Problem 56 - College Algebra

Question

Use this data for the exercises that follow: In 2013 , there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over). \(^{[34]}\) If you meet a U.S. citizen, what is the percent chance that the person is elderly? (Round to the nearest tenth of a percent.)

Step-by-Step Solution

Answer

12.6% chance.

Step 1: Identify Total and Elderly Populations

The total population of U.S. citizens in 2013 is given as 317 million, and the number of elderly citizens (aged 65 and over) is 40 million.

Step 2: Set Up the Probability Formula

To find the percent chance that a randomly met U.S. citizen is elderly, use the formula for probability: \( \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \). Here, the 'favorable outcome' is meeting an elderly person, and the total number of outcomes is the total population.

Step 3: Calculate the Probability

Substitute the appropriate values into the probability formula: \( \text{Probability} = \frac{40,000,000}{317,000,000} \).

Step 4: Convert to Percentage

To convert the probability to a percent, multiply the result by 100: \( \text{Percent chance} = \left( \frac{40,000,000}{317,000,000} \right) \times 100 \).

Step 5: Round to the Nearest Tenth

Calculate the exact percent value and round it to the nearest tenth of a percent. The value is approximately 12.6%.

Key Concepts

Percent Chance Calculation

Calculating the percent probability involves determining how likely an event is to occur, given a set of conditions. Here, we are interested in finding the chance of meeting an elderly U.S. citizen. To calculate this, follow these steps:

• Identify the total possible outcomes, which in this context is the entire U.S. population of 317 million people.
• Determine the number of favorable outcomes, which is the elderly population, counted as 40 million people.
• Use the probability formula: \( \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \).
• Once you have this fraction, convert it to a percentage by multiplying by 100.

By substituting our values, we see: \( \text{Probability} = \frac{40,000,000}{317,000,000} \). Multiply this fraction by 100 to convert it to a percentage, resulting in an approximate 12.6% chance.
This method helps us understand how often specific outcomes occur relative to the total number of possibilities.

Elderly Population Statistics

Understanding the demographics of an aging population is crucial in many aspects of planning and services in a country. In 2013, the elderly population in the United States reached about 40 million people among a total of 317 million citizens.
Elderly citizen data is essential for:

• Healthcare planning, as older people typically require more medical services.
• Social services and support systems, including pensions and housing.
• Assessing economic impacts since the elderly population might affect workforce dynamics and consumer spending trends.

By analyzing these statistics, policymakers can allocate resources more effectively and plan for the future. It is a way to predict and prepare for shifts in demographic trends as the population ages. This form of study helps ensure that the society caters to its members' needs equitably.

Rounding to the Nearest Tenth

Rounding is a mathematical technique that simplifies numbers to make them easier to work with. When rounding to the nearest tenth, we focus on the digit in the tenths place (the first digit after the decimal point).
To round a number to the nearest tenth, use these guidelines:

• Look at the digit in the hundredths place (the second digit after the decimal point).
• If this digit is 5 or more, increase the tenths place by one.
• If this digit is less than 5, leave the tenths place unchanged.

For example, in the calculated percentage 12.621793, the tenths digit is 6 and the hundredths is 2. Since the hundredths digit is less than 5, the tenths digit stays the same. Thus, the rounded value is 12.6%.
This simple yet powerful concept is widely used in statistics to report concise results without unnecessary detail.