E: v4=27v - 4 = 27
    This would imply v=31v = 31, but this is not quite the right setup for the margin of error situation. This is not correct.

  • Option F: 27v=427 - v = 4
    This represents the minimum percentage, since v=274=23v = 27 - 4 = 23. This is a correct option.

Correct answers:

  • A: 274=v|27 - 4| = v
  • D: v27=4v - 27 = 4
  • F: 27v=427 - v = 4

Do you have any questions or want more details?

Related questions:

  1. How is the margin of error typically calculated in polls?
  2. What is the meaning of the absolute value notation x|x| in real-life contexts?
  3. What would the result be if the margin of error increased to 5%?
  4. How do we handle rounding when dealing with percentages and polling?
  5. What other statistical factors affect the accuracy of poll results?

Tip: When working with percentages, always keep an eye on possible rounding errors, especially in large datasets.

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Math Problem Analysis

Mathematical Concepts

Algebra
Absolute Value
Linear Equations

Formulas

v = 27 ± 4
|27 - 4| = v
v - 27 = 4
27 - v = 4

Theorems

Absolute Value Properties
Linear Equation Solving

Suitable Grade Level

Grades 8-10