Problem 4 - Calculus Single Variable

Question

Convert the decimal to a rational fraction. (Ellipses are included in some exercises to indicate repetition.) \(2.222 \ldots\)

Step-by-Step Solution

Answer

\(2.222\ldots = \frac{20}{9}\).

Step 1: Understand Repeating Decimal

The decimal number given is \(2.222\ldots\), which is a repeating decimal. The "2" repeats indefinitely.

Step 2: Set up the Equation

Let \(x = 2.222\ldots\). This means \(x\) represents the repeating decimal.

Step 3: Multiply to Shift the Decimal

To eliminate the repeating part, multiply both sides of the equation by 10, the power of 10 needed to shift the repeating part one place to the left: \(10x = 22.222\ldots\).

Step 4: Subtract to Eliminate the Repeating Decimal

Subtract the original equation \(x = 2.222\ldots\) from the multiplied equation \(10x = 22.222\ldots\):\[10x - x = 22.222\ldots - 2.222\ldots\]This yields \(9x = 20\).

Step 5: Solve for \(x\)

Divide both sides of the equation by 9 to solve for \(x\):\[x = \frac{20}{9}\]

Step 6: Simplify the Fraction

The fraction \(\frac{20}{9}\) is already in its simplest form, as 20 and 9 have no common factors other than 1.

Key Concepts

Converting Decimals to Fractions

Converting a decimal, especially a repeating one, into a fraction might seem tricky at first, but it's quite straightforward with a little practice. Repeating decimals are those that have one or more digits that repeat indefinitely. In the given example, the decimal number is \(2.222\ldots\), which clearly shows that the digit '2' endlessly repeats.

The first step to convert such a decimal into a fraction is to set up an equation where you equate the repeating decimal to a variable, usually denoted as \(x\). For instance, let \(x = 2.222\ldots\). Then, to move the decimal point right and align the repeating part, we multiply both sides of this equation by 10. This operation aids in shifting the decimal point so that the repeating sequence aligns neatly for elimination through subtraction.

Remember:

• Set the repeating decimal equal to \(x\).
• Multiply by a power of 10 to shift the repeating part.
• Perform subtraction to eliminate the repeating decimal.

Following these steps results in a clean equation that isolates the repeating portion, laying the groundwork to express it as a rational fraction.

Simplifying Fractions

Once the decimal has been converted into a fraction, simplification becomes the next task. Simplifying a fraction means making it as concise as possible without changing its value. The fraction is in its simplest form when the numerator and denominator are integers that have no common divisors other than 1.

In our example, after converting \(2.222\ldots\) to a fraction, we obtained \(\frac{20}{9}\). To check if this fraction is already simplified, analyze the greatest common divisor (GCD) of the numerator and denominator. Since 20 and 9 only have 1 as a common divisor, \(\frac{20}{9}\) is already simplified.

Simplification tips:

• Find the GCD of the numerator and denominator.
• Divide both by their GCD.
• Ensure the result is a fraction with no common divisors except 1.

By ensuring steps adhere to these fundamentals, you obtain a fraction that is as simple as possible, making it easier to understand and work with.

Mathematical Equations

Mathematical equations are essential tools used to express quantities and relationships between numbers. In this example, they played a vital role in converting a repeating decimal into a fraction. The interplay between setting up the initial equation \(x = 2.222\ldots\) and performing operations like multiplication and subtraction showcases the power of equations in solving real-life math problems.

An equation represents balance; the two sides are equal, and tools like multiplication or subtraction transform these balances into simpler, more workable forms. Here is what the main steps involved:

• Formulate the equation by equating \(x\) to the repeating decimal.
• Manipulate the equation by multiplying to adjust decimal placement.
• Subtract to eliminate repeating parts, simplifying the equation.

Understanding how to craft and solve equations is invaluable in mathematics. They not only help simplify numerical relationships but also enhance problem-solving skills crucial in many scientific and engineering fields.