Problem 109 - Precalculus with Limits

Question

In Exercises 107 - 110, find the rational number representation of the repeating decimal. \( 0.3\overline{18} \)

Step-by-Step Solution

Answer

The rational number representation of the repeating decimal 0.3\overline{18} is \( \frac{103}{330} \).

Step 1: Define the Decimal as a Variable

Let's define a variable \( x \) to represent the repeating decimal \( 0.3\overline{18} \). So, \( x = 0.3\overline{18} \).

Step 2: Create an Equation with No Decimals

Next, we multiply \( x \) by a power of 10 that shifts the decimal place to the right, just before the start of repetition, i.e., 100x = 31.\overline{18}. The goal here is to set up an equation such that, when subtracted, the repeating decimals cancel out.

Step 3: Subtract the Two Equations

You then subtract the equation from Step 1 from the equation in Step 2. This gives us 100x - x = 31.\overline{18} - 0.3\overline{18}. Simplifying this gives us 99x = 30.9.

Step 4: Solve for x

Solving the equation from step 3 for \( x \) will give us the rational number equivalent of the repeating decimal. Dividing both sides by 99, we get \( x = \frac{30.9}{99} \).

Step 5: Simplify the Fraction

Lastly, the fraction should be reduced to its simplest form. The fraction \( \frac{30.9}{99} \) equals \( \frac{309}{990} \) in terms of integers, which simplifies to \( \frac{103}{330} \).

Key Concepts

Repeating Decimals

Repeating decimals are numbers that have one or more digits that repeat infinitely. They occur when you divide a number and the result isn't a whole number. Instead, a group of digits repeats endlessly. For example, the repeating decimal \( 0.3\overline{18} \) has "18" as the repeating block.

To find a rational number equivalent for a repeating decimal, you can set the decimal equal to a variable, then create an algebraic equation. By appropriately multiplying the variable by a power of ten, it shifts the decimal to a position where its repeating nature can later be canceled out through subtraction. This process effectively turns a complex repeating decimal into a manageable algebraic equation with no repeating parts.

The repeating part of a decimal can be expressed with a horizontal line above the digits (also known as the vinculum), which informs you which digits are repeating. Identifying these repeating parts is a crucial first step in converting such decimals into rational forms.

Equation Solving

Equation solving in the context of repeating decimals involves manipulating the decimal so that their repeating sections can be eliminated. This is achieved through strategic multiplication.

Here's how it works: given a repeating decimal represented by a variable \( x \), like \( x = 0.3\overline{18} \), we multiply \( x \) by a power of 10. The goal is to shift the decimal point over until one repetition of digits lies right after the decimal of a new variable expression, e.g., \( 100x = 31.\overline{18} \). This setup is essential because when you subtract the original \( x \) from this new equation, the repeating decimals cancel out.

The subtraction step is key. It should look like:

• Original Equation: \( x = 0.3\overline{18} \)
• Modified Equation: \( 100x = 31.\overline{18} \)
• Subtraction: \( 100x - x = 31.\overline{18} - 0.3\overline{18} \)

This leaves us with a simple linear equation that can easily be solved for \( x \), thus transforming the repeating decimal into a rational number form.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest terms, which means the numerator and denominator don't have any common factors other than 1.

After you find the equivalent rational number for a repeating decimal, you may not have a completely simplified fraction right away. In the context of converting \( 0.3\overline{18} \) into a fraction, we first reach a point where \( x = \frac{30.9}{99} \) after solving our equation in the previous steps.

Since fractions typically use whole numbers, we convert this to \( \frac{309}{990} \). It turns out that both 309 and 990 have common factors that can be divided out. Finding the greatest common divisor (GCD), which is 3 in this case, lets you simplify \( \frac{309}{990} \) to \( \frac{103}{330} \). This form is the simplest, as both 103 and 330 have no other common divisors, thus completing the transformation from decimal to fraction.