Problem 25 - Intermediate Algebra

Question

According to the Insurance Information Institute, \(6.5 \%\) of the \(4,730\) recreational boating accidents in 2009 involved alcohol use. How many boating accidents were alcohol-related? Round up to the nearest one accident. (Source: U.S. Coast Guard)

Step-by-Step Solution

Answer

308 boating accidents were alcohol-related.

Step 1: Understanding the Problem

We are given that 6.5% of 4,730 boating accidents involved alcohol use. We need to determine how many of those accidents were alcohol-related.

Step 2: Identify Important Values

The total number of boating accidents is 4,730, and the percentage that involved alcohol is 6.5%.

Step 3: Convert Percentage to Decimal

To use the percentage in calculations, convert it to a decimal by dividing by 100: \(6.5\% = 0.065\).

Step 4: Calculate the Alcohol-Related Accidents

Multiply the total number of accidents by the decimal form of the percentage: \[ 4,730 \times 0.065 = 307.45 \]

Step 5: Round Up to the Nearest Whole Number

Since the question asks for the number of accidents and 307.45 accidents isn't possible, round up to the nearest whole number, which is 308.

Key Concepts

Understanding Percentage Calculation

Percentage calculations are an essential part of solving problems in intermediate algebra, especially when dealing with real-world applications. In our exercise, you are asked to find how many boating accidents were alcohol-related from a total using a specified percentage.

• First, recognize that a percentage is a way of expressing a number as a fraction of 100. Therefore, 6.5% is equivalent to 6.5 out of 100, or in more concrete terms, 6.5 parts out of every 100 parts.
• Because of this, calculating a percentage share of a quantity involves converting the percentage to a decimal.

This is achieved by dividing the percentage by 100, giving:

• 6.5% = 6.5/100 = 0.065

Once you have the decimal, the next step is straightforward: you multiply this number by the total quantity involved, which was 4,730 in our case. This multiplication process helps us determine the number of accidents linked to alcohol use:

• 4,730 \( \times \) 0.065 = 307.45

This result, though, needs a bit of adjustment to become a whole number, which leads us to rounding, the next core concept.

Mastering the Art of Rounding Numbers

Rounding numbers is a skill that allows us to express a number in a simpler form with a specified number of significant digits. It is crucial when numbers do not easily translate to real-world contexts, such as fractions of an accident.

• To round a number, look at the digit right after the place value to which you're rounding. If this digit is 5 or greater, you round up. If it's less than 5, you round down.

In our example:

• The calculated result was 307.45. Since we are looking to round to the nearest whole number and the digit after the decimal is 4, we would normally round down. But since the context (3l07.45 accidents) isn't feasible, as you can't have a fraction of an accident in this practical scenario, we follow the instruction to round up, giving us 308.

Remember, when specific instructions tell you to round up, as they did here, always apply that direction to fit the context.

Steps to Decimal Conversion

Decimal conversion is the process of changing a percentage into a decimal form so it can be utilized in mathematical operations. This conversion is crucial for efficient computation and ensures precision in calculations.

• To perform a decimal conversion, divide the percentage by 100. This conversion drops the percent sign and places the figure in a form that can be more easily manipulated mathematically.

For example:

• 6.5% becomes 0.065 when you divide by 100.

Understanding decimal conversion is especially useful not only for cases like our exercise but also in various fields involving finance, science, and statistical calculations. By consistently using this approach, you secure greater accuracy in operations that demand calculated precision such as budgeting, data analysis, and scaling issues.