Problem 44 - Elementary Algebra

Question

If 2 out of every 7 voters approve of a sales tax increase then determine the number of voters out of the 588 surveyed who do not support the increase.

Step-by-Step Solution

Answer

420 voters do not support the increase.

Step 1: Identify the fraction who approve

We are given that 2 out of every 7 voters approve of the sales tax increase. This means that the fraction of voters who approve is \( \frac{2}{7} \).

Step 2: Determine the fraction who do not approve

Since 2 out of 7 approve, 5 out of 7 do not approve (as there are 7 voters in total). Hence, the fraction of voters who do not approve is \( \frac{5}{7} \).

Step 3: Calculate the number of voters who do not approve

To find the number of surveyed voters who do not support the increase, multiply the total number of surveyed people by the fraction who do not approve: \[ \frac{5}{7} \times 588 \] Calculate this to get the number of voters who do not support the increase.

Step 4: Perform the multiplication

Carrying out the multiplication, we have:\[ \frac{5}{7} \times 588 = \frac{5 \times 588}{7} = \frac{2940}{7} = 420 \]Therefore, 420 voters do not support the sales tax increase.

Key Concepts

Ratios

Understanding ratios is fundamental in many real-world applications. A ratio is a way to compare two quantities by showing the relative size of one quantity to another. They express how much of one thing there is compared to another. In this context, if 2 out of every 7 voters approve a change, it can be represented as a simple ratio of 2:7.

• The first number (2) refers to the part of the whole that agrees or approves.
• The second number (7) indicates the total parts under consideration, which in this case is all voters.

Ratios can also be expressed as fractions, decimals, or even percentages. In our example, a ratio of 2:7 converts to a fraction, \( \frac{2}{7} \), showing clear proportion. Ratios are particularly useful when you want to predict outcomes for different sample sizes from the same population.

Voter Survey

Surveys are a common method to gauge public opinion or gather information from individuals within a specific group. In this exercise, a voter survey involves collecting opinions from a group of people regarding a sales tax increase. The main purpose is to understand how many agree versus how many disagree. This data often guides decision-makers.

• Sample Size: In our context, 588 people form the sample size, representing the larger population.
• Survey Results: The survey setup reveals that only 2 out of every 7 surveyed support the increase.
• Results Interpretation: These ratios provide a quick glance at sentiment without needing every single opinion.

Understanding the surveyed data helps in extrapolating the behavior of the larger group by using fractions derived from ratios.

Percentage Calculation

Calculating percentages is very useful for comprehending part-to-whole relationships. It's an essential tool used to express ratios and fractions in an easily digestible form, particularly in surveys and statistical data.To find the percentage of voters in favor or against something, you can convert the related fraction (or ratio) into a percentage.For instance, if \( \frac{2}{7} \) voters approve, we can find this percentage by converting the fraction. Here's how:

• Multiply the fraction by 100:

\[\frac{2}{7} \times 100 = 28.57\]This calculation shows that approximately 28.57% of the voters support the tax increase. Similarly, the percentage against it, using \( \frac{5}{7} \) who do not support, would be:

• \[\frac{5}{7} \times 100 = 71.43\]This shows 71.43% are not in favor.

Such calculations allow people to understand proportions in an intuitive way, which is invaluable for presentations and decision making.