Problem 11 - Prealgebra

Question

Solve each of the following problems by first restating it as one of the three basic percent problems of Section 7.2 . In each case, be sure to show the equation. Voting In the 2004 Presidential election, George Bush received \(53.25 \%\) of the total electoral votes and John Kerry received \(46.75 \%\) of the total electoral votes. If there were 537 total votes cast by the Electoral College how many electoral votes did each candidate receive?

Step-by-Step Solution

Answer

George Bush received 286 votes; John Kerry received 251 votes.

Step 1: Identify Problem Type for George Bush

First, we restate the problem for George Bush using the basic percent problem: We need to find the part (electoral votes received by Bush), given the percentage (53.25%) and the whole (537 total votes). The problem is of the type "what is the part, given the part percent and the whole."

Step 2: Setup Equation for George Bush

Use the formula for finding the part: \( \text{Part} = \frac{\text{Percent}}{100} \times \text{Whole} \). For George Bush, this becomes: \[ \text{Votes for Bush} = \frac{53.25}{100} \times 537 \]

Step 3: Calculate Electoral Votes for George Bush

Perform the calculations: \[ \text{Votes for Bush} = 0.5325 \times 537 = 285.5025 \]. Rounding to the nearest whole number, since votes cannot be fractional, Bush received 286 votes.

Step 4: Identify Problem Type for John Kerry

Similarly, restate the problem for John Kerry using the basic percent problem: We need to find the part (electoral votes received by Kerry), given the percentage (46.75%) and the whole (537 total votes).

Step 5: Setup Equation for John Kerry

Use the same formula for finding the part: \( \text{Part} = \frac{\text{Percent}}{100} \times \text{Whole} \). For John Kerry, this becomes: \[ \text{Votes for Kerry} = \frac{46.75}{100} \times 537 \]

Step 6: Calculate Electoral Votes for John Kerry

Perform the calculations: \[ \text{Votes for Kerry} = 0.4675 \times 537 = 251.4975 \]. Rounding to the nearest whole number, Kerry received 251 votes.

Key Concepts

Understanding the Percent Equation

When dealing with percentage problems, the percent equation is a powerful tool to understand the relationship between parts, wholes, and percentages. In each case,

• the part is what we are trying to find,
• the percent is a given percentage,
• the whole is a known total amount.

To find the part using the percent and whole, the equation can be represented as:\[ \text{Part} = \frac{\text{Percent}}{100} \times \text{Whole} \]This equation allows us to calculate quantities directly by plugging in the specific values. For transaction simplicity, always ensure to convert the percentage to its decimal form by dividing it by 100. The conversion step is crucial for finding the precise value of a part when given a whole and its percentage, as illustrated in the calculation for electoral votes in the 2004 Presidential election problem.

Exploring Electoral Votes

The electoral vote system in the United States is a unique process employed during presidential elections. Unlike popular votes, where each vote contributes directly to a candidate's tally, electoral votes are assigned based on rules that vary by state.

• Each state has a certain amount of electoral votes, which contributes to a total (like the 537 votes in the exercise).
• The number of electoral votes per state is relative to its population and congressional representation.
• To win an election, a candidate generally must secure a majority of the electoral votes.

In the provided exercise, the electoral votes given for Bush and Kerry are determined by calculating a fraction of the whole, matched against their respective percentages. Understanding how the electoral college functions is essential for grasping how presidential winners are decided beyond just the popular vote counts. This knowledge provides a real-world application to our percent equation use case.

Percentage Calculation Breakdown

Calculating percentages involves an essential mathematical skill that allows comparisons and proportional evaluations across different contexts. When performing any percentage calculation, the steps generally include:

• Identify the desired outcome, which is often the part we seek.
• Refer to a known whole, which is the total or maximum possible quantity.
• Use the given percent value, expressed as a number between 0 and 100, and convert it to decimal form for calculation ease.

For example, in the context of electoral votes discussed earlier, these calculations determine how many votes one receives based on their share of a fixed total. By substituting relevant values into the percent equation, anyone can find the accurate parts to corresponding wholes, verifying real-world quantities like votes and other statistical data. Moreover, rounding can also play a critical role, as seen in the need to round electoral votes to the nearest whole number since fractional votes are not feasible in this context.