Problem 65 - Prealgebra

Question

Write as a percent. Round to the nearest tenth of a percent. $$\frac{1}{3}$$

Step-by-Step Solution

Answer

\(\frac{1}{3}\) as a percent is 33.3%.

Step 1: Convert Fraction to Decimal

To begin solving the exercise, convert the given fraction \( \frac{1}{3} \) into a decimal by dividing the numerator by the denominator. Calculate: \( 1 \div 3 = 0.3333 \ldots \) (The decimal repeats indefinitely as this is a recurring decimal).

Step 2: Convert Decimal to Percent

Now, convert the decimal obtained in Step 1 to a percentage by multiplying it by 100. So, \( 0.3333 \ldots \times 100 = 33.3333 \ldots \).

Step 3: Round the Percentage

Finally, round the calculated percentage to the nearest tenth of a percent. The number 33.3333 rounds to 33.3 when considering the digit in the tenths place.

Key Concepts

Fractions to Decimals Conversion

The journey from fractions to decimals starts with understanding the roles of numerators and denominators. In any fraction, such as \(\frac{1}{3}\), the numerator (1) is divided by the denominator (3). This division transforms the fraction into a decimal. When you do this division using a calculator or manually, you get the decimal \(0.3333\ldots\).This process sounds more complicated than it actually is.

• Simply think of the fraction bar \(\left( / \right)\) as a division symbol.
• Divide the top number (numerator) by the bottom number (denominator).

For some fractions, like \( \frac{1}{3} \), the result is a decimal that goes on forever. Don't let this infinite part scare you—recurring decimals are common in math. It's all about understanding that dividing fractions is the first step in many math processes.

Recurring Decimals

What happens when your division doesn’t end neatly? That’s when you encounter recurring decimals. These are decimals where one or more digits repeat infinitely. The decimal \(0.3333\ldots\) is a perfect example. The number \(3\) repeats forever after the decimal point.When you see this repeated pattern, you can express it with a line above the recurring digit—known as the vinculum. For example, \(0.3333\ldots\) may be written as \(0.\overline{3}\). Recurring decimals are common with fractions where the denominator doesn’t easily divide into a whole number.

• Recognize the pattern: Look for numbers repeating at regular intervals.
• Use the vinculum to note the recurring cycle.

Converting these to percentages is important, especially when rounding numbers or needing a clear, simpler version for practical use.

Rounding Numbers

After transforming a fraction to a recurring decimal and then a percent, you often need to deal with rounding — especially if the numbers go on infinitely. Rounding helps by simplifying numbers to the required precision. For instance, 33.3333...% needs to be rounded to the nearest tenth (one decimal place). Here’s how you do it:

• Identify the tenths place — for 33.3333..., it's the first 3 after the decimal.
• Look at the next digit (also 3). If it is 5 or more, you increase the tenths place by one. If less than 5, you leave it as it is.

With 33.3333..., the tenths place remains 3, resulting in 33.3%. Rounding ensures numbers are both definitive and user-friendly, especially critical when discussing percentages in practical scenarios.