91Ó°ÊÓ

When performing operations like addition or subtraction with fractions, they need to have the same denominator. This common denominator allows us to easily combine the fractions.
For example, let's consider the fractions \( \frac{3}{4} \) and \( \frac{2}{5} \). Adding these fractions directly isn't straightforward because their denominators are different. To solve this, we convert them to fractions with a common denominator.
To find the common denominator, we look for the least common multiple (LCM) of the denominators. In this case, the LCM of 4 and 5 is 20. We can now rewrite these fractions with a denominator of 20:
\[ \frac{3}{4} = \frac{3 \cdot 5}{4 \cdot 5} = \frac{15}{20} \]
\[ \frac{2}{5} = \frac{2 \cdot 4}{5 \cdot 4} = \frac{8}{20} \]
Now that both fractions have a common denominator, they can be added or subtracted with ease.

The least common multiple (LCM) is essential for finding a common denominator. The LCM of two or more numbers is the smallest number that is a multiple of all the numbers. Let's explore how to find the LCM.
For the numbers 4 and 5, we list their multiples:
- Multiples of 4: 4, 8, 12, 16, 20, 24, ...
- Multiples of 5: 5, 10, 15, 20, 25, 30, ...
The first common multiple is 20, so the LCM of 4 and 5 is 20.
For our exercise, we needed the LCM of 1, 3, and 6, which is 6. This LCM helps us convert each fraction so they all share the same denominator.

When adding fractions, it's crucial that they share a common denominator. Consider the example from our exercise:
\[ \frac{5}{1} + \frac{5}{3} \]
After finding the common denominator (which we found to be 6), we converted them:
\[ \frac{5}{1} = \frac{30}{6}, \ \frac{5}{3} = \frac{10}{6} \]
Now, we can add the fractions by simply adding their numerators:
\[ \frac{30}{6} + \frac{10}{6} = \frac{40}{6} \]
This step ensures that the fractions are combined correctly without altering their values.

Subtraction of fractions works similarly to addition. We need a common denominator to make the operation straightforward.
After adding in our exercise:
\[ \frac{30}{6} + \frac{10}{6} \]
We also need to subtract:
\[ \frac{40}{6} - \frac{5}{6} \]
Both fractions have the same denominator, so we can simply subtract the numerators:
\[ \frac{40 - 5}{6} = \frac{35}{6} \]
This gives us the final answer, \( \frac{35}{6} \), in its simplest form.