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Chapter 2: Problem 9

In a resort area during the summer months, only one out of eight people is a year-round resident. The others are there on vacation. If the year-round population of the area is 3000 , how many people are there in the summer?

Short Answer

Expert verified
The total population in the area during the summer is 24,000.

Step by step solution

01

Understanding the Ratio of Residents to Total Population

The problem states that only one out of eight people is a year-round resident. This means that for every resident, there are eight people total in the resort area during the summer.
02

Identifying the Number of Year-Round Residents

The problem provides that the year-round population (residents) is 3,000.
03

Setting Up the Equation

Given the ratio, we know that 1 out of 8 people is a resident. Therefore, the ratio translates to: \( \frac{1}{8} \text{ of total population} = \text{residents} \). Let \( x \) be the total summer population. Thus, \( \frac{1}{8}x = 3000 \).
04

Solving for the Total Population

To find \( x \), multiply both sides by 8: \[ x = 3000 \times 8 \] \[ x = 24000 \]
05

Conclusion

The total population in the area during the summer months is 24,000.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ratio and Proportion
Understanding ratios and proportions is fundamental in solving various algebraic word problems. A **ratio** compares two numbers, expressing how much of one thing exists relative to another.
For example, the ratio of residents to the total population often helps in calculating aggregate numbers in demographic studies and scenarios like the given problem.
A **proportion** occurs when two ratios are equal, providing the means to solve for unknown values within these connected sets. Consider a situation involving a known ratio between year-round residents and the total summer population. You can use proportions to find missing information, like the total population.
  • Ratio: A comparison of two numbers. In the exercise, this is 1 (resident) to 8 (total population).
  • Proportion: An equation that states that two ratios are equal. Used here to set up the primary equation aiding in the solution.
To solve the provided problem, we use the ratio of 1:8 to understand the relationship between residents and the total population. This guides the creation of the equation to solve for the total number of people.
Equations
Equations in algebra help us translate real-world situations into solvable mathematical expressions. By setting relationships as equations, we find unknown quantities given some variables.
In the context of our problem, an equation is set to find the total summer population using the known ratio.The equation derived from the problem's scenario is:\[ \frac{1}{8}x = 3000 \]This translates real-world conditions 鈥� describing how one part (in this case, year-round residents) relates to a whole (total summer population) 鈥� into a form that is easily manipulated mathematically. To isolate the unknown quantity, multiply both sides by 8, giving:\[ x = 3000 \times 8 \]Then, solving:\[ x = 24000 \]By manipulating the equation properly, you arrive at understanding key dynamics like the total population being 24,000 during summer. Equations are vital in framing and answering such population-related queries effectively.
Population Calculation
Population calculation involves understanding how many individuals are present in a given area within specific conditions.
It often requires manipulating ratios and equations to estimate or count populations accurately. In our scenario, calculation involves converting the provided ratio into usable information about populations. Given a known number of year-round residents, the goal is to figure out the total population during the summer when visitors are also present.
The solution uses integral steps:
  • Identify the ratio of residents to the entire population.
  • Formulate a proportion or equation to represent this relationship.
  • Solve the equation step-by-step to find the desired total population, in this case: 24,000.
By accurately applying the ratio of 1 resident to every 8 members of the population, you can derive complete counts, highlighting the dependence on relationships between portion and total numbers in demographic computations.

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