Problem 23 - Prealgebra

Question

Solve each problem using the percent equation. Round to the nearest tenth. What is \(28.5 \%\) of \(84 ?\)

Step-by-Step Solution

Answer

28.5% of 84 is 23.9.

Step 1: Identify the Percent Equation

The percent equation is generally given by \( \text{Part} = \frac{\text{Percent}}{100} \times \text{Whole} \). We need to find 28.5% of 84.

Step 2: Plug in Values

Substitute the known values into the percent equation. Here, the percent is 28.5 and the whole is 84. This gives us \( \text{Part} = \frac{28.5}{100} \times 84 \).

Step 3: Calculate the Fraction

First, convert the percent to a decimal by dividing by 100: \( \frac{28.5}{100} = 0.285 \).

Step 4: Multiply by the Whole

Now multiply the decimal by the whole number: \( 0.285 \times 84 \).

Step 5: Compute the Result

Perform the multiplication: \( 0.285 \times 84 = 23.94 \).

Step 6: Round the Result

Round the result to the nearest tenth: 23.94 rounded to the nearest tenth is 23.9.

Key Concepts

Percent Conversion

Converting a percentage to a decimal is a crucial step in solving problems that involve percentages, as in our given task to find 28.5% of 84. The conversion process simplifies calculations and makes them easier to handle.
To convert a percentage into a decimal, divide the percentage by 100. This method works because "percent" literally means "per one hundred," so dividing by 100 converts the percentage to its decimal form.
For example, if we take 28.5% and divide by 100, we get 0.285. This straightforward operation turns the percentage into a form that's ready for arithmetic operations like multiplication.
In general, when working with percent problems, always remember:

• Remove the percent sign (%).
• Divide by 100 to find the decimal equivalent.
• Utilize the decimal form in further calculations like multiplying by the whole number.

Decimal Multiplication

Once you have converted the percentage into a decimal, the next step is to multiply by the whole number—the original value you're calculating the percentage of, which is 84 in our problem. Decimal multiplication might seem a bit complex, but it's straightforward if you break it down into smaller steps.
Consider the expression 0.285 multiplied by 84. Here are a few helpful tips to make the multiplication process smoother:

• Line up the numbers based on their significant digits, ignoring the decimal points initially.
• Perform multiplication as if they were whole numbers.
• Count the total number of decimal places in the original numbers to place the decimal point correctly in the product. In this case, three decimal places lead to the answer, 23.94.

Decimal multiplication allows us to calculate the required part value from the whole—effectively simplifying the percent equation into an arithmetic problem.

Rounding Decimals

Rounding decimals is often necessary to simplify results and present them in a more digestible form, especially in the context of percent calculations. In our example, the calculated value (23.94) needs to be rounded to the nearest tenth. This is a common practice that helps in reporting a clean single-digit after the decimal point.
Here's how rounding typically works:

• Identify the digit in the place you are rounding to, and look at the next digit to the right.
• If that digit is 5 or higher, increase the rounding digit by 1; if it's 4 or less, keep the rounding digit the same.
• Finalize your number by cutting off all digits after the rounding point.

In our case, since we are rounding 23.94 to the nearest tenth, we look at the value '94'. The '4' is less than 5, so we do not increase the digit in the tenth place, resulting in the rounded number 23.9. This makes the final number clearer and leaves it properly formatted for most practical applications.