Problem 85 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=.\) (Skills Review pp. \(763,770,771\) ) $$\frac{9}{4} ? 2 \frac{3}{4}$$

Step-by-Step Solution

Answer

\( \frac{9}{4} < 2 \frac{3}{4} \)

Step 1: Convert Mixed Numbers

First, we need to convert \(2 \frac{3}{4}\) into an improper fraction. An improper fraction is a fraction where the numerator (top) is greater than or equal to the denominator (bottom). This is done by multiplying the whole number (2) by the denominator of the fractional part (4) and adding the numerator of the fractional part (3).

Step 2: Compute the Conversion

After converting, we get \(2 \times 4 + 3 = 11\). Thus, \(2 \frac{3}{4}\) is equivalent to \( \frac{11}{4} \).

Step 3: Compare the Fractions

Now we can compare \( \frac{9}{4} \) and \( \frac{11}{4} \). Because they have the same denominator, we just need to compare the numerators. Here, \(9 < 11\). So, \( \frac{9}{4} < \frac{11}{4} \).

Key Concepts

Improper Fractions

Improper fractions are a type of fraction where the numerator is larger than or equal to the denominator. For example, in the fraction \( \frac{9}{4} \), 9 is greater than 4. Such fractions always represent values greater than or equal to 1. They are essential when performing operations like addition, subtraction, multiplication, or division of fractions because they allow for easier calculations.Here's how you can spot or convert to an improper fraction:

• Check if the numerator (top number) is greater than or equal to the denominator (bottom number).
• If you're working with a mixed number, multiply the whole number by the denominator. Then, add the numerator of the fractional part.

Improper fractions are convenient for calculations but can be converted back to mixed numbers for easier interpretation when presenting a final answer.

Mixed Numbers

Mixed numbers combine a whole number and a fractional part, like \( 2 \frac{3}{4} \). They are a straightforward way to understand quantities more than 1 but not whole. To convert a mixed number into an improper fraction, follow these steps:

• Multiply the whole number portion by the fraction's denominator.
• Add the result to the numerator of the fractional part. Place this sum over the original denominator to form an improper fraction.

For our example, \( 2 \frac{3}{4} \) becomes \( \frac{11}{4} \) because \( 2 \times 4 + 3 = 11 \). Mixed numbers are easy to visualize, making them useful in everyday contexts such as cooking or shopping.

Numerator and Denominator

Fractions are composed of two essential parts: the numerator and the denominator. Understanding them is crucial for comparing and calculating fractions.

• The numerator is the top part of a fraction and denotes how many parts of a whole are considered.
• The denominator is the bottom part, indicating into how many parts the whole is divided.

When fractions have the same denominator, comparing them becomes straightforward because you only need to look at the numerators. For instance, with \( \frac{9}{4} \) and \( \frac{11}{4} \), the denominator is the same (4). This means you only compare 9 and 11 to see which fraction is larger.

Inequality Symbols

Inequality symbols such as \( <, >, \) and \( = \) are used to compare two values or expressions. They are fundamental in mathematics for expressing relationships:

• \( < \) means 'less than.'
• \( > \) means 'greater than.'
• \( = \) means 'equal to.'

When you compare fractions like \( \frac{9}{4} \) and \( \frac{11}{4} \), you use these symbols to show their relationship. Since 9 is less than 11, you write \( \frac{9}{4} < \frac{11}{4} \). These symbols are vital in expressing solutions to problems and in making clear comparisons between numerical values.