91Ó°ÊÓ

When working with fractions, combining them often requires a common denominator. This is especially crucial in addition or subtraction of fractions because it standardizes the fractions to a shared base which simplifies the process.

To find a common denominator:

• Identify the least common multiple (LCM) of the denominators.
• Rewrite each fraction equivalent to the new denominator by converting each fraction's denominator to the LCM and adjusting the numerator accordingly.

For example, in our exercise, the fractions \(\frac{2}{3}\) and \(\frac{1}{9}\):
1. The LCM of 3 and 9 is 9.
2. Rewrite \(\frac{2}{3}\) as \(\frac{6}{9}\). Now both fractions have a denominator of 9, making subtraction straightforward: \(\frac{6}{9} - \frac{1}{9} = \frac{5}{9}\).
Similarly, for the fractions \(\frac{3}{4}\) and \(\frac{5}{6}\):
1. The LCM of 4 and 6 is 12.
2. Rewrite \(\frac{3}{4}\) as \(\frac{9}{12}\) and \(\frac{5}{6}\) as \(\frac{10}{12}\). Adding them together gives \(\frac{9}{12} + \frac{10}{12} = \frac{19}{12}\). Familiarizing yourself with finding common denominators will significantly ease your handling of fraction operations.

Multiplying fractions is straightforward compared to addition or subtraction.

Here's how you do it:

• Multiply the numerators of the fractions together to get the new numerator.
• Multiply the denominators of the fractions together to get the new denominator.

In our problem, after simplifying the expressions in both the numerator and the denominator, we rewrite the complex fraction \(\frac{\frac{5}{9}}{\frac{19}{12}}\) as a multiplication by taking the reciprocal of the denominator fraction. So, \(\frac{\frac{5}{9}}{\frac{19}{12}}\) becomes \(\frac{5}{9} \times \frac{12}{19}\). Now we multiply as follows:

• Step 1: Multiply the numerators: 5 × 12 = 60
• Step 2: Multiply the denominators: 9 × 19 = 171

The resulting fraction is \(\frac{60}{171}\). Remember to always look for common factors to see if there's an opportunity to simplify. In this case, \(\frac{60}{171}\) is already in its simplest form since 60 and 171 have no common factors other than 1.

Subtracting fractions entails bringing them to a common denominator, just as when adding them, but then we subtract the numerators instead.

The steps are:

• Find a common denominator, often the least common multiple (LCM) of the denominators.
• Convert the fractions to equivalent forms with the common denominator.
• Subtract the numerators of the equivalent fractions while keeping the common denominator.

Let's take the initial problem in the numerator: \(\frac{2}{3} - \frac{1}{9}\). Here's how to approach it:

• The LCM of 3 and 9 is 9.
• Convert \(\frac{2}{3}\) to \(\frac{6}{9}\).
• Now the subtraction is \(\frac{6}{9} - \frac{1}{9}\).
• Subtract the numerators: 6 - 1 = 5.

So, \(\frac{6}{9} - \frac{1}{9} = \frac{5}{9}\). Having a grasp of common denominators will certainly simplify fraction subtraction and prevent mistakes.