Problem 22 - Algebra 2 and Trigonometry

Question

In \(13-22,\) write each decimal as a common fraction. $$ 0.15 \overline{90} $$

Step-by-Step Solution

Answer

The decimal \(0.15\overline{90}\) as a fraction is \(\frac{7159}{45000}\).

Step 1: Identify the Periodic and Non-Periodic Parts

The given decimal is \(0.15\overline{90}\). This means that \(0.15\) is the non-repeating part and \(0.90\) is the repeating or periodic part of the decimal.

Step 2: Assign Variables to the Decimal

Let \(x = 0.1590909090\ldots\), where \(0.90\) repeats. This will help in forming equations to eliminate the repeating part.

Step 3: Eliminate Non-Repeating Part

Multiply \(x\) by 100 to shift the non-repeating part left of the decimal: \[100x = 15.90909090\ldots\]

Step 4: Eliminate Repeating Part

Now multiply \(x\) by 1000 (since 90 has 2 digits) to shift one complete period left of the decimal: \[1000x = 159.0909090\ldots\]

Step 5: Subtract Equations to Eliminate Repeating Decimal

Subtract the equation from Step 3 from the equation in Step 4: \[1000x - 100x = 159.0909090\ldots - 15.90909090\ldots\]\[900x = 143.18\]

Step 6: Solve for x

Divide both sides of the equation by 900 to solve for \(x\): \[x = \frac{143.18}{900}\]Which can be simplified, but represents the fraction form of the decimal.

Step 7: Simplify the Fraction

To simplify \(\frac{143.18}{900}\), recognize that converting the decimal to an integer makes it easier:\[x = \frac{14318}{90000}\]Simplify by finding the greatest common divisor (GCD) which simplifies the fraction to \(\frac{7159}{45000}\).

Key Concepts

Periodic Decimals

Periodic decimals are decimals that have a repeating pattern of digits. In the example provided, we see a decimal written as \(0.15\overline{90}\). Here, the period, also known as the repeating part, is "90." This means that after a certain point in the decimal, the sequence "90" repeats indefinitely.
Understanding periodic decimals is essential as it helps in converting these infinite decimals into finite fractional numbers. A key feature of periodic decimals is the line or bar placed over the repeating section.

• Non-Periodic Part: The numbers before the repeating cycle, such as the "0.15" in our case.
• Periodic Part: The repeating sequence, "90," which follows the non-periodic digits.

When dealing with periodic decimals, the challenge lies in effectively eliminating the repeating cycle, which may seem complex at first, but making use of algebraic methods turns this into a structured process.

Common Fractions

Common fractions are one of the simplest and most intuitive ways to express numbers. They consist of a numerator and a denominator, written as \(\frac{a}{b}\), where "a" is a whole number and "b" is a non-zero whole number.

• Numerator: Represents the number of parts we have.
• Denominator: Denotes the total number of equal parts the whole is divided into.

The process of converting a repeating decimal to a common fraction is exceptionally useful. In the example of \(0.15\overline{90}\), using algebraic techniques, specifically creating and solving equations, allows us to express this repeating decimal as the common fraction \(\frac{7159}{45000}\). This conversion not only provides a more complete representation of the decimal but is also useful in arithmetic operations and understanding the decimal's value in precise terms.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means expressing the fraction such that the numerator and denominator are as small as possible while still maintaining the same value.
For example, consider the fraction \(\frac{14318}{90000}\) which results from the repeating decimal in the solution steps. Simplifying it involves both recognizing and applying the greatest common divisor (GCD) of the numerator and the denominator.

Find GCD: The greatest number that can divide both the numerator and the denominator without a remainder.

Divide: Each by the GCD to simplify the fraction fully.

In our case, the fraction \(\frac{14318}{90000}\) can be simplified using its GCD to \(\frac{7159}{45000}\). Simplified fractions are important because they offer a cleaner, more understandable representation of numbers.