Problem 102 - Elementary Algebra

Question

For exercises \(85-108\), write \(>\) or \(<\) between the numbers to make a true statement. $$ \frac{3}{5} \quad \frac{5}{9} $$

Step-by-Step Solution

Answer

\(\frac{3}{5} > \frac{5}{9}\)

Step 1: Understanding the Problem

To determine if \(\frac{3}{5}\) is greater than or less than \(\frac{5}{9}\), compare the two fractions.

Step 2: Find a Common Denominator

To compare fractions, it's often helpful to have a common denominator. The denominators are 5 and 9. The least common multiple (LCM) of 5 and 9 is 45.

Step 3: Convert to Equivalent Fractions

Convert each fraction to an equivalent fraction with 45 as the denominator. \(\frac{3}{5} = \frac{3 \times 9}{5 \times 9} = \frac{27}{45}\). Now \(\frac{5}{9} = \frac{5 \times 5}{9 \times 5} = \frac{25}{45}\).

Step 4: Compare the Numerators

Now that both fractions have a common denominator, compare the numerators. Since \(\frac{27}{45} > \frac{25}{45}\), it follows that \(\frac{3}{5}\) is greater than \(\frac{5}{9}\).

Step 5: Write the Conclusion

Based on the comparison, write the appropriate symbol between the original fractions. \(\frac{3}{5} > \frac{5}{9}\).

Key Concepts

least common multiple

When comparing fractions, it's crucial to find a common denominator. The Least Common Multiple (LCM) of the denominators lets you compare fractions easily. The LCM is the smallest number that is a multiple of both denominators.
For instance, to compare \( \frac{3}{5} \) and \( \frac{5}{9} \), we need the LCM of 5 and 9.
To find the LCM, list the multiples of each number:
* Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45...
* Multiples of 9: 9, 18, 27, 36, 45...
The least common multiple is 45.
This means 45 is the smallest number both denominators can divide into without leaving a remainder.

equivalent fractions

Equivalent fractions represent the same part of a whole, even if they appear different.
For example, \( \frac{3}{5} \) can be expressed with a denominator of 45:
* Multiply both numerator and denominator by the same number (in this case, 9): \( \frac{3 \times 9}{5 \times 9} = \frac{27}{45} \)
Similarly, convert \( \frac{5}{9} \) to a fraction with denominator 45:
* Multiply both numerator and denominator by the same number (in this case, 5): \( \frac{5 \times 5}{9 \times 5} = \frac{25}{45} \)
Now both fractions have the same denominator, making it easier to compare them directly.

numerator comparison

Once the fractions have the same denominator, comparing them is simple. Look at the numerators.
For example, to compare \( \frac{27}{45} \) and \( \frac{25}{45} \), notice that they have the same denominator (45).
Compare the numerators: 27 and 25.
Since 27 is greater than 25, \( \frac{27}{45} > \frac{25}{45} \).
Hence, \( \frac{3}{5} \) is greater than \( \frac{5}{9} \).
This method of numerator comparison makes it easy to see which fraction is larger when their denominators are equal.