Question

Write each decimal number as a percent. $$0 . \overline{47}$$

Short answer

Question: Convert the repeating decimal number 0.47 (with the 47 repeating) to a percent. Answer: The repeating decimal number 0.47 can be written as approximately \(47.\overline{47} \%\).

Step-by-step solution

Convert the repeating decimal to fraction

To convert the repeating decimal number 0.47 to a fraction, let's set the number as x and use the following process: $$ x = 0. \overline{47} $$ $$ 100x = 47.\overline{47} $$ Next, subtract the first equation from the second equation: $$ 100x - x = 47.\overline{47} - 0.\overline{47} $$ $$ 99x = 47 $$ Now, divide both sides of the equation by 99 to get the value of x: $$ x = \frac{47}{99} $$ So the repeating decimal number 0.47 can be represented as the fraction 47/99.

Convert the fraction to a percent

Now that we have a fraction, we can easily convert it to percent. To do this, we'll multiply the fraction by 100 and add the % symbol. $$ \frac{47}{99} \times 100 = 47.\overline{47} % $$ So the decimal number \(0 . \overline{47}\) can be written as approximately \(47.\overline{47} \%\).

Key concepts

Understanding Repeating Decimals

Repeating decimals are numbers that have digits after the decimal point that repeat infinitely. In this case, the decimal number is \(0.\overline{47}\), meaning the sequence "47" repeats indefinitely.

Recognizing a repeating decimal can be quite straightforward:

• Look for a period or an overline above the repeating digits, such as \(0.\overline{47}\).
• The digits under the overline are the ones that repeat continuously.

Repeating decimals are interesting because they represent rational numbers, which means they can be expressed as fractions. Identifying and converting these decimals to fractions is a key mathematical skill that helps in various calculations and simplifications.

Fraction to Percent Conversion

Fraction to percent conversion is a simple but important process. A fraction gives you a part of a whole, while a percent is a way of expressing that part in terms of 100.

Here's how you do it:

• Start with a fraction, like \(\frac{47}{99}\), obtained from the repeating decimal \(0.\overline{47}\).
• Multiply the fraction by 100 to convert it to a percentage: \(\frac{47}{99} \times 100\).
• This multiplication expresses the part represented by the fraction in terms of a whole divided into 100 parts.
• The final result for this conversion is approximately \(47.\overline{47}\%\), meaning the repeating pattern appears even in the percentage.

This conversion is widely used in various fields like finance, statistics, and everyday calculations, where you need a clear representation of fractions as percentages.

Basic Arithmetic Operations in Conversion

Arithmetic operations such as multiplication and division are essential in the process of converting between decimals, fractions, and percentages.

Here's why these operations matter:

• Multiplication: It helps us convert fractions to percentages by scaling the fraction to 100. For instance, \(\frac{47}{99}\times 100\) converts the fraction to a percentage.
• Division: It's used in finding the fraction equivalent of a repeating decimal. For example, simplifying \(99x = 47\) by dividing both sides by 99 gives us the value of \(x\) as a fraction, \(\frac{47}{99}\).

Understanding these operations not only aids in accurate conversion but also builds a strong foundation for more complex mathematical calculations. Sharpening these arithmetic skills is like acquiring a toolset for solving mathematical problems efficiently.