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Determining Relative Frequencies as Percentages

Residents of three counties were polled to find the percentage of residents who are Republican or Democrat.
\begin{tabular}{|c|c|c|c|}
\cline { 2 - 4 } \multicolumn{1}{c|}{} & Democrat & Republican & Total \\
\hline \begin{tabular}{c}
County \\
[tex]$1$[/tex]
\end{tabular} & 4,050 & 2,470 & 6,520 \\
\hline \begin{tabular}{c}
County \\
[tex]$2$[/tex]
\end{tabular} & 3,925 & 6,010 & 9,935 \\
\hline \begin{tabular}{c}
County \\
[tex]$3$[/tex]
\end{tabular} & 2,100 & 5,950 & 8,050 \\
\hline Total & 10,075 & 14,430 & 24,505 \\
\hline
\end{tabular}

Find the following relative frequencies to the nearest tenth of a percent.

1. Republican residents of County 1: [tex]$\square$[/tex] [tex]$\%$[/tex]

2. Democratic residents of County 1: [tex]$\square$[/tex] [tex]$\%$[/tex]

3. Total Republican residents: [tex]$\square$[/tex] [tex]$\%$[/tex]

4. Total Democratic residents: [tex]$\square$[/tex] [tex]$\%$[/tex]



Answer :

GinnyAnswer
Let's solve the problem step by step.

### Given Data
- County 1:
- Democrat: 4050
- Republican: 2470
- Total: 6520

- County 2:
- Democrat: 3925
- Republican: 6010
- Total: 9935

- County 3:
- Democrat: 2100
- Republican: 5950
- Total: 8050

- Overall Totals:
- Total Democrats: 10075
- Total Republicans: 14430
- Total Population: 24505

### Step-by-Step Calculations

#### 1. Relative Frequency for Republican Residents in County 1
We need to find the percentage of Republican residents in County 1 relative to the total population of County 1.
[tex]\[ \text{Relative Frequency} = \left(\frac{\text{Number of Republicans in County 1}}{\text{Total Population of County 1}}\right) \times 100 \][/tex]

Substitute the given values:
[tex]\[ \text{Relative Frequency} = \left(\frac{2470}{6520}\right) \times 100 \][/tex]

Calculating this,
[tex]\[ \text{Relative Frequency} = 37.9\% \][/tex]

So, the percentage of Republican residents in County 1 is [tex]\(37.9\%\)[/tex].

#### 2. Relative Frequency for Democratic Residents in County 1
Similarly, we need to find the percentage of Democratic residents in County 1 relative to the total population of County 1.
[tex]\[ \text{Relative Frequency} = \left(\frac{\text{Number of Democrats in County 1}}{\text{Total Population of County 1}}\right) \times 100 \][/tex]

Substitute the given values:
[tex]\[ \text{Relative Frequency} = \left(\frac{4050}{6520}\right) \times 100 \][/tex]

Calculating this,
[tex]\[ \text{Relative Frequency} = 62.1\% \][/tex]

So, the percentage of Democratic residents in County 1 is [tex]\(62.1\%\)[/tex].

#### 3. Total Republican Residents as a Percentage of the Total Population
We need to find the percentage of the total Republican residents relative to the total population.
[tex]\[ \text{Relative Frequency} = \left(\frac{\text{Total Number of Republicans}}{\text{Total Population}}\right) \times 100 \][/tex]

Substitute the given values:
[tex]\[ \text{Relative Frequency} = \left(\frac{14430}{24505}\right) \times 100 \][/tex]

Calculating this,
[tex]\[ \text{Relative Frequency} = 58.9\% \][/tex]

So, the percentage of total Republican residents is [tex]\(58.9\%\)[/tex].

#### 4. Total Democratic Residents as a Percentage of the Total Population
We need to find the percentage of the total Democratic residents relative to the total population.
[tex]\[ \text{Relative Frequency} = \left(\frac{\text{Total Number of Democrats}}{\text{Total Population}}\right) \times 100 \][/tex]

Substitute the given values:
[tex]\[ \text{Relative Frequency} = \left(\frac{10075}{24505}\right) \times 100 \][/tex]

Calculating this,
[tex]\[ \text{Relative Frequency} = 41.1\% \][/tex]

So, the percentage of total Democratic residents is [tex]\(41.1\%\)[/tex].

### Summary of Results
- Republican residents of County 1: [tex]\(37.9\%\)[/tex]
- Democratic residents of County 1: [tex]\(62.1\%\)[/tex]
- Total Republican residents: [tex]\(58.9\%\)[/tex]
- Total Democratic residents: [tex]\(41.1\%\)[/tex]

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