Problem 79 - Precalculus: Mathematics for Calculus

Question

Express the repeating decimal as a fraction. $$\begin{aligned} &0.030303 \dots\\\ &1 \end{aligned}$$

Step-by-Step Solution

Answer

The repeating decimal 0.030303... as a fraction is \( \frac{1}{33} \).

Step 1: Assign a Variable

Let the repeating decimal be denoted by \( x \). Therefore, \( x = 0.030303\ldots \). This is the decimal number we want to convert into a fraction.

Step 2: Understand the Repetition

Identify that the repeating part of the decimal is "03". The decimal \( 0.030303\ldots \) can be written as \( 0.\overline{03} \).

Step 3: Multiply to Remove Repetition

Since the repeating part "03" has two digits, multiply the decimal by \( 100 \) to shift the decimal point two places to the right: \( 100x = 3.0303\ldots \).

Step 4: Set Up an Equation

You now have two equations: 1. \( x = 0.0303\ldots \)2. \( 100x = 3.0303\ldots \)Subtract the first equation from the second equation to eliminate the repeating part.

Step 5: Solve the Equation

Subtract the first equation from the second: \( 100x - x = 3.0303\ldots - 0.0303\ldots \) This simplifies to: \( 99x = 3 \)

Step 6: Divide to Find the Fraction

Solve for \( x \) by dividing both sides of the equation by 99: \( x = \frac{3}{99} \). Simplify this fraction by finding the greatest common divisor of 3 and 99, which is 3, so \( x = \frac{1}{33} \).

Key Concepts

Understanding Repeating Decimals

A repeating decimal is a decimal number that has one or more digits that repeat infinitely. For instance, the decimal number given in the exercise is 0.030303..., where the sequence "03" repeats endlessly. These numbers are often represented with a bar over the digits that repeat, like so: \( 0.\overline{03} \).
This notation makes it simpler to identify the repeated section quickly and is essential when converting repeating decimals to fractions. Recognizing the repeating sequence is the initial step in conversion, as it allows us to set up the necessary equations.

• Write the decimal with a bar over the repeating section, e.g., \( 0.\overline{03} \).
• Remember, understanding the repetition helps in determining how to manipulate the decimal using multiplication to remove the repeating part.

Steps in Fraction Conversion

Converting repeating decimals into fractions involves a straightforward method. Here's how you do it using the given example of \( 0.030303... \).
Start by letting \( x = 0.030303... \). The repeated section here, "03", contains two digits. We multiply the equation by \( 100 \) to shift the decimal point over these two digits, obtaining a new equation: \( 100x = 3.0303... \).
Now set up a system of equations:

• Equation 1: \( x = 0.0303... \)
• Equation 2: \( 100x = 3.0303... \)

With these, we can eliminate the repeating part by subtracting the first equation from the second. This results in \( 99x = 3 \).
Finally, solving for \( x \) gives us the fraction: \( x = \frac{3}{99} \). Simplifying this fraction is the last step in a fraction conversion process.

Simplifying with the Greatest Common Divisor

After converting a repeating decimal to a fraction, the final step is to simplify the fraction. This involves finding the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that divides both the numerator and denominator without a remainder.
In the example \( \frac{3}{99} \), both the numerator and denominator can be divided by 3, since 3 is their GCD. Dividing the numerator and the denominator by 3, we simplify the fraction to \( \frac{1}{33} \).

• Use the GCD to find the simplest form of any fraction.
• Always check both the numerator and denominator to ensure there are no further divisors.

Simplifying fractions makes them easier to understand and use in further calculations.