Problem 60 - Algebra and Trigonometry

Question

Express the repeating decimal as a fraction. $$ 0.123123123 \ldots $$

Step-by-Step Solution

Answer

The fraction equivalent is \( \frac{41}{333} \).

Step 1: Define the repeating decimal

Let's define the repeating decimal as a variable. Set \( x = 0.123123123 \ldots \) which is the decimal we want to convert into a fraction.

Step 2: Eliminate the repeating portion

Multiply \( x \) by 1000 to shift the decimal point three places to the right, moving the repeating part to align after the decimal. So we have \( 1000x = 123.123123123 \ldots \).

Step 3: Set up an equation to eliminate repetition

Now subtract \( x = 0.123123123 \ldots \) from \( 1000x = 123.123123123 \ldots \). This gives: \( 1000x - x = 123.123123123 \ldots - 0.123123123 \ldots \).

Step 4: Simplify the equation

The subtraction from the previous step results in \( 999x = 123 \). The repeating parts cancel out because they are the same in both terms.

Step 5: Solve for x

Now solve for \( x \) by dividing both sides of the equation by 999. Thus, \( x = \frac{123}{999} \).

Step 6: Simplify the fraction

Simplify the fraction \( \frac{123}{999} \). Find the greatest common divisor (GCD) of 123 and 999, which is 3. Dividing both the numerator and denominator by 3 gives \( \frac{41}{333} \).

Key Concepts

Converting Decimals to Fractions

Converting decimals to fractions can seem daunting at first, but with practice it becomes a lot easier. Repeating decimals have a unique conversion method compared to non-repeating ones. For instance, with a repeating decimal like 0.123123123..., our goal is to transform it into a simpler fraction form.

• Identify the repeating sequence: In this case, it is "123" which repeats indefinitely.
• Set this repeating decimal as a variable: Here, we chose to set it as \( x = 0.123123123... \).
• Multiply by a power of 10 to align the repeating sections: Multiply the decimal by 1000 (since "123" has three digits) to shift the decimal and prepare for subtraction.

By following these steps, we establish clear equations that help eliminate the repeating part, moving us closer to the fraction form.

Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is very useful when simplifying fractions. It's the largest positive integer that divides two or more integers without leaving a remainder.

• To simplify a fraction like \( \frac{123}{999} \), first find the GCD.
• Use methods like the Euclidean algorithm for efficiency.
• Here, the GCD of 123 and 999 is 3.

This value helps in reducing fractions to their simplest form by dividing both the numerator and the denominator by this common divisor.

Simplifying Fractions

Simplifying fractions means reducing them to the smallest possible integer values for the numerator and the denominator while maintaining the value of the fraction. It involves dividing both parts of the fraction by their GCD.

• Take the fraction \( \frac{123}{999} \) as an example.
• We found the GCD to be 3.
• Divide both numerator and denominator by this number: \( \frac{123 \div 3}{999 \div 3} = \frac{41}{333} \).

This process results in the simplified fraction \( \frac{41}{333} \), which is equivalent to the original decimal form.

Numerator and Denominator

In any fraction, the numerator and denominator have distinct roles. Understanding these roles is crucial for converting and simplifying fractions effectively.

• The numerator, located above the fraction line, indicates how many parts of the whole are being considered.
• The denominator, below the fraction line, describes the total number of equal parts that make up the whole.

For example, in \( \frac{123}{999} \), 123 is the numerator, showing how many parts we have, and 999 is the denominator, showing the whole from which those parts are taken. These components are key in performing operations like simplification.