Problem 52 - Calculus: Early Transcendentals

Question

Express the number as a ratio of integers. \(0 . \overline{46}=0.46464646 \ldots\)

Step-by-Step Solution

Answer

The ratio is \( \frac{46}{99} \).

Step 1: Assign Variable to the Repeating Decimal

Let's assign a variable to the repeating decimal. Let \( x = 0.\overline{46} \). This implies \( x = 0.46464646 \ldots \).

Step 2: Multiply to Eliminate the Repeating Part

Multiply both sides of the equation by 100 to shift the decimal point two places to the right, matching the repeat cycle length: \( 100x = 46.464646 \ldots \).

Step 3: Subtract the Original Equation

Subtract the original equation \( x = 0.464646 \ldots \) from the new equation \( 100x = 46.464646 \ldots \). This gives:\( 100x - x = 46.464646 \ldots - 0.464646 \ldots \).

Step 4: Solve for x

Subtract to get \( 99x = 46 \). From this equation, solve for \( x \) by dividing both sides by 99: \( x = \frac{46}{99} \).

Step 5: Simplify the Fraction

Check if the fraction \( \frac{46}{99} \) can be simplified. Since 46 and 99 have no common divisors other than 1, \( \frac{46}{99} \) is already in its simplest form.

Key Concepts

Understanding Ratios and Proportions

When we express a repeating decimal like \(0 . \overline{46}\) as a ratio of integers, we're essentially finding a fraction that equals the decimal. A ratio compares two numbers, often expressing the relationship between them. In this case, the goal is to express the repeating decimal as a ratio of two integers.

• Repeating decimals have an pattern that goes on forever. It's crucial to identify this repeating cycle, which helps in converting it to a fraction.
• By setting a variable to the repeating decimal and manipulating it (like multiplying by a power of ten), we can create an equation that allows us to solve for the decimal as a fraction.

Understanding the connection between decimals and fractions is essential for problem-solving. With repeating decimals, the ability to turn them into ratios is a powerful skill for both mathematical calculations and real-world applications.

Mastering Fraction Simplification

After deriving the fraction \(\frac{46}{99}\) from the repeating decimal, it's important to check if it can be simplified. Simplifying fractions is about reducing them to their most straightforward form by dividing both the numerator and the denominator by their greatest common divisor.Finding the Greatest Common Divisor (GCD):

• The GCD of 46 and 99 is 1 since they have no other common divisors. Hence, \(\frac{46}{99}\) is already in its simplest form.

Keeping fractions in their simplest form makes them easier to work with, whether you’re performing operations or making comparisons. Simplification helps in understanding the underlying relationship between the numbers represented more effectively.

Basic Algebra in Action

The process of converting a repeating decimal like \(0 . \overline{46}\) to a fraction relies heavily on basic algebra principles. Here's how it works:Using Variables and Equations:

• Assign a variable to the repeating decimal. For instance, let \(x = 0.\overline{46}\)
• Create a new equation by multiplying the original variable equation by a power of 10 that aligns with the repeating length, e.g., \(100x = 46.464646\ldots\)
• Subtract the original equation from this new equation to eliminate the repeating part. This subtraction isolates the integer part of the expression.
• Finally, solve for \(x\) by isolating it, which involves dividing by the coefficient that appears in front of \(x\).

Through these algebraic steps, we transform a repeating decimal into a neat fraction form. This method highlights the power of algebra in turning complex numerical problems into more familiar forms, making them simpler to interpret and handle.