Problem 44 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true sentence. $$\frac{34}{99} \circ 0.3 \overline{4}$$

Step-by-Step Solution

Answer

\(\frac{34}{99} < 0.3\overline{4}\)

Step 1: Convert Decimal to Fraction

First, we need to express the decimal \(0.3\overline{4}\) as a fraction. Let \(x = 0.3\overline{4}\). This can be expressed as \(x = 0.344444...\). Since the decimal repeats every single digit after the decimal point, we set \(10x = 3.44444...\), and \(100x = 34.4444...\). Now, subtract the equations: \(100x - 10x = 34.4444... - 3.4444...\), which simplifies to \(90x = 31\). Thus, \(x = \frac{31}{90}\).

Step 2: Compare the Fractions

Now, we have the fractions \(\frac{34}{99}\) and \(\frac{31}{90}\). To compare them, find a common denominator or cross-multiply. Using cross-multiplication, we calculate \(34 \times 90 = 3060\) and \(31 \times 99 = 3069\).

Step 3: Determine the Relationship

The results show \(3060 < 3069\), which indicates that \(\frac{34}{99} < \frac{31}{90}\). Therefore, \(\frac{34}{99} < 0.3\overline{4}\).

Key Concepts

Understanding Recurring Decimals

Recurring decimals are decimals that have a digit or group of digits that repeat infinitely. For example, in the decimal \(0.3\overline{4}\), the number \(4\) continues repeating. Being able to identify and work with recurring decimals is crucial in various mathematical problems, particularly when converting these decimals to fractions.
One of the first steps in dealing with recurring decimals is recognizing the repeating part. The notation \(0.3\overline{4}\) makes it clear that the digit \(4\) is the repeating part. By understanding and noting this repetition, you can facilitate the process of converting the decimal into a more manageable form, such as a fraction.
When you convert a recurring decimal to a fraction, you rely on the understanding that the repetitive sequence adds a regular pattern to the decimal. This pattern can often be expressed as a simple fraction, making it much easier to work with in mathematical calculations.

The Cross-Multiplication Technique

When comparing two fractions, cross-multiplication provides an efficient way to determine which is greater without needing to find a common denominator. It's a quick comparison technique that helps compare the values of fractions.
In cross-multiplication, you multiply the numerator of one fraction by the denominator of the other and vice versa. For instance, to compare \(\frac{34}{99}\) and \(\frac{31}{90}\), you compute \(34 \times 90\) and \(31 \times 99\). The cross products are \(3060\) and \(3069\), respectively. Since \(3060 < 3069\), this tells us that \(\frac{34}{99} < \frac{31}{90}\).
Cross-multiplication provides a straightforward comparison without the need to simplify the fractions first. This method is especially helpful when you're dealing with fractions that do not have obvious common denominators and can save a substantial amount of time in your calculations.

Converting Decimals to Fractions

Converting decimals to fractions involves expressing the decimal number as a ratio of two integers. This conversion is particularly useful when dealing with recurring decimals.
For a decimal like \(0.3\overline{4}\), you start by assigning a variable \(x\) to the repeating decimal. By creating an equation for \(10x\) and \(100x\), you eliminate the repeating part through subtraction. In our example, \(10x = 3.4444...\) and \(100x = 34.4444...\). Subtracting the first equation from the second gives \(90x = 31\), simplifying to \(x = \frac{31}{90}\).
Understanding how to convert decimals into fractions expands your mathematical toolkit, allowing for more versatile applications in algebra and number theory. The conversion process simplifies the problem and opens up new ways of analyzing and solving for different mathematical operations. This skill thus provides a clear advantage in many quantitative situations.