Problem 60 - College Algebra and Calculus: An Applied Approach

Question

The percent of people in a research study that have a particular health risk is \(39.5 \%\). The total number of people in the study is 12,857 . How many people have the health risk?

Step-by-Step Solution

Answer

The short answer is that, approximately, 5,078 people in the study have this health risk.

Step 1: Convert the Percentage to a Decimal

To perform calculations with percentages, it's often easiest to convert them to decimals. This can be done by dividing the percentage by 100. So, \(39.5 \%\) becomes \(0.395\).

Step 2: Multiply the Decimal by the Total Number

Once you've converted the percentage to a decimal, you can multiply it by the total number of people in the study to find how many people have the health risk. So, you multiply \(0.395\) by \(12,857\).

Step 3: Round to the Nearest Whole Number

Because you can't have a fraction of a person, you need to round your answer to the nearest whole number. After performing the multiplication, you should round your answer appropriately.

Key Concepts

Decimal Conversion

When working with percentages, converting them to decimals helps make mathematical calculations simpler. A percentage represents a number out of 100. To convert a percentage to a decimal, divide the percentage value by 100. This is a universal method:
- For example, to convert 39.5% to a decimal, divide 39.5 by 100, resulting in 0.395.
This step is crucial for calculations involving percentages, such as finding a part of a whole. By transforming percentages into decimals, we streamline the process of determining proportions in various contexts, whether in academic exercises or real-world scenarios.

Multiplication

Multiplication is a fundamental operation that allows us to determine the total effect of repeated addition. In this example, we use it to find out how many individuals have a health risk out of a total group. After converting the percentage to a decimal, we multiply this decimal by the total number of subjects:
- Multiply 0.395 (the decimal form of 39.5%) by 12,857 (the total number of participants).
This multiplication tells us the precise number of people that fall into the specified category—in this case, those with the health risk. It is an essential step in translating percentages into tangible numbers.

Rounding Numbers

After multiplication, the result might not always be a whole number. Since we are dealing with people, it's logical to round to the nearest whole number. The rounding process involves:
- Identifying the digit at the first decimal place after obtaining your raw result.
- If this digit is 5 or greater, round up.
- If it is less than 5, round down. For instance, if the multiplication yields 5083.615, you would round this number to 5084. Rounding ensures the final answer is both accurate and realistic, especially when whole units are required, as it is impossible to have a fraction of a person.