Problem 21 - Algebra 2 and Trigonometry

Question

In \(13-22,\) write each decimal as a common fraction. $$ 0.1 \overline{36} $$

Step-by-Step Solution

Answer

The decimal \( 0.1\overline{36} \) is equal to \( \frac{3}{22} \).

Step 1: Understand the Decimal Notation

The decimal \( 0.1 \overline{36} \) means \( 0.1363636\ldots \) where the digits \( 36 \) repeat infinitely. We need to express this decimal as a fraction.

Step 2: Set Up an Equation for the Repeating Decimal

Let \( x = 0.1\overline{36} \). Thus, \( x = 0.1363636\ldots \).

Step 3: Isolate the Repeating Part

To eliminate the repeating decimal, multiply both sides of the equation \( x = 0.1363636\ldots \) by a power of 10 that matches the length of the repeating part. Here, multiply by 1000 because \( 36 \) has 2 digits. So, \( 1000x = 136.363636\ldots \).

Step 4: Subtract to Eliminate the Repeating Part

Now multiply the original \( x = 0.1363636\ldots \) by 10 to align the repeating parts, giving \( 10x = 1.363636\ldots \). Subtract the \( 10x \) equation from the \( 1000x \) equation: \( 1000x - 10x = 136.363636\ldots - 1.363636\ldots \). This simplifies to \( 990x = 135. \)

Step 5: Solve for \( x \)

Divide both sides by 990 to solve for \( x \):\[ x = \frac{135}{990} \]

Step 6: Simplify the Fraction

Simplify \( \frac{135}{990} \) by finding the greatest common divisor, which is 45, and dividing both the numerator and denominator by it:\[ \frac{135 \div 45}{990 \div 45} = \frac{3}{22} \]

Step 7: Conclusion

Thus, the decimal \( 0.1\overline{36} \) can be written as the fraction \( \frac{3}{22} \).

Key Concepts

Repeating Decimals

Repeating decimals are numbers that contain a set of digits that repeat infinitely after the decimal point. In mathematical notation, a repeating decimal is often denoted by a line, or bar, over the digits that repeat. For example, in the expression \(0.1\overline{36}\), the digits \(36\) repeat endlessly. Understanding repeating decimals is crucial because they provide a compelling connection between the world of decimals, which is endless and continuous, and fractions, which are finite and discrete.
To convert a repeating decimal into a fraction, we use a clever algebraic trick. The idea is to express the decimal as an equation and then manipulate it to eliminate repetition, allowing us to solve for an equivalent fraction. This process allows us to uncover the hidden fraction behind the endless numbers by applying multiplication and subtraction strategically.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form, where the numerator and the denominator are coprime, meaning they have no common factors other than 1. When you simplify a fraction, you make it easier to understand, compare, and work with.
To simplify a fraction, follow these steps:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• What remains is the simplest form of the fraction.

By simplifying a fraction, you transform a potentially complex or large fraction into a version that is more direct and tells the same story as the decimal it originated from in a more digestible way.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that divides two or more numbers without leaving a remainder. Calculating the GCD is essential in simplifying fractions, where you need to reduce fractions to their simplest forms.
To find the GCD of two numbers, you can use several methods:

• Prime Factorization: Break down both numbers into their prime factors and identify the common factors. Multiply these to get the GCD.
• Euclidean Algorithm: This efficient method uses a series of division steps; divide the larger number by the smaller one, and continue with the remainder until it reaches zero. The last non-zero remainder is the GCD.

Understanding the GCD is not just about numerical manipulation; it's about recognizing patterns and relationships within numbers, which are at the heart of mathematics.

Algebraic Equations

Algebraic equations serve as tools to express unknown values in a structured way. They employ variables, constants, and mathematical operations to express relationships. When dealing with repeating decimals, algebraic equations offer a strategic way to unlock the equivalent fraction.
In our example \( x = 0.1\overline{36}\), we set up a system of two equations to isolate the non-repeating and repeating parts:

• First, write the repeating decimal as an equation: \( x = 0.1363636\ldots \).
• Multiply by a power of 10 to shift the repeating part; in this case, multiply by 1000: \( 1000x = 136.363636\ldots \).
• Also multiply by 10 to align potential repeated parts: \( 10x = 1.363636\ldots \).
• Subtract the second equation from the first to remove the repeating section and solve for \( x \).

This manipulation demonstrates the power of algebraic thinking to transform complex, infinite decimals into understandable, finite fractions.