Problem 43 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true statement. $$\frac{8}{9} \circ 0.888 \dots$$

Step-by-Step Solution

Answer

\(\frac{8}{9} = 0.888\ldots\)

Step 1: Understand the Problem

We need to compare the fraction \(\frac{8}{9}\) and the repeating decimal \(0.888\ldots\), represented as \(0.\overline{8}\). We will decide whether \(\frac{8}{9}\) is less than, greater than, or equal to \(0.888\ldots\).

Step 2: Convert Fraction to Decimal

First, let's convert the fraction \(\frac{8}{9}\) into a decimal. To do this, divide 8 by 9. When you do the long division or use a calculator, you'll find that \(\frac{8}{9} = 0.\overline{8}\), which translates to the repeating decimal \(0.888\ldots\).

Step 3: Compare the Two Decimals

Now that we have \(\frac{8}{9}\) as \(0.888\ldots\), we can compare it to the decimal expression given in the problem: \(0.888\ldots\). Since both are equal expressions of the same repeating decimal, it follows that \(\frac{8}{9} = 0.888\ldots\).

Step 4: Replace the Symbol

Based on our comparison, we replace \(\circ\) with \(=\). Thus, the true statement is \(\frac{8}{9} = 0.888\ldots\).

Key Concepts

Fraction to Decimal Conversion

Converting fractions to decimals is a fundamental skill in prealgebra. It involves dividing the numerator by the denominator using long division or a calculator. This is helpful when you want to compare fractions with decimals or understand their decimal equivalents.
For example, to convert \( \frac{8}{9} \) into a decimal, you divide 8 by 9. The process of dividing might involve several steps, but once you complete it, you'll see that \( \frac{8}{9} = 0.888\ldots \). This decimal goes on infinitely and has a repeating pattern of 8.
Understanding this conversion helps because it allows you to express fractions in a familiar form—decimals—which are often easier to comprehend in everyday scenarios.

Repeating Decimals

A repeating decimal is a decimal number where some digit or group of digits repeats indefinitely. It's common when converting fractions into decimals. In our example, \( 0.888\ldots \) is a repeating decimal where the digit 8 repeats continuously.
To denote a repeating decimal, a line or a bar is placed over the repeating digit or group of digits. Thus, \( 0.888\ldots \) can also be written as \( 0.\overline{8} \). This shorthand notation makes it clear which digits repeat.
Identifying and writing repeating decimals is crucial because it confirms when two values, such as a fraction and its decimal representation, are indeed equal. By comparing these formats, you ensure you have the same value expressed differently.

Comparing Fractions and Decimals

When comparing fractions and decimals, converting one into the other format is often necessary. This alignment simplifies the comparison. Once in the same form, like decimals, comparing them becomes straightforward.
In our case, both \( \frac{8}{9} \) and \( 0.888\ldots \) were written as decimals, \( 0.888\ldots \) and \( 0.\overline{8} \). This approach shows clearly that they are the same number, making \( \frac{8}{9} = 0.888\ldots \).
Understanding how to switch between and compare these numbers reinforces essential skills in mathematical reasoning and enhances your ability to work with different numeric expressions efficiently.