Problem 100 - Elementary Algebra

Question

For exercises \(85-108\), write \(>\) or \(<\) between the numbers to make a true statement. $$ \frac{3}{5} \quad \frac{2}{3} $$

Step-by-Step Solution

Answer

\( \frac{3}{5} < \frac{2}{3} \)

Step 1 - Find a common denominator

To compare the fractions \(\frac{3}{5}\) and \(\frac{2}{3}\), find a common denominator. The common denominator of 5 and 3 is 15.

Step 2 - Convert fractions

Convert \(\frac{3}{5}\) and \(\frac{2}{3}\) to have the common denominator of 15. Multiply the numerator and denominator of \(\frac{3}{5}\) by 3 to get \(\frac{9}{15}\). Multiply the numerator and denominator of \(\frac{2}{3}\) by 5 to get \(\frac{10}{15}\).

Step 3 - Compare fractions

Now compare the two fractions \(\frac{9}{15}\) and \(\frac{10}{15}\). Since 9 is less than 10, \(\frac{9}{15}\) is less than \(\frac{10}{15}\).

Step 4 - Write the inequality

Therefore, the inequality \(\frac{3}{5} \) and \(\frac{2}{3}\) is \(\frac{3}{5} < \frac{2}{3}\).

Key Concepts

common denominator

When comparing fractions, finding a common denominator is an essential step. A common denominator is a shared multiple of the denominators of the fractions involved. In our exercise, the fractions are \(\frac{3}{5}\) and \(\frac{2}{3}\).
First, identify the denominators: 5 and 3.
Next, find the least common multiple (LCM) of these denominators. The LCM of 5 and 3 is 15.
This common denominator makes it easier to compare the two fractions as it converts them into like fractions.
A like fraction means the denominators are the same, and we can directly compare the numerators.
In this way, finding a common denominator simplifies comparing fractions, allowing for a clearer and more accurate comparison.

fraction conversion

Once you find the common denominator, the next step is fraction conversion. This means adjusting each fraction so that they share the same denominator.
For the fractions \(\frac{3}{5}\) and \(\frac{2}{3}\), we convert them both to have the denominator of 15.

Steps to Convert Fractions:

• Take \(\frac{3}{5}\) and multiply both the numerator and the denominator by 3, since 15 divided by 5 equals 3. This gives us \(\frac{9}{15}\).
• Take \(\frac{2}{3}\) and multiply both the numerator and the denominator by 5, since 15 divided by 3 equals 5. This gives us \(\frac{10}{15}\).

After converting, \(\frac{3}{5}\) becomes \(\frac{9}{15}\) and \(\frac{2}{3}\) becomes \(\frac{10}{15}\).
This process allows a direct comparison of the two fractions since their denominators are the same.

inequality

Inequality is a mathematical way to compare two values or expressions. When comparing fractions, inequality symbols such as \(>\) (greater than) or \(<\) (less than) are used.
After conversion, our fractions are \(\frac{9}{15}\) and \(\frac{10}{15}\).

Comparing Fractions:

• Since the denominators are the same, compare the numerators directly.
• Here, 9 is less than 10.
• Thus, \(\frac{9}{15}\) < \(\frac{10}{15}\).

This shows that \(\frac{3}{5}\) is less than \(\frac{2}{3}\).
Using the inequality symbol, we write the final statement as \(\frac{3}{5} < \frac{2}{3}\).
This process of converting to a common denominator and then comparing makes understanding inequalities between fractions straightforward and clear.