Problem 51 - Elementary Algebra Within Reach

Question

Polling Results In a poll, 624 people from a sample of 1100 indicated they would vote for the Republican candidate. How many votes can the candidate expect to receive from 40,000 votes cast?

Step-by-Step Solution

Answer

The predicted number of votes for the Republican candidate based on the poll data would be calculated using the proportion of the polling sample that indicated they would vote for the Republican candidate multiplied by the total number of votes.

Step 1: Calculate The Proportions

Firstly, find the proportion of people that would vote for the Republican candidate. This is done by dividing the number of people who would vote for the candidate by the total number polled. In mathematics symbols, the proportion would be calculated as: proportion = \( \frac{624}{1100} \)

Step 2: Apply The Proportion

Next, this proportion is applied to the total number of votes to determine how many the candidate could expect. This is done by multiplying the proportion by the total number of votes. In mathematical terms: expected_votes = proportion * total_votes. In this case, the total_votes is given as 40,000.

Step 3: Calculate The Expected Votes

Finally, complete the calculation using the proportion from step 1 and the total votes from step 2. Now, substitute these values into the formula from step 2. This will give the final answer to the problem.

Key Concepts

Polling Results

Understanding polling results is key when predicting an election outcome. In the context of this problem, a poll surveyed 1100 people, out of which 624 indicated they would support the Republican candidate. This creates a "sample" from a larger "population". The sample is used to infer trends in the entire voting population, which in this case is projected to be 40,000 voters. By analyzing this smaller group, we can anticipate how the larger group might behave, a common approach in statistical predictions.

Expected Votes

Expected votes are the projected number of votes a candidate might receive based on polling data. To estimate this number, we use the proportion derived from the sample to make predictions about the population.

In this scenario:

• The sample size is 1100 people.
• The votes in favor of the candidate are 624.
• The larger voting population is 40,000.

By applying the proportion of favorable poll responses to the overall expected voters, we calculate the expected votes. This prediction helps in understanding probable election outcomes.

Mathematical Symbols

Mathematical symbols are crucial in expressing complex ideas simply. In our exercise, they help us to succinctly communicate the steps required to solve the problem.

• The proportion is calculated using the division symbol: \( \frac{624}{1100} \).
• The multiplication symbol (\( * \)) helps apply this proportion to the total number of expected voters: expected_votes = proportion \( * \) total_votes.

These symbols make the procedure precise and reduce word count, letting numbers speak for themselves in a universal language. Grasping them is beneficial for anyone working with mathematical calculations.

Step-by-Step Solution

Breaking down a problem into clear, smaller steps can make comprehension easier. Here’s how we approach this specific calculation:- **Step 1: Calculate the Proportion** - First, find the proportion, \( \frac{624}{1100} \), which represents those who favor the Republican candidate.- **Step 2: Apply the Proportion** - Multiply this by the total number of voters, 40,000, to get the expected votes.- **Step 3: Calculate the Expected Votes** - Following the formula, expected_votes = proportion \( * \) 40,000, gives us 22,690 (rounded appropriately).Understanding this kind of step-by-step process is critical when tackling mathematical problems, as it ensures everything is calculated methodically and correctly.