Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

Change each repeating decimal to a ratio of two integers \(2.56565656 \ldots\)

Step-by-Step Solution

Answer

The repeating decimal \(2.56565656\ldots\) is \(\frac{254}{99}\) as a ratio.

Step 1: Express the repeating decimal with variables

Let \( x = 2.56565656\ldots \). Notice that the digits '56' repeat indefinitely.

Step 2: Eliminate the repeating part by multiplication

Since two digits repeat, multiply \( x \) by 100 to shift the repeating part: \[ 100x = 256.565656\ldots \]

Step 3: Subtract to isolate the repeating part

Subtract the equation from Step 1 from the equation in Step 2:\[ 100x - x = 256.565656\ldots - 2.565656\ldots \]\[ 99x = 254 \]

Step 4: Solve for x

Divide both sides of the equation by 99:\[ x = \frac{254}{99} \]

Step 5: Simplify the fraction

Check if \( \frac{254}{99} \) can be simplified. Since 254 and 99 have no common factors other than 1, the fraction is already in simplest form.

Key Concepts

Repeating Decimals

Repeating decimals are numbers with one or more digits after the decimal point that repeat indefinitely. For example, in the number 2.56565656..., the group of digits '56' repeats over and over. Understanding repeating decimals is a crucial step in converting them into fractions. Many students wonder how these numbers can be accurately represented as fractions. Repeating decimals often help in identifying numbers that are actually rational numbers. When working with repeating decimals:

• Identify the repeating sequence of digits.
• Use a variable to represent the entire decimal number.
• Multiply the variable through by an appropriate power of ten to "slide" the repeating sequence to the left of the decimal point.

This method allows you to effectively eliminate the repeating portion and convert it into a fraction, which can then be simplified.

Rational Numbers

A key concept is understanding that repeating decimals are a way to express rational numbers. Rational numbers are numbers that can be expressed as the quotient or ratio of two integers, where the denominator is not zero.Examples of rational numbers include whole numbers, such as 3 or -7, and fractions like \( \frac{1}{2} \) or \( \frac{254}{99} \). Decimal numbers that terminate or repeat indefinitely also fall under the category of rational numbers because they can be converted into fractions.Converting a repeating decimal to a rational number involves recognizing the decimal structure and translating this periodic repetition into the numerator and denominator of a fraction. This process showcases the intriguing nature of numbers and offers insight into the deep connection between decimals and fractions.

Ratio of Integers

The ratio of integers is a fundamental mathematical concept that is essential in expressing rational numbers. This is any pair of two integers where one integer is divided by another.For example:

• In \( \frac{254}{99} \), 254 is the numerator and 99 the denominator. This creates a ratio that meaningfully represents a repeating decimal.
• If you encounter a repeating decimal like 0.333..., it translates to the ratio \( \frac{1}{3} \).

Creating a ratio of integers from a decimal allows you to express the number as a precise fraction. This can be useful for exact calculations or to better understand the number's properties, as fractions often provide clearer insights than long decimals.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form. This means ensuring that the numerator and the denominator have no common factors other than 1.To simplify a fraction:

• Identify any common factors in both the numerator and denominator.
• Divide both terms by their greatest common divisor.

In the given example, the fraction \( \frac{254}{99} \) is already in its simplest form because 254 and 99 have no common factors other than 1. This means there is nothing further to reduce.Simplifying fractions helps in making calculations easier and achieving the most precise and understandable form of a ratio, offering a clearer picture of the overall value.