Problem 59 - Algebra and Trigonometry

Question

Express the repeating decimal as a fraction. $$ 0 . \overline{112} $$

Step-by-Step Solution

Answer

The repeating decimal \( 0.\overline{112} \) as a fraction is \( \frac{112}{999} \).

Step 1: Identify the Repeating Decimal

First, let's identify the repeating part of the decimal. Here, the repeating decimal is given as \( 0.\overline{112} \), which means that "112" is the repeating sequence.

Step 2: Set up an Equation with a Variable

Next, set the repeating decimal equal to a variable, say \( x \). Thus, we write: \[ x = 0.112112112 ext{{...}} \] This will help us in expressing the decimal as a fraction.

Step 3: Multiply by a Power of 10 to Remove the Repeating Part

Since the repeating block is "112" and has three digits, we need to multiply both sides of the equation by \( 10^3 \) (which is 1000) so that the decimal point moves three places: \[ 1000x = 112.112112 ext{{...}} \]

Step 4: Subtract the Original Equation

Subtract the original equation \( x = 0.112112112 ext{{...}} \) from the equation obtained after multiplication: \[ 1000x - x = 112.112112 ext{{...}} - 0.112112112 ext{{...}} \] Simplifying this gives us: \[ 999x = 112 \]

Step 5: Solve for x

Now solve for \( x \) by dividing both sides of the equation by 999: \[ x = \frac{112}{999} \] This is the fraction form of the repeating decimal 0.\overline{112}.

Step 6: Check for Simplification

Check if the fraction \( \frac{112}{999} \) can be simplified. The greatest common divisor of 112 and 999 is 1, meaning this fraction is already in its simplest form.

Key Concepts

Fraction Conversion

Repeating decimals can seem tricky at first, but converting them into fractions demystifies them significantly. When you see a repeating decimal like \( 0.\overline{112} \), your aim is to express it as a fraction easily.
Here's a simple way to do it:

• Start with identifying the repeating part of the decimal. In this case, it's "112".
• Use a variable to represent the entire repeating decimal sequence. For example, let \( x = 0.112112112\ldots \).
• This method turns the decimal into an algebra problem, offering a straightforward way to find the exact fractional representation.

This process not only converts decimals to fractions but also offers a deeper understanding of the relationship between decimals and fractions.

Algebra

Algebra comes in to save the day when dealing with repeating decimals. By assigning a variable to the decimal, like \( x = 0.112112112\ldots \), you effectively set up an equation that will help in the conversion process. This involves crucial steps:

• Establish an equation: Write it in terms of a variable \( x \) to manipulate the decimal effectively.
• Multiply by a power of 10 that matches the length of the repeating block. For \( 0.\overline{112} \), we multiply by 1000 to align the decimals.
• Ultimately, subtraction helps cancel the repeating part. This results in a clean fraction form.

This algebraic method might feel like a magic trick, but it’s pure logic in action that makes working with repeating decimals manageable.

Long Division

Long division might not be directly used while converting the repeating decimal to a fraction, but it helps in understanding why they appear as they do.
When you divide numbers like 1 by a number that isn’t a factor of 10, you often get decimals that repeat. Here’s how long division fits in:

• Understanding repeating decimals gives insight into divisibility and division process outcomes.
• Imagine dividing 112 by 999. It shows how the remainders repeat, which causes the decimal sequence to repeat over and over.
• Long division serves as a background check, confirming that repeating decimals can indeed be expressed as fractions accurately.

In essence, long division helps build the foundational knowledge that complements conversion techniques involving fractions and algebra, enriching your problem-solving toolkit.