Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

change each repeating decimal to a ratio of two integers. $$ \text { 3.929292... } $$

Step-by-Step Solution

Answer

The repeating decimal 3.929292... can be written as the fraction \( \frac{389}{99} \).

Step 1: Identify the repeating part of the decimal

First, observe the decimal number given: 3.929292.... Here, we can see that the digits '92' repeat indefinitely. This is the repeating part of the decimal.

Step 2: Set up an equation to represent the number

Let \( x = 3.929292... \). This equation represents our repeat decimal as a variable \( x \).

Step 3: Multiply to move the repeating part

Multiply \( x \) by 100 to move the repeating part two decimal places to the right (since '92' is repeated every two digits). This gives us: \( 100x = 392.929292... \).

Step 4: Subtract to eliminate the repeating part

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

Step 5: Solve for x

To find the value of \( x \), divide both sides of the equation by 99: \[x = \frac{389}{99}\].Thus, the repeating decimal 3.929292... is equivalent to the fraction \( \frac{389}{99} \).

Step 6: Simplify the fraction

Check if the fraction \( \frac{389}{99} \) can be simplified by finding the greatest common divisor (GCD) of 389 and 99. Since the GCD is 1, the fraction is already in its simplest form.

Key Concepts

Rational numbers

When we talk about rational numbers, we're diving into a fundamental concept in mathematics. At its core, a rational number is any number that can be expressed as a fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. For instance, - The number 1/2 is a rational number because it is the ratio of two integers. - The same goes for -5/3 and 8. Rational numbers can either terminate after a few decimal places or they can repeat in a certain pattern indefinitely. In the example of 3.929292..., this repeating decimal showcases that it's a rational number. The beauty of rational numbers lies in their ability to represent quantities precisely using simple fractions, making them immensely useful for arithmetic, algebra, and many real-world applications. They're quite practical anytime you communicate or compute exact values.

Decimal representation

Numbers often have different forms of representation, and decimal form is one of the most common. Decimals can either terminate, like 0.75, or they can go on indefinitely—often repeating. Repeating decimals, such as 3.929292..., can be converted to fractions because they represent rational numbers. To identify the repeating part: look for a sequence of digits that keeps recurring. Here '92' repeats. This pattern is crucial in setting up an equation to convert the decimal into a fraction. Understanding decimal representation is key to mastering mathematical skill sets as it assists in precisely determining and handling repeating patterns in numbers, which then can be used in further mathematical computations or real-world analyses.

Fraction simplification

Fraction simplification is an important skill that makes calculations easier and clearer. Once you've obtained a fraction like \( \frac{389}{99} \), the next step is to simplify it. Simplifying means expressing a fraction in its smallest possible terms by finding the greatest common divisor (GCD) of the numerator and the denominator. - First, determine if there's a number greater than 1 that both the numerator and denominator can be divided by.- In this case, the GCD of 389 and 99 is 1, meaning the fraction \( \frac{389}{99} \) is already simplified.Simplification is especially handy in calculations or when comparing different fractions. It ensures you're working with the most straightforward form, easing interpretation and further mathematical manipulation. Mastering this concept strengthens mathematical understanding and efficiency.