Question

\(37-42\), change each repeating decimal to a ratio of two integers. $$ 0.123123123 \ldots $$

Short answer

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

Step-by-step solution

Assign a Variable to the Repeating Decimal

Let \( x = 0.123123123 \ldots \). This helps us keep track of the repeating decimal part as we perform operations to find the fraction.

Identify the Repeating Block

The repeating block here is \(123\). This repeating block spans 3 digits.

Multiply to Eliminate the Repeating Decimal

Since the repeating block has 3 digits, multiply \(x\) by \(1000\) to shift the decimal point 3 places to the right: \( 1000x = 123.123123 \ldots \).

Set Up an Equation to Subtract

Write a separate equation for the initial \(x\) as well: \( x = 0.123123 \ldots \). Now, subtract the latter from the former: \(1000x - x = 123.123123 \ldots - 0.123123 \ldots \).

Solve for x

Perform the subtraction: \( 999x = 123 \). Next, solve for \(x\) by dividing both sides by 999: \( x = \frac{123}{999} \).

Simplify the Fraction

Simplify \( \frac{123}{999} \) by finding the greatest common divisor of 123 and 999, which is 3: \( \frac{123}{999} = \frac{41}{333} \).

Conclusion

The repeating decimal \(0.123123123 \ldots\) can be expressed as the fraction \( \frac{41}{333} \).

Key concepts

Understanding the Ratio of Integers

When we talk about expressing a repeating decimal as a ratio of integers, we mean representing the decimal in the form of a fraction. A fraction is essentially a ratio where one integer is divided by another. For example, \( \frac{41}{333} \) is a ratio.
To convert a repeating decimal like \(0.123123123\ldots\) into a ratio of two integers, we follow a systematic process:

• Assign a variable to the repeating decimal, which will represent our eventual fraction.
• Determine the repeating block of numbers.
• Use multiplication to shift the decimal point, setting up an equation to eliminate the repeating portion.
• Perform subtraction and simple arithmetic to isolate and solve for the original decimal as a fraction.

This process transforms a potentially daunting task into manageable steps, leading us to an integer ratio that precisely represents our original repeating decimal.

Steps in Fraction Conversion

Converting a repeating decimal to a fraction involves a few key steps. Let's detail each one to clarify the process:
Firstly, identify and label the repeating decimal with a variable \(x\). For instance, if you have \( x = 0.123123123\ldots \), the repeating block is \(123\) here.
Next, to manage the repeating sequence, multiply \(x\) by a power of 10 that suits the length of the repeating block. Since our block is 3 digits long, we multiply by \(1000\) to shift the decimal point three places: \(1000x = 123.123123\ldots\).
Now, you create a second equation with the original decimal: \(x = 0.123123\ldots\). Subtract this from the multiplied equation to eliminate the repeating part: \(1000x - x = 123.123123\ldots - 0.123123\ldots\).
By simplifying the result (e.g., \(999x = 123\)), you isolate \(x\) and end up with the fraction representation \(x = \frac{123}{999}\). This fraction can be further simplified by dividing the numerator and denominator by their greatest common divisor, which is \(3\) here, to get \(\frac{41}{333}\).
This meticulous breakdown in fraction conversion helps ensure precision and accuracy in translating repeating decimals to fractions.

Decimal to Fraction Transformation

The transformation of a decimal into a fraction involves transferring the decimal's numerical value into a form expressed by two integers. This transition can seem tricky with repeating decimals, but the process is straightforward when broken down.
Firstly, recognize the pattern in the repeating decimal. For \(0.123123123\ldots\), the sequence "123" repeats indefinitely. This regularity allows us to structure the transformation process effectively.
Next, manipulate the decimal to eliminate the repeating part using multiplication. By aligning the decimals up to a common repeated state through multiplication and subtraction, we obtain an integer numerator.
In the process, we end with a simple fraction, like \(\frac{123}{999}\). However, to find the most reduced form, we need to simplify this fraction. Find a common divisor, which might be found using the Euclidean algorithm, and divide both numerator and denominator: here, dividing by \(3\) gives us \(\frac{41}{333}\).
By converting decimals to fractions, we uncover the exact ratio of integers that perfectly replicates the value of the original decimal, providing clarity and precision in mathematical representations.