Problem 39 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true sentence. $$3 \frac{4}{9} \circ 3 . \overline{4}$$

Step-by-Step Solution

Answer

Replace with \(=\). Both are equal.

Step 1: Understand the numbers

We have two numbers: \(3 \frac{4}{9}\) and \(3 . \overline{4}\). \(3 \frac{4}{9}\) is a mixed number, which means \(3 + \frac{4}{9}\). \(3 . \overline{4}\) is a repeating decimal, which means \(3.4444\dots\), where the \(4\) repeats indefinitely.

Step 2: Convert the Mixed Number to Decimal

Convert \(3 \frac{4}{9}\) to a decimal by first rewriting it as an improper fraction: \(\frac{27}{9} + \frac{4}{9} = \frac{31}{9}\). Now, divide \(31\) by \(9\) to get approximately \(3.4444\). The decimal is also repeating, \(3.\overline{4}\).

Step 3: Compare the Two Values

Now we compare \(3.\overline{4}\) from the mixed number, with \(3.\overline{4}\), the repeating decimal. Since both numbers are equal, place the equal sign \(=\) between them.

Key Concepts

Mixed Numbers

Mixed numbers are a combination of whole numbers and fractions. They are useful because they provide a clear way to represent quantities that are more than a whole but not quite another whole. For instance, in the problem, we have the mixed number \(3 \frac{4}{9}\), which combines the whole number \(3\) and the fraction \(\frac{4}{9}\). This means "three and four-ninths."

• To work with mixed numbers, you can convert them into improper fractions. An improper fraction is where the numerator is larger than the denominator. It makes calculations easier especially when converting to decimals or comparing numbers.
• For example, \(3 \frac{4}{9}\) becomes \(\frac{31}{9}\) when converted into an improper fraction. This helps in comparing it with other values such as decimals.

Repeating Decimals

Repeating decimals are decimals in which one or more digits repeat infinitely. Recognizing repeating patterns in decimals helps to understand numbers beyond simple fractions. In our example, the repeating decimal is \(3.\overline{4}\), which stands for \(3.4444\ldots\) with the digit '4' repeating forever.

• Repeating decimals can often be linked back to fractions. For example, \(\frac{1}{3}\) equals \(0.\overline{3}\) or \(0.3333\ldots\)
• They can usually be written in shorthand form by placing a line over the repeating digit or digits, which indicates those digits continue forever.

Decimal Conversion

Decimal conversion involves changing numbers from one form, like a fraction or mixed number, into a decimal form. This is essential when we want to compare different types of numbers accurately.

• For a mixed number like \(3 \frac{4}{9}\), after converting it to an improper fraction, dividing the numerator by the denominator gives the decimal equivalent. Here, \(\frac{31}{9}\) converts to approximately \(3.4444\), which matches \(3.\overline{4}\) as a repeating decimal.
• Understanding how to convert between these forms allows better comprehension of the relationships and equivalencies between fractions, decimals, and mixed numbers.

Remember, converting and comparing these forms, like we did in this exercise, leads to understanding whether numbers are truly equal or different.