Problem 46 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true statement. $$4 . \overline{63} \circ 4 \frac{5}{8}$$

Step-by-Step Solution

Answer

The correct symbol is \(>\), so \(4 .\overline{63} > 4 \frac{5}{8}\).

Step 1: Understand the Numbers

The problem provides two numbers in different forms:1. The number \(4.\overline{63}\) is a repeating decimal, which means \(4.\overline{63} = 4.636363...\).2. The number \(4 \frac{5}{8}\) is a mixed number, where \(4 \frac{5}{8} = 4 + \frac{5}{8}\).

Step 2: Convert the Repeating Decimal to a Fraction

To compare, it might be easier to convert the repeating decimal \(4.\overline{63}\) into a fraction.Let \(x = 4.\overline{63}\).Then, \(100x = 463.6363...\) (since repeating decimal moves two places on multiplication by 100).Subtract: \(100x - x = 463.6363... - 4.6363... \), resulting in \(99x = 459\), thus \(x = \frac{459}{99}\), simplifying to \(\frac{153}{33}\). Further simplifying gives us \(\frac{51}{11}\).

Step 3: Convert the Mixed Number to an Imperfect Fraction

Convert \(4 \frac{5}{8}\) into an improper fraction for easier comparison.\(4 \frac{5}{8} = \frac{4 \times 8 + 5}{8} = \frac{32 + 5}{8} = \frac{37}{8}\).

Step 4: Compare the Fractions

To compare \(\frac{51}{11}\) and \(\frac{37}{8}\), we need a common denominator. A suitable choice could be 88 (which is the least common multiple of 11 and 8).Convert both fractions:- \(\frac{51}{11} = \frac{51 \times 8}{11 \times 8} = \frac{408}{88}\)- \(\frac{37}{8} = \frac{37 \times 11}{8 \times 11} = \frac{407}{88}\).Now, compare \(\frac{408}{88}\) and \(\frac{407}{88}\).

Step 5: Determine the Relationship

Since \(\frac{408}{88} > \frac{407}{88}\), it follows that \(4.\overline{63} > 4 \frac{5}{8}\).

Key Concepts

Repeating Decimals

In mathematics, repeating decimals are decimals that have one or more repeating digits indefinitely. For example, the number \(4.\overline{63}\) is a repeating decimal where the digits '63' keep repeating. These types of decimals often arise when dividing numbers that result in infinite decimal expansions.

Here are some key things to remember about repeating decimals:

• A repeating decimal is indicated by placing a bar over the digits that repeat. In \(4.\overline{63}\), the bar over '63' means that this sequence repeats indefinitely.
• Repeating decimals can be converted to fractions by using algebraic methods. For example, by setting \(x = 4.\overline{63}\) and using multiplication and subtraction, we deduced that it equals \(\frac{51}{11}\).
• This conversion is useful for comparisons, as fractions are often easier to work with than infinite decimal sequences.

Understanding these details can simplify how you approach problems involving repeating decimals and aid in accurate calculations.

Mixed Numbers

Mixed numbers are a combination of whole numbers and fractions, which express quantities greater than one but not whole numbers alone. For instance, in the problem discussed, \(4 \frac{5}{8}\) is a mixed number, made up of the whole number 4 and the fraction \(\frac{5}{8}\).

Here are some important aspects of mixed numbers:

• To perform operations involving mixed numbers (like addition or comparison), it's often needed to convert them to improper fractions. This helps in simplifying calculations.
• For example, we converted \(4 \frac{5}{8}\) into an improper fraction by calculating \(\frac{32 + 5}{8} = \frac{37}{8}\).
• Mixed numbers are useful for expressing everyday quantities in a more intuitive way, like measuring ingredients or distances.

Grasping how to handle mixed numbers can make dealing with such practical problems much more manageable.

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. They represent values greater than or equal to one. In many mathematical operations, converting numbers like mixed numbers or repeating decimals into improper fractions simplifies calculations.

Let's explore some essential points about improper fractions:

• An improper fraction like \(\frac{37}{8}\) implies the total of 37 parts where each part is one-eighth. Thus, it represents a value over one whole.
• Converting mixed numbers into improper fractions involves multiplying the whole number by the denominator and adding the numerator of the fractional part, which restructures numbers for straightforward arithmetic operations.
• For comparison, it's easier to work with improper fractions when finding a common denominator, as seen with our conversion to compare \(\frac{51}{11}\) and \(\frac{37}{8}\).

Recognizing the characteristics and benefits of improper fractions allows for enhanced insight and problem-solving capabilities when working with complex mathematical scenarios.