Problem 68 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=\). \(1.5 ? \frac{5}{5}\)

Step-by-Step Solution

Answer

1.5 > 1

Step 1: Simplify the fraction

Firstly, evaluate the fraction to simplify it. \( \frac{5}{5} \) simplifies to 1.

Step 2: Compare Numbers

Next, 1.5 and 1 are compared. 1.5 is larger than 1.

Step 3: Write the correct statement

The correct statement is '1.5 > 1', since 1.5 is larger than 1.

Key Concepts

Comparing Numbers

When comparing numbers, our primary goal is to determine which number is greater, smaller, or if both numbers are equal. In mathematics, we use specific symbols to express these relationships clearly:

• The symbol \(<\) indicates that the first number is less than the second.
• The symbol \(>\) indicates that the first number is greater than the second.
• The symbol \(=\) indicates that both numbers are equal.

To compare numbers like 1.5 and 1, straightforward observation or subtraction is often the trick. Subtracting them can help us realize that 1.5 - 1 gives 0.5, which is a positive number. This result tells us that 1.5 is indeed greater than 1. Another simple method is to visualize or use a number line, which helps confirm that 1.5 is positioned to the right of 1, reinforcing that it is larger.

Simplifying Fractions

Simplifying fractions is a foundation in mathematics that involves reducing a fraction to its simplest form. A fraction is in simplest form when the numerator and the denominator have no common factors other than 1. For instance, the fraction \(\frac{5}{5}\) is considered equal to 1 because 5 divided by 5 equals 1.

The simplification process can be streamlined using the following steps:

• First, find the greatest common divisor (GCD) of the numerator and the denominator.
• Then, divide both the numerator and the denominator by the GCD.

In our example, both the numerator and denominator in \(\frac{5}{5}\) are the same, hence their GCD is 5. Dividing both by this number simplifies the fraction to 1, which makes it more manageable and easier to compare with other values.

Mathematical Statements

Mathematical statements are expressions that always yield a true or false result. They are important in forming logical conclusions based on numerical data. In the exercise provided, the aim is to express a relationship between two numbers using an inequality symbol.

Here’s how we crafted our mathematical statement:

• We started with comparing two numbers, 1.5 and the simplified value of \(\frac{5}{5}\), which is 1.
• Then, using the comparison results, we confidently state that 1.5 is greater than 1.
• Finally, we write the statement as '1.5 > 1' to correctly show this comparison.

Mathematical statements like these help form the backbone of logical reasoning, essential for deeper mathematical understanding and problem-solving.