Problem 101 - Elementary Algebra

Question

For exercises \(85-108\), write \(>\) or \(<\) between the numbers to make a true statement. $$ \frac{3}{4} \quad \frac{5}{9} $$

Step-by-Step Solution

Answer

\( \frac{3}{4} > \frac{5}{9} \)

Step 1 - Find a common denominator

To compare fractions, first find a common denominator. The denominators are 4 and 9. The least common multiple (LCM) of 4 and 9 is 36.

Step 2 - Convert to equivalent fractions

Convert each fraction to an equivalent fraction with the common denominator. For \(\frac{3}{4}\): multiply the numerator and the denominator by 9 to get \(\frac{27}{36}\). For \(\frac{5}{9}\): multiply the numerator and the denominator by 4 to get \(\frac{20}{36}\).

Step 3 - Compare the numerators

Now compare the numerators of the equivalent fractions. Since \(\frac{27}{36}\) has a larger numerator than \(\frac{20}{36}\), \(\frac{3}{4}\) is greater than \(\frac{5}{9}\).

Step 4 - Write the correct inequality symbol

Based on the comparison, write \(>\) between the original fractions: \(\frac{3}{4} > \frac{5}{9}\).

Key Concepts

Least Common Multiple (LCM)

When comparing fractions, it's necessary to have the same denominator. This is where the Least Common Multiple (LCM) comes in handy. The LCM of two numbers is the smallest number that both can divide into without a remainder. For example, the LCM of 4 and 9 is 36. To find the LCM, list the multiples of each number:

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
Multiples of 9: 9, 18, 27, 36, ...
The smallest common multiple is 36. Now use that to convert each fraction to have the same denominator.

Equivalent Fractions

To compare fractions with different denominators, convert them into equivalent fractions with a common denominator. An equivalent fraction of a given fraction \(\frac{a}{b}\) can be found by multiplying the numerator and the denominator by the same number. For instance, let's take the fractions \(\frac{3}{4}\) and \(\frac{5}{9}\):

• To convert \(\frac{3}{4}\) to have a denominator of 36, multiply both the numerator & denominator by 9: \(\frac{3 \times 9}{4 \times 9} = \frac{27}{36}\).


• To convert \(\frac{5}{9}\) to have a denominator of 36, multiply both the numerator & denominator by 4: \(\frac{5 \times 4}{9 \times 4} = \frac{20}{36}\).

The fractions \(\frac{27}{36}\) and \(\frac{20}{36}\) are now equivalent fractions with the same denominator.

Inequalities

To compare fractions with the same denominator, look at the numerators:

Which is greater, 27 or 20? Since 27 is larger, \(\frac{27}{36}\) is greater than \(\frac{20}{36}\). Thus, \(\frac{3}{4} > \frac{5}{9}\). Remember, when writing inequality symbols, the opening of the symbol faces the larger number.

• Use \(>\) if the fraction on the left is greater
• Use \(<\) if the fraction on the left is smaller

This way, you can correctly compare and order fractions using inequalities.