Problem 41 - Calculus Early Transcendentals: Pearson New International Edition

Question

change each repeating decimal to a ratio of two integers. $$ 0.199999 \ldots $$

Step-by-Step Solution

Answer

The repeating decimal \(0.199999\ldots\) is equivalent to the fraction \(\frac{1}{5}\).

Step 1: Understand the Problem

The exercise requires expressing the repeating decimal \(0.199999\ldots\) as a fraction or the ratio of two integers. Here, \(0.199999\ldots\) can be understood as the repeating decimal \(0.1\overline{9}\).

Step 2: Introduce a Variable

Let's denote \(x\) as the repeating decimal: \(x = 0.199999\ldots\). Our goal is to solve for \(x\) in terms of a fraction.

Step 3: Eliminate the Repeating Part

Multiply \(x = 0.199999\ldots\) by 10 to shift the decimal point: \(10x = 1.99999\ldots\).

Step 4: Set Up an Equation

Now, set up an equation using \(x\): \(10x = 1.99999\ldots\). Subtract the original \(x\) from this equation: \(10x - x = 1.99999\ldots - 0.199999\ldots\).

Step 5: Solve the Simple Equation

The subtraction gives \(9x = 1.8\). Divide both sides by 9 to solve for \(x\): \(x = \frac{1.8}{9}\).

Step 6: Convert the Decimal Numerator to a Fraction

The numerator \(1.8\) can be expressed as a fraction: \(1.8 = \frac{18}{10}\) (by shifting the decimal one place to the right and dividing by 10). Simplify \(\frac{18}{10}\) to \(\frac{9}{5}\).

Step 7: Solve for x Fully in Fraction Form

Substitute the simplified fraction back to find \(x\): \(x = \frac{9}{5} \times \frac{1}{9}\). This simplifies to \(x = \frac{9}{45} = \frac{1}{5}\) after simplifying \(\frac{18}{90}\) to \(\frac{1}{5}\).

Step 8: Verify the Final Ratio

Finally, verify that \(\frac{1}{5}\) correctly represents the decimal \(0.199999\ldots\) by converting \(\frac{1}{5}\) back to a decimal (\(0.2\)) and ensuring it's approximately equal to the original decimal value.

Key Concepts

Repeating Decimals

Repeating decimals are numbers with a digit or group of digits that endlessly repeat after the decimal point. In simple terms, when you see a number like \(0.199999\ldots\), the digit 9 is repeating over and over again. This is often represented as \(0.1\overline{9}\), where the line over the 9 indicates that it keeps repeating.

Understanding repeating decimals is crucial because they frequently occur in mathematical problems, and converting them into fractions can simplify calculations. Sometimes, these decimals are non-terminating, meaning they go on infinitely, yet they still represent a specific fraction. This conversion is our goal.

To convert a repeating decimal into a fraction, we typically use algebraic equations to "capture" the repeating part by multiplying and subtracting equations, as we'll see in the upcoming steps.

Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form. When you simplify a fraction, you make it easier to understand and work with without changing its value.

For instance, simplifying \(\frac{18}{90}\) involves finding the greatest common divisor (GCD) of 18 and 90, which is 9. Divide both the numerator and the denominator by 9 to get \(\frac{2}{10}\), and then further simplify to \(\frac{1}{5}\).

Here's a quick guide to simplifying fractions:

• Identify the GCD of the numerator and the denominator.
• Divide both the numerator and denominator by the GCD.
• The result is your fraction in simplest form, retaining the same proportion or value.

Fraction simplification is vital because it provides the most straightforward expression of the value, which is especially helpful when adding, subtracting, or comparing fractions.

Decimal to Fraction Steps

Converting a decimal, especially a repeating one, into a fraction involves a systematic approach:

1. **Introduce a Variable:** Start by letting \(x = 0.199999\ldots\). This sets up the foundation for the operation.
2. **Eliminate the Repeating Part:** Multiply \(x\) by 10 so that you shift the decimal point: \(10x = 1.99999\ldots\). This helps align the repeating decimal part for elimination.
3. **Set Up an Equation:** By setting these equal, you can remove the repeating part through subtraction: \(10x - x = 1.99999\ldots - 0.199999\ldots\). This results in a much simpler equation \(9x = 1.8\).
4. **Solve the Equation:** From \(9x = 1.8\), solve for \(x\) by dividing both sides by 9, giving \(x = \frac{1.8}{9}\).
5. **Convert to Fraction:** Express the decimal \(1.8\) as a fraction, \(\frac{18}{10}\), and simplify to \(\frac{9}{5}\). Combine this to find: \(x = \frac{9}{5} \times \frac{1}{9}\), which simplifies further to \(x = \frac{1}{5}\).

This sequence is a main method to convert any repeating decimal into a fraction accurately. It's a powerful tool for when decimals aren't quite enough.