Problem 79 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=.\) (Skills Review pp. \(763,770,771\) ) $$\frac{8}{3} ? 2 \frac{1}{3}$$

Step-by-Step Solution

Answer

The completed statement should read: \(\frac{8}{3} > 2 \frac{1}{3}\)

Step 1: Convert the mixed number \(2 \frac{1}{3}\) to an improper fraction

In order to convert a mixed number to an improper fraction, multiply the whole number (in this case, 2) by the denominator of the fractional part (in this case, 3), and then add the numerator of the fractional part (in this case, 1). This result will be the numerator of the improper fraction, and the denominator will be same as in the original fractional part. Following these steps, \(2 \frac{1}{3}\) can be converted to \(\frac{7}{3}\).

Step 2: Compare the fractions

Now that both numbers are in fractions, they can be easily compared. We have \(\frac{8}{3}\) and \(\frac{7}{3}\). Since 8 is greater than 7, \(\frac{8}{3}\) is greater than \(\frac{7}{3}\). Therefore, the correct symbol to use is \(>\).

Key Concepts

Mixed Numbers

Mixed numbers are numbers that consist of two parts: a whole number and a fraction. They offer a way to express values that are more than a whole but less than the next increment. For instance, the mixed number \(2 \frac{1}{3}\) represents the whole number 2 and the fraction \(\frac{1}{3}\).

• Components: The whole number part and the fractional part.
• Representation: Written as a whole number followed by a fraction, such as \(5 \frac{3}{4}\).
• Usage: Mixed numbers are often used in everyday contexts where measurements require an additional fractional value, like in distances, weights, or recipes.

It's important to recognize that mixed numbers can also be converted into improper fractions, especially for mathematical operations like addition or comparison to make calculations easier.

Improper Fractions

An improper fraction is a type of fraction where the numerator (the top number) is larger than the denominator (the bottom number). This might seem a bit unusual because we often think of fractions as parts of a whole. In improper fractions, the numerator can be seen as encompassing more than one whole part.

• Transformation: You can convert mixed numbers to improper fractions by multiplying the whole number by the denominator, then adding the numerator to this product. The denominator remains the same. For example, for \(2 \frac{1}{3}\), multiply 2 by 3, add 1, resulting in \(\frac{7}{3}\).
• Benefits: Improper fractions are helpful in mathematical calculations, like operations of addition, subtraction, and comparison, as they allow for easier manipulation of numbers.

Improper fractions might look a bit confusing, but they are quite powerful in expressing quantities greater than one whole. They are pivotal in comparing such values straightforwardly.

Fraction Comparison

Comparing fractions involves determining which fraction is greater, lesser, or if they are equal. To make comparisons easier, it's beneficial to have fractions with like denominators. Here’s how you can compare fractions:

• Same Denominators: If two fractions share the same denominator, compare the numerators directly. The fraction with the larger numerator is the greater one. For example, comparing \(\frac{8}{3}\) and \(\frac{7}{3}\), since 8 is greater than 7, \(\frac{8}{3}\) is the larger fraction.
• Different Denominators: Make the denominators the same using techniques like finding a common denominator or cross-multiplication, then compare the numerators as above.

Fraction comparison is a fundamental skill in mathematics, helping us understand the relationships between different quantities.