Problem 44 - Algebra and Trigonometry with Analytic Geometry

Question

Find the rational number represented by the repeating decimal. $$3.2 \overline{394}$$

Step-by-Step Solution

Answer

The rational number is \(\frac{599}{1665}\).

Step 1: Understanding the Decimal

The decimal given is \(3.2\overline{394}\), which means the decimal repeats with '394'. This can be split into two parts: the non-repeating part '3.2' and the repeating decimal part '0.0\overline{394}'.

Step 2: Express as a Sum

Express \(3.2\overline{394}\) as the sum of 3.2 and the repeating part: \(3.2\overline{394} = 3.2 + 0.0\overline{394}\). We'll focus on finding the fraction representation of the repeating part next.

Step 3: Define the Variable for the Repeating Part

Let \(x = 0.0\overline{394}\). This means \(x = 0.0394394\ldots\), where '394' repeats indefinitely.

Step 4: Eliminate the Repeating Part by Using Multiplication

Multiply \(x = 0.0394394\ldots\) by 1000 (since '394' has 3 digits) to move the decimal point and eliminate the repeating part:\(1000x = 39.4394394\ldots\).

Step 5: Setup an Equation to Solve for x

We now have:\[ 1000x = 39.4394394\ldots \]\[ x = 0.0394394\ldots \]Subtract the second equation from the first:\[ 1000x - x = 39.4394394 \ldots - 0.0394394 \ldots \]\(999x = 39.4\).

Step 6: Solve for x

Divide both sides of the equation by 999 to solve for \(x\):\[ x = \frac{39.4}{999} \].Since we started with \(x = 0.0\overline{394}\), we found \(x\) as a fraction.

Step 7: Combine with Non-Repeating Part

Add this result to the non-repeating part (3.2): \[3.2 + \frac{39.4}{999}\].Convert 3.2 to a fraction with a denominator of 1, or 10\(\frac{32}{10}\), for the purpose of adding fractions.

Step 8: Simplify the Expression

To add the fractions, you need a common denominator. Convert \(3.2 = \frac{32}{10}\) to have a common denominator with \(\frac{39.4}{999}\). This gives:\[ \frac{32}{10} = \frac{3200}{9990}\].Add the fractions: \[ \frac{3200}{9990} + \frac{394}{9990} = \frac{3594}{9990} \].

Step 9: Simplify the Fraction Further

Simplify the fraction \(\frac{3594}{9990}\) by finding the greatest common divisor (GCD) of 3594 and 9990, which is 6: \[ \frac{3594 \div 6}{9990 \div 6} = \frac{599}{1665} \].

Step 10: Write the Final Answer

Thus, the rational number represented by the repeating decimal 3.2\overline{394} is \(\frac{599}{1665}\), after simplification.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the ratio of two integers, where the denominator is not zero. These numbers include integers, fractions, and repeating or terminating decimals. They are crucial for understanding how to represent various quantities in a more exact form.

Key characteristics of rational numbers include:

• The ability to express them as a fraction, i.e., \frac{p}{q}\, where both \(p\) and \(q\) are integers, and \(q eq 0\).
• In decimal form, a rational number either terminates (e.g., 3.5, -2.0) or repeats (e.g., 3.666..., -7.111...).

Understanding rational numbers allows us to handle numbers with repeating decimal patterns, like in the original problem where repeating digits "394" lead to a specific fraction. They provide a bridge between simple whole numbers and more complex number forms, facilitating operations and comparisons.

Fractions

Fractions represent a part of a whole and are written as the ratio of two integers, where the numerator is the top number, and the denominator is the bottom number, not equal to zero. Converting repeating decimals to fractions can sometimes seem tricky, but they follow systematic algebraic steps, as demonstrated in the solution.

In the provided problem, the repeating decimal \(3.2\overline{394}\) is first represented as the sum of a non-repeating part \(3.2\) and a repeating part \(0.0\overline{394}\). Turning the repeating part into a fraction involves several steps:

• Identifying the repeating segment of the decimal allows us to set up an algebraic expression.
• Multiplying the equation involving the repeating decimal by a power of 10 shifts the decimal point.
• A subtraction event between the original and shifted equations eliminates the repeating portion, leaving a solvable fraction.

This process makes apparent how mathematical techniques can be applied to produce a fraction that perfectly equates to the repetitive nature of the decimal, preserving its value in a different form.

Algebra

Algebra plays a significant role when converting repeating decimals into fractions. By using variables to represent unknown quantities, algebra creates a framework for solving equations that arise from these conversions.

In our example, algebraic techniques involve:

• Introducing a variable \(x\) to improve clarity in relation to the repeating decimal \(0.0\overline{394}\).
• Creating equations by multiplying the variable by powers of 10, aligning the repeating parts appropriately.
• Forming and solving an equation by subtracting these expressions, enabling the isolation of \(x\).

This mathematical manipulation of equations highlights the power of algebra in solving real-world numerical problems, such as finding the fractional equivalent of decimals with infinite repetitions. Algebra thus simplifies otherwise complex patterns, providing exact values and solutions.