Question

Write each decimal number as a percent. $$0 . \overline{3}$$

Short answer

Question: Convert the repeating decimal 0.𝟑 to a percent. Answer: The repeating decimal 0.𝟑 can be written as approximately 33.33%.

Step-by-step solution

Convert the repeating decimal to a fraction

To convert the repeating decimal $$0 . \overline{3}$$ to a fraction, we'll let $$x$$ represent the decimal. Since the number $$3$$ is repeating, we can write the decimal as a geometric series: $$x = 0.333... = 3 \times 10^{-1} + 3 \times 10^{-2} + 3 \times 10^{-3} + ...$$ Now, we can use the formula for the sum of an infinite geometric series: $$S = \frac{a}{1-r}$$ Where $$S$$ is the sum of the series, $$a$$ is the first term, and $$r$$ is the common ratio between terms. In this case, $$a = 3 \times 10^{-1}$$(which is 0.3) and $$r = 10^{-1}$$(which is 0.1). Plugging these values into the formula, we get: $$x = \frac{3 \times 10^{-1}}{1 - 10^{-1}}$$ $$x = \frac{3 \times 10^{-1}}{9 \times 10^{-1}}$$ Simplifying, we find that the fraction representation of $$x$$ is: $$x = \frac{3}{9}$$ $$x = \frac{1}{3}$$

Convert the fraction to a percent

To convert the fraction $$\frac{1}{3}$$ to a percent, we will multiply it by $$100\%$$. Percent simply means "per hundred," so when we multiply by $$100\%$$, we are showing the fraction as a proportion of $$100$$: $$\frac{1}{3} \times 100\% = \frac{100}{3}\%$$ Approximately, the percent representation of the repeating decimal $$0 . \overline{3}$$ is: $$33.33\%$$ So, the repeating decimal $$0 . \overline{3}$$ can be written as approximately $$33.33\%$$.

Key concepts

Repeating Decimals

Repeating decimals are numbers that have a digit or group of digits after the decimal point that repeat endlessly. For example, in the decimal \(0.\overline{3}\), the digit '3' repeats indefinitely. Recognizing repeating decimals is crucial because they often represent fractional values. Here’s a quick guide to understanding them:

• Identify the repeating part of the decimal (e.g., in \(0.\overline{3}\), the repeating part is '3').
• Convert the repeating decimal to a fraction by expressing the repeating part as a series.
• This repeating series can be converted into a geometric series for easier calculation.

The repeating pattern helps us understand the exact value of these numbers as fractions, allowing for precise mathematical manipulation.

Geometric Series

A geometric series is a sum of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Repeating decimals can be expressed as geometric series, making them easier to convert into fractions. Here’s how to work with them:

• The first term \(a\) is the first digit or group of digits in the decimal (e.g., 0.3 for \(0.\overline{3}\)).
• The common ratio \(r\) is the base increment for the decimal position (usually 0.1 for repeating decimals like \(0.\overline{3}\)).
• The formula for the sum of an infinite geometric series is \( S = \frac{a}{1 - r} \), which helps you find the exact fraction representation.

Understanding this framework allows for effective conversion of repeating decimals into their true fractional forms.

Fraction to Percent Conversion

Changing a fraction to a percent involves expressing the fraction as a part of 100. It's often used to make the information more relatable and easy to understand. Here's how it works:

• Take the fraction you want to convert (e.g., \(\frac{1}{3}\) from our previous example).
• Multiply the fraction by 100 to find its percent equivalent: \(\frac{1}{3} \times 100\% = \frac{100}{3}\%\).
• For better understanding, you can convert it to a decimal percent: approximately 33.33%.

This conversion is helpful in comparing different values and understanding the proportions they represent.