Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

Change each repeating decimal to a ratio of two integers \(3.929292 \ldots\)

Step-by-Step Solution

Answer

The repeating decimal \(3.929292\ldots\) as a ratio is \( \frac{389}{99} \).

Step 1: Express the Decimal as an Equation

Let \( x = 3.929292\ldots \), which repeats every two digits.

Step 2: Multiply to Isolate the Repeating Part

Since the repeating block consists of two digits, multiply both sides of the equation by 100 to shift the decimal point: \( 100x = 392.929292\ldots \).

Step 3: Subtract to Eliminate Repetition

Subtract the original equation from the above equation to eliminate the repeating decimal: \( 100x - x = 392.929292\ldots - 3.929292\ldots \). This simplifies to \( 99x = 389 \).

Step 4: Solve for x

Divide both sides by 99 to find \( x \): \( x = \frac{389}{99} \).

Step 5: Simplify the Fraction

Check if \( \frac{389}{99} \) can be simplified. Since 389 is a prime number and does not divide evenly into 99, this fraction is already in its simplest form.

Key Concepts

Convert Decimal to Fraction

Converting a repeating decimal into a fraction can seem tricky, but it's a systematic process. It involves transforming the decimal into a mathematical equation. Let's break this down into simple steps. Start by assigning the repeating decimal to a variable. For instance, let’s set the repeating decimal \(x = 3.929292\ldots\).

• Notice that the repeating part here is "92", which consists of two digits.
• To manage this repetition, multiply both sides by 100. This multiplication shifts the decimal two places to the right. So, you get: \(100x = 392.929292\ldots\).

You’ll see that the repeated sequence "92" remains the same one step further. By forming these equations, you're essentially preparing to simplify your work into a straightforward subtraction task.

Ratio of Two Integers

At this stage, we aim to express the repeating decimal as a ratio of two integers—this is the ultimate objective of converting a decimal into a fraction.To do this, subtract the original equation from the obtained equation to eliminate the repeating decimal part.

• The repetitive segment "92" cancels itself out, simplifying your equation:
• From \(100x - x = 392.929292\ldots - 3.929292\ldots\) you get \(99x = 389\).

This step effectively reduces the looping decimal to a solvable algebraic equation, representing it as a ratio of whole numbers. Finding this ratio is critical because it turns an infinitely repeating decimal into a manageable and simplified mathematical expression.

Simplify Fractions

Once you have the fraction, it’s important to simplify it if possible. Simplification makes the expression easier to understand and work with. In our example, we found the fraction:\(x = \frac{389}{99}\).

• Now, check whether this fraction can be simplified.
• To do this, determine whether the numerator and the denominator share any common factors other than 1.

The numerator, 389, is a prime number, which means its only divisors are 1 and 389. Unfortunately, 389 does not divide evenly into 99, indicating that the fraction is already in its simplest form.Simplifying fractions is an essential skill because it reduces complexity and improves clarity in mathematical problems, allowing for easier comparison, addition, or subtraction of fractions with other fractions.