Problem 12 - Prealgebra

Question

Express each decimal or fraction as a percent. Round to the nearest tenth, if necessary. $$\frac{16}{3}$$

Step-by-Step Solution

Answer

\( \frac{16}{3} \) as a percent is \( 533.3\% \).

Step 1: Convert Fraction to Decimal

Divide the numerator by the denominator to convert the fraction to a decimal. For the fraction \( \frac{16}{3} \), perform the division \( 16 \div 3 = 5.3333... \).

Step 2: Convert Decimal to Percent

To convert a decimal to a percent, multiply the decimal by 100. Take \( 5.3333... \) and calculate \( 5.3333... \times 100 = 533.33... \% \).

Step 3: Round the Percent

Round the percent to the nearest tenth. Here, \( 533.33... \% \) becomes \( 533.3\% \) when rounded to the nearest tenth.

Key Concepts

Converting Fractions to Decimals

Let's begin by understanding how to convert a fraction into a decimal. A fraction consists of two components: a numerator (the top part) and a denominator (the lower part). To convert a fraction to a decimal, you simply divide the numerator by the denominator. For instance, with the fraction \( \frac{16}{3} \), you perform the division \( 16 \div 3 \). This operation will result in an ongoing decimal, \( 5.3333... \).

• The division shows that \( 16 \) divided by \( 3 \) gives \( 5 \) as a whole number, while the remainder continues in a repeating pattern.
• Repeating decimals are common when dealing with fractions, and it's useful to write them with a bar or ellipsis to show they continue.

Whenever you divide and spot a recurring sequence, you know you're dealing with a repeating decimal. This skill is very important as it is the first step towards understanding percentages.

Rounding Decimals

Once you have a decimal, you may need to round it to make it easier to work with. Rounding is the process of reducing the precision of a number, while keeping its value close to what it was. It’s particularly helpful when the number has many decimal places.

Here's a simple way to round decimals to the nearest tenth:

• Look at the number in the tenths place (the first digit to the right of the decimal point).
• Then, observe the number immediately following it, in the hundredths place.
• If this digit is 5 or higher, round the tenths digit up by one.
• If it is less than 5, keep the tenths digit the same.

In our example, the decimal \( 5.3333... \) would specifically be rounded to \( 5.3 \), after following these steps.

Calculating Percentages

Now, we focus on transforming a decimal into a percentage. Converting decimals to percentages is straightforward.

To convert a decimal to a percent, multiply the decimal by 100. The rationale behind this is:

• Percent means per hundred, so multiplying by 100 shifts the value two places to the left.
• It changes a number like \( 0.75 \) to \( 75\% \), making it more intuitive as it relates directly to parts of a hundred.

In our case with \( 5.3333... \), multiplying by 100 gives \( 533.33...\% \).

Finally, rounding this result applies the earlier discussed method to give \( 533.3\% \). Proper rounding ensures clarity, especially when dealing with calculations that require a high level of precision.