Problem 47 - College Algebra

Question

Express each repeating decimal as a fraction in lowest terms. $$0 . \overline{47}=\frac{47}{100}+\frac{47}{10,000}+\frac{47}{1,000,000}+\cdots$$

Step-by-Step Solution

Answer

\( 0 .\overline{47} = \frac{47}{99} \)

Step 1: Set up equation

Set the repeating decimal as an equation. Let's call the decimal \( x \). Therefore, \( x = 0.474747... \)

Step 2: Eliminate the repeating part

To eliminate the repeating part, we multiply the equation \( x = 0.474747... \) by 100. Through this step, we get \( 100x = 47.474747... \). This operation shifts the decimal point two places to the right.

Step 3: Subtract original equation

Now, subtract the original equation \( x = 0.474747... \) from the equation \( 100x = 47.474747..., \) yielding \( 99x = 47 \). This step eliminates the repeating decimal on the right.

Step 4: Solve the equation

By solving the equation \( 99x = 47, \) find \( x = \frac{47}{99}. \)

Step 5: Reduce to lowest terms

The equation \( x = \frac{47}{99} \) is already in the reduced form, as 47 and 99 do not have any common factors besides 1.

Key Concepts

Fractions

Fractions are a fundamental part of mathematics, representing parts of a whole. They consist of two numbers divided by a slash, where the top number is called the numerator, and the bottom number is called the denominator. Fractions can be used to express values that are not whole numbers.
- A fraction with a denominator of 1 is simply equal to its numerator. For example, \( \frac{5}{1} = 5 \).- Equivalent fractions are fractions that represent the same value even though they may have different numerators and denominators. For instance, \( \frac{2}{4} \) and \( \frac{1}{2} \) are equivalent because they both equal 0.5.
- To simplify a fraction, you divide both the numerator and the denominator by their greatest common factor (GCF). Simplifying doesn't change its value but reduces it to the lowest terms. For example, \( \frac{12}{16} \) simplifies to \( \frac{3}{4} \) by dividing both by 4.

Expressing Repeating Decimals as Fractions

Repeating decimals are decimal numbers where one or more digits repeat infinitely. A classic example is 0.333..., where the digit "3" goes on forever. Even though they appear endless, repeating decimals can be expressed as fractions that capture their exact value.
The process of converting a repeating decimal into a fraction involves a few key steps:

• Identify the repeating sequence. For example, in 0.474747..., "47" is the repeating part.
• Set the repeating decimal equal to a variable, say \( x \), e.g., \( x = 0.474747... \).
• Multiply \( x \) by a power of ten (10, 100, etc.) that moves the decimal point to the right of the repeating sequence.
• Subtract the original equation from this new equation to eliminate the repeating part, solving for \( x \).
• Simplify the resulting fraction to get the expression in the simplest form possible.

By following these steps, a repeating decimal can be neatly expressed as a simple fraction.

Converting Decimals to Fractions

Converting decimals, whether repeating or non-repeating, into fractions can seem tricky, but it follows a structured process. For non-repeating decimals, you process them as a whole differently compared to repeating ones.

• For non-repeating decimals, count how many places are after the decimal point. This will determine the power of 10 you'll use in the denominator. For example, 0.25 has two decimal places, thus its fraction form is \( \frac{25}{100} \), which simplifies to \( \frac{1}{4} \).
• For repeating decimals, identify the repeating part, as described before, and use equations to eliminate the repeat. This was done in the original exercise where a repeating decimal like 0.474747... was converted to the fraction \( \frac{47}{99} \).

In both cases, ensure to simplify the fraction to its lowest terms. This practice not only gives the most precise form but also aids clearer understanding and communication of values.