Problem 42 - Basic College Mathematics

Question

Use \(=\) or \(\neq\) for \(\square\) to write a true sentence. $$ \frac{2}{6} \square \frac{6}{18} $$

Step-by-Step Solution

Answer

\(\frac{2}{6} = \frac{6}{18}\)

Step 1: Simplify the Fractions

First, simplify both fractions to their lowest terms. For \(\frac{2}{6}\), divide both the numerator and the denominator by their greatest common divisor, which is 2. \(\frac{2 \/ 2}{6 \/ 2} = \frac{1}{3}\). Next, simplify \(\frac{6}{18}\) by dividing both the numerator and denominator by their greatest common divisor, which is 6. \(\frac{6 \/ 6}{18 \/ 6} = \frac{1}{3}\).

Step 2: Compare the Simplified Fractions

Compare \(\frac{1}{3}\) and \(\frac{1}{3}\). Since both fractions are equal, \eq\ can be used to write a true sentence.

Step 3: Insert the Correct Symbol

Finally, \eq\ is the correct symbol to insert between \(\frac{2}{6}\) and \(\frac{6}{18}\): \(\frac{2}{6} = \frac{6}{18}\).

Key Concepts

simplifying fractions

One of the first steps in working with fractions is simplifying them. Simplifying a fraction means reducing it to its smallest form, where the numerator and denominator can no longer be divided by the same number (other than 1). This process makes fractions easier to understand and compare. To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator. Then, you divide both the numerator and the denominator by this number.
For example, to simplify \(\frac{2}{6}\), find the GCD of 2 and 6, which is 2. Divide both by 2: \(\frac{2 \/ 2}{6 \/ 2} = \frac{1}{3}\). Notice how \(\frac{2}{6}\) is reduced to \(\frac{1}{3}\).
Simplifying fractions can make your math work more straightforward and help in comparing different fractions. Let's move on to understanding how to find the GCD.

greatest common divisor

The greatest common divisor (GCD) is the highest number that can evenly divide both the numerator and the denominator of a fraction. Finding the GCD is a crucial step for simplifying fractions. There are several methods to find the GCD, but one of the simplest ways is the

comparing fractions

Comparing fractions can be a bit tricky, especially when the denominators are different. However, once fractions are simplified, it becomes much easier. In the case of our exercise, we simplified \(\frac{2}{6}\) and \(\frac{6}{18}\) to \(\frac{1}{3}\) each. When both fractions have the same denominator or are simplified to the same form, you can easily see that they are equal.
If fractions have different denominators, there are other methods to compare them, such as finding a common denominator or converting them into decimals. These methods allow you to see which fraction is larger or if they are equal. Knowing how to compare fractions can save you time and help you make accurate decisions in your math problems.