Problem 66 - Beginning and Intermediate Algebra

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make a true statement. See Examples 6 through 8 . $$ \frac{8}{2} \quad \frac{12}{3} $$

Step-by-Step Solution

Answer

The correct statement is \(\frac{8}{2} = \frac{12}{3}\).

Step 1: Simplify the Left Fraction

Start by simplifying the fraction \(\frac{8}{2}\). Divide the numerator (8) by the denominator (2): \[\frac{8}{2} = 4\].

Step 2: Simplify the Right Fraction

Next, simplify the fraction \(\frac{12}{3}\). Divide the numerator (12) by the denominator (3): \[\frac{12}{3} = 4\].

Step 3: Compare the Simplified Values

Now compare the simplified results from both fractions. Both simplify to 4, which means they are equal: \(4 = 4\). Therefore, insert the equality sign (=) between the fractions.

Key Concepts

Fraction Simplification

Simplifying fractions is the process of reducing a fraction to its simplest form. This means finding a fraction equivalent to the original, but with smaller numbers.
To simplify, divide both the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD).
For example, in the fraction \( \frac{8}{2} \), both 8 and 2 can be divided by 2, simplifying the fraction to \( 4 \).
Similarly, for the fraction \( \frac{12}{3} \), since both numbers can be divided by 3, it also simplifies to \( 4 \). Simplifying fractions makes them easier to understand and compare.

Numerator and Denominator

A fraction consists of two main components: the numerator and the denominator. The numerator is the top part of the fraction, while the denominator is the bottom part. These components are separated by a slash.
For the fraction \( \frac{8}{2} \), 8 is the numerator, indicating the number of parts we are considering. 2 is the denominator, showing the total number of equal parts in a whole.
Understanding these components helps in performing operations like addition, subtraction, multiplication, and especially simplification, which relies on knowing which numbers to divide.
Pay attention to both parts for correct fraction manipulation.

Equality of Fractions

Two fractions are considered equal if they have the same value or represent the same proportion of a whole when simplified. This is a key step in comparing fractions.
In the exercise, both fractions \( \frac{8}{2} \) and \( \frac{12}{3} \) simplified to 4. This means these fractions are equal because they represent the same number.
When comparing fractions, always simplify them first.

• If the simplified numerators are the same, then the fractions are equal.
• If they differ, use the appropriate symbol (<, >) to indicate which is larger or smaller.

Simplifying first makes the comparison straightforward and accurate.