Problem 42 - Calculus Early Transcendentals: Pearson New International Edition

Question

change each repeating decimal to a ratio of two integers. $$ 0.399999 \ldots $$

Step-by-Step Solution

Answer

0.399999\ldots = \frac{2}{5}.

Step 1: Understand the Problem

We need to express the repeating decimal \(0.399999\ldots\) as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers.

Step 2: Define the Variable

Let \(x = 0.399999\ldots\). This equation represents the repeating decimal.

Step 3: Multiply to Isolate the Repetition

By multiplying both sides of \(x = 0.399999\ldots\) by 10, we shift the decimal one place to the right. This gives us \(10x = 3.99999\ldots\).

Step 4: Subtract to Eliminate the Repeating Part

Subtract the original equation from the new equation: \(10x - x = 3.99999\ldots - 0.39999\ldots\). This simplifies to \(9x = 3.6\).

Step 5: Solve for the Variable

Divide both sides by 9 to get \(x = \frac{3.6}{9}\).

Step 6: Simplify the Fraction

Simplify \(\frac{3.6}{9}\) by converting 3.6 to a fraction and then simplifying it: \(\frac{36}{90} = \frac{4}{10} = \frac{2}{5}\). Thus, \(0.399999\ldots = \frac{2}{5}\).

Key Concepts

Converting Repeating Decimals to Fractions

Have you ever looked at a repeating decimal and wondered how it could be expressed as a fraction? Repeating decimals have a digit or a group of digits after the decimal point that keeps occurring infinitely. In the example of 0.399999..., the number 9 repeats forever. Transforming such a repeating decimal into a fraction is a handy skill to learn. Here’s a simple and friendly way to do it:

• First, identify the repeating part of the decimal.
• Then, set up an equation where the repeating decimal equals a variable, like x.
• Multiply the entire equation by a power of 10 that moves the decimal point to the right, aligning it just past one full cycle of the repeating digits.
• Subtract the original equation from the new, shifted equation to eliminate the repeating portion.
• Solve the resulting simple equation for x to find it as a fraction.

By following these steps, the infinite repetition vanishes as if by magic, leaving you with a beautiful simple fraction. It’s an insightful peek into the neatness of numbers!

Simplifying Fractions

After you convert a repeating decimal into a fraction, the next step is fraction simplification. This is the process of rewriting the fraction in its simplest form. Simplifying involves dividing both the numerator and the denominator by their greatest common divisor (GCD).Simplifying might seem trivial, but it’s essential:

• A simplified fraction is easier to understand and use in further calculations.
• It reveals the true ratio in the smallest whole numbers.

Let’s take the fraction \(\frac{36}{90}\) as an example. To simplify it, you can follow these steps:

• List the factors of the numerator (36: 1, 2, 3, 4, 6, 9, 12, 18, 36) and the denominator (90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90).
• Identify the largest factor they have in common, which is 18 in this case.
• Divide both numerator and denominator by 18, resulting in \(\frac{2}{5}\).

A fraction like \(\frac{2}{5}\) is now in its simplest form, representing the same value with lesser complexity.

Step-by-Step Math Solutions

Step-by-step math solutions break down mathematical problems into their most basic parts, making them easier to comprehend. This methodical approach is especially helpful for complex problems, like converting repeating decimals to fractions.Why use step-by-step solutions? Here are a few reasons:

• They make challenging problems more manageable by isolating each underlying concept.
• You can see the logic and reasoning behind every move, promoting deeper understanding.
• Each step builds upon the previous one, giving confidence that you won’t miss any critical detail.

In our example problem with the decimal \(0.399999\ldots\), each step was designed to solve a piece of the puzzle all the way from identifying the repeating decimal, setting up the initial equation, through to simplifying the final fraction. This logical progression turns a seemingly daunting problem into something entirely doable! By using this approach, students learn not just to get the right answer, but also understand the underlying mathematical principles at play.