Problem 37 - Basic College Mathematics

Question

Use \(=\) or \(\neq\) for \(\square\) to write a true sentence. $$ \frac{3}{4} \square \frac{9}{12} $$

Step-by-Step Solution

Answer

\( \frac{3}{4} = \frac{9}{12} \)

Step 1: Simplify the first fraction

The first fraction is \( \frac{3}{4} \). This fraction is already in its simplest form.

Step 2: Simplify the second fraction

To simplify \( \frac{9}{12} \), find the greatest common divisor (GCD) of 9 and 12, which is 3. Divide the numerator and the denominator by their GCD: \[ \frac{9 \div 3}{12 \div 3} = \frac{3}{4} \]

Step 3: Compare the simplified fractions

Both fractions \( \frac{3}{4} \) and \( \frac{3}{4} \) are equal. Therefore, the correct operator is \( = \).

Key Concepts

equal fractions

Fractions are considered equal if they represent the same value or proportion, even if they look different at first glance. In the problem above, you need to determine whether \( \frac{3}{4} \) and \( \frac{9}{12} \) are equal. To do this:

• Simplify both fractions.
• Compare their simplest forms.

After simplifying both fractions to \( \frac{3}{4} \), it's clear that they are equal. Therefore, the correct operator between \( \frac{3}{4} \) and \( \frac{9}{12} \) is \( = \). This shows that both fractions represent the same part of a whole.

simplest form

A fraction is in its simplest form when the numerator and the denominator have no common divisors other than 1. To simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both parts of the fraction by the GCD.

For example, in the given problem, \( \frac{9}{12} \) is simplified by dividing both the 9 and 12 by their GCD, which is 3. This gives us: \[ \frac{9 \div 3}{12 \div 3} = \frac{3}{4} \] Simplified fractions are easier to work with and compare.

greatest common divisor (GCD)

The greatest common divisor (GCD) is the largest number that can exactly divide both the numerator and the denominator of a fraction. Here's how to find it:

• List all the factors of the numerator and the denominator.
• Identify the largest factor that appears in both lists.

For the fraction \( \frac{9}{12} \), the factors of 9 are 1, 3, and 9; the factors of 12 are 1, 2, 3, 4, 6, and 12. The largest common factor is 3. Therefore, the GCD of 9 and 12 is 3. Using the GCD to simplify fractions ensures they are in their simplest form. It streamlines the process of comparing different fractions to see if they are equal.