Problem 60 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The repeating decimal \(0.123123123\ldots\) as a fraction is \(\frac{41}{333}\).

Step 1: Understand the Repeating Pattern

Identify the repeating sequence in the decimal. In this case, the decimal repeats every three digits as \(123\). This is written as \(0.\overline{123}\).

Step 2: Set up the Equation

Let \(x = 0.\overline{123}\). This implies that \(x = 0.123123123\ldots\).

Step 3: Eliminate the Repeating Decimals

Multiply both sides of the equation by 1000 to shift the decimal to the right of the repeating part: \[ 1000x = 123.123123\ldots \] This shifts the decimal point three places to the right.

Step 4: Create Another Equation

We already have \(x = 0.123123123\ldots\). Now we will subtract the first equation (Step 2) from the equation in Step 3 to eliminate the repeating decimals:\[ 1000x - x = 123.123123\ldots - 0.123123123\ldots \]

Step 5: Simplify the Equation

Simplify the equation from Step 4:\[ 999x = 123 \] By subtracting, the repeating part is removed.

Step 6: Solve for x

Divide both sides of the equation by 999:\[ x = \frac{123}{999} \]

Step 7: Simplify the Fraction

Simplify the fraction \( \frac{123}{999} \) by dividing the numerator and the denominator by their greatest common divisor, which is 3:\[ \frac{123 \div 3}{999 \div 3} = \frac{41}{333} \]

Key Concepts

Fraction Conversion

Converting a repeating decimal into a fraction might seem tricky at first, but it's a straightforward process once you understand the steps. The main objective is to transform a decimal, which repeats indefinitely, into a fraction using basic algebra.
Start by identifying the pattern of repetition in the decimal. For instance, in the number \(0.123123123\ldots\), the repeating sequence is \(123\). We express this as \(0.\overline{123}\).
Assign a variable, such as \(x\), to represent the repeating decimal: \(x = 0.\overline{123}\). Multiply both sides of the equation by a power of ten that matches the length of the repeating sequence to shift the decimal point rightward across the repeating digits. In this case, multiplying by 1000 is appropriate:

• \(1000x = 123.123123\ldots\)

Then, subtract the original equation from this new equation, which effectively eliminates the repeating portion and isolates the non-repeating part of the number.
This creativity with multiplication and subtraction allows you to reframe an otherwise infinite decimal into a neat fractional value.

Simplifying Fractions

Simplifying fractions is the process of expressing a fraction in its simplest form, meaning the numerator and denominator have no common factors other than one. To simplify the fraction obtained from a repeating decimal, such as \(\frac{123}{999}\), you seek the greatest common divisor (GCD) of both numbers.
Once you find the GCD, divide both the numerator and denominator by this number to reduce the fraction:

• For \(\frac{123}{999}\), we identified the GCD to be 3.
• Thus, \(\frac{123 \div 3}{999 \div 3} = \frac{41}{333}\).

It's essential to ensure that the fraction cannot be simplified further. Check if the numerator and denominator have any other common factors. If no more common factors exist, you have reached the simplest form. Hence, \(\frac{41}{333}\) is the simplified version of the initial fraction.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is a concept used to simplify fractions by determining the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD involves a bit of division and logic.
To determine the GCD of two numbers, start with the smaller number and find which larger number wholly divides both values. For our example, \(123\) and \(999\) share no common factors larger than \(3\), making \(3\) our GCD.
Here’s a simple method to find the GCD:

• List the factors of each number.
• Identify the largest common factor that appears in both lists.

Alternatively, use the Euclidean algorithm, an efficient method that subtracts the smaller number from the larger repeatedly until reaching zero, at which point the last non-zero remainder becomes the GCD.
Knowing how to find the GCD enables you to simplify fractions efficiently, essential for converting repeating decimals to their simplest fractional form. Understanding and using the GCD streamlines mathematical processes, allowing for clear and concise expressions of even complex repeating decimals like \(0.\overline{123}\).