Problem 27 - Prealgebra

Question

Write each decimal as a fraction or mixed number in simplest form. $$-0.333 \ldots$$

Step-by-Step Solution

Answer

The decimal \(-0.333\ldots\) as a fraction in simplest form is \(-\frac{1}{3}\).

Step 1: Identify the Repeating Decimal

The decimal given is \(-0.333\ldots\) which is a repeating decimal. The digit '3' repeats indefinitely.

Step 2: Set Up the Equation

Let \( x = -0.333\ldots \). This is our original repeating decimal.

Step 3: Eliminate the Repeating Part

Multiply both sides by 10 to shift the decimal point: \( 10x = -3.333\ldots \).

Step 4: Subtract the Equations

Subtract the original equation from this new equation to eliminate the repeating part: \( 10x - x = -3.333\ldots - (-0.333\ldots) \). This gives \( 9x = -3 \).

Step 5: Solve for x

Divide both sides by 9 to isolate \( x \): \( x = -\frac{3}{9} \).

Step 6: Simplify the Fraction

Simplify \(-\frac{3}{9}\) by dividing the numerator and denominator by their greatest common divisor, which is 3: \(-\frac{3 \div 3}{9 \div 3} = -\frac{1}{3}\).

Step 7: Verify the Solution

Check that \(-0.333\ldots\) equals \(-\frac{1}{3}\) by converting \(-\frac{1}{3}\) back to a decimal: \(-1 \div 3\) results in \(-0.333\ldots\), confirming our solution.

Key Concepts

Simplifying Fractions

Simplifying fractions is a crucial skill in mathematics, especially when converting decimals to fractions. A fraction becomes simplified when it is reduced to its smallest form, meaning the numerator and the denominator have no common factors other than 1.
To simplify a fraction, you should follow these steps:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.
• This will give you the fraction in its simplest form.

For example, in the problem where the fraction \(-\frac{3}{9}\) was simplified, the GCD of 3 and 9 is 3. Dividing the numerator and the denominator by 3 yields \(-\frac{1}{3}\), which is the simplest form of the fraction.

Repeating Decimals

Repeating decimals occur when one or more digits in a decimal keep repeating indefinitely. Understanding repeating decimals is essential when converting them into fractions.
Here's how you can deal with repeating decimals like \(-0.333\ldots\):

• Recognize the repeating pattern, which is often indicated by an overline or ellipsis (e.g., \(-0.3\overline{3}\) or \(-0.333\ldots\)).
• Set the repeating decimal equal to a variable, such as \(x\).
• Multiply by a power of 10 that corresponds to the number of repeating digits. For \(-0.333\ldots\), multiply by 10 to shift the decimal point.
• Set up equations to eliminate the repeating part, making it possible to solve for the variable.

This approach not only helps in understanding repeating decimals but also in converting them into neat and tidy fractions.

Mathematical Problem-Solving

Mathematical problem-solving involves breaking down a complex task into manageable steps. This approach can be applied to various mathematical concepts, including converting decimals to fractions.
Solving the original problem of turning \(-0.333\ldots\) into a fraction involved engaging several problem-solving steps:

• Identify the nature of the problem, such as recognizing a repeating decimal.
• Formulate a strategy, like setting up an equation with a variable to represent the decimal.
• Solve the problem step-by-step, ensuring to follow through with operations like multiplication and subtraction of equations.
• Verify the results for accuracy, such as turning the fraction back into a decimal to check your work.

Using this structured approach not only aids in solving mathematical problems but also enhances understanding and confidence in working with numbers.