Problem 64 - Algebra and Trigonometry

Question

Express the repeating decimal as a fraction. $$ 0.2 \overline{53} $$

Step-by-Step Solution

Answer

The repeating decimal \( 0.2 \overline{53} \) as a fraction is \( \frac{251}{990} \).

Step 1: Express the Decimal as an Equation

Start by setting the repeating decimal equal to a variable. Let \( x = 0.2535353\ldots \) where '53' is the repeating part.

Step 2: Eliminate the Repeating Part

Multiply both sides of the equation by 100 to shift the decimal point two places to the right, since '53' has two digits. This gives \( 100x = 25.353535\ldots \).

Step 3: Subtract to Remove the Decimal

Now subtract the original \( x = 0.2535353\ldots \) from \( 100x = 25.353535\ldots \) to eliminate the repeating part. This gives: \[\begin{aligned} 100x - x &= 25.353535\ldots - 0.2535353\ldots \ 99x &= 25.1 \end{aligned}\]

Step 4: Solve for x

Solve the equation for \( x \) by dividing both sides by 99: \[ x = \frac{25.1}{99} \].

Step 5: Convert to Fraction

Convert 25.1 into a fraction, recognizing that 25.1 = 25 + 0.1 = \(\frac{251}{10} \). Substitute back into the equation: \[ x = \frac{\frac{251}{10}}{99} = \frac{251}{990}.\]

Step 6: Simplify the Fraction

To simplify \( \frac{251}{990} \), check for any common factors. Since 251 is a prime number and does not divide 990, the fraction is already in its simplest form.

Key Concepts

Fraction Conversion

Converting a repeating decimal into a fraction involves expressing the decimal in terms of a mathematical equation. Think of this process as identifying hidden fractions within the decimal. For repeating decimals, we first need to set the decimal equal to a variable. This helps us to manage the decimal part systematically. Once we have assigned a variable, we perform operations to isolate and convert the repeating section into a usable format. The key is "capturing" the repeating numbers by moving the decimal point accurately. This method efficiently transfers the repeating numbers into the fraction format, which is much easier to work with in further calculations.

Decimal Representation

Decimal representation is a standard way of showing numbers that have fractional parts. Decimals can be either terminating or repeating. Terminating decimals end after a few digits, such as 0.5 or 0.125. Repeating decimals, like 0.2\overline{53}, have one or more digits after the decimal point that repeat indefinitely.
To express repeating decimals, a line or bar is commonly placed above the repeated numbers (e.g., \( 0.2\overline{53} \)). This signals that the sequence of digits '53' recurs over and over. It's critical to recognize these patterns when trying to convert decimals to fractions since this recognition plays a significant role in systematically solving and simplifying the expression.

Simplification of Fractions

Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator have no common factors other than 1. For example, our resultant fraction \( \frac{251}{990} \) is already in its simplest form, as 251 is a prime number. This means it cannot divide evenly into 990.
When simplifying fractions:

• Identify any common factors between the numerator and the denominator. If there are any, divide them out.
• If the numerator or denominator is a prime number, chances are, the fraction is already simplified.
• A simplified fraction provides cleaner and more understandable results for further calculations or interpretations.

Algebraic Equations

When dealing with repeating decimals, algebraic equations become essential tools. These equations help us transform a repeating decimal into a fraction by providing a structured method to isolate and capture the repeating parts.
Creating equations like \( x = 0.2535353\ldots \) involves understanding how to operate on the equation to liberate the repeating string.
Here's a step-by-step on how this can be done:

• Assign the repeating decimal to a simple variable, like \( x \).
• Multiply \( x \) by a power of 10 that matches the number of repeating digits. This aligns the repeating digits for subtraction.
• Subtract the original equation from this new equation to eliminate the repeating part.
• Solve for \( x \) to find its fractional form.

This method of manipulating the equation unlocks the repeating decimal, revealing the fraction within.