Adding Fractions with Unlike Denominators

Study Notes

Common denominators are necessary to add fractions; bottom numbers must be the same.
Finding the least common multiple (LCM) of denominators helps in determining the common denominator.
Multiples of 9: 9, 18, 27, 36
Multiples of 3: 3, 6, 9; Common multiple found is 9, which is the LCM.
Example:
- For 4/9: Already has a denominator of 9, remains as 4/9.
- For 1/3: Convert by multiplying both numerator and denominator by 3 to get 3/9.
After renaming, add the numerators: 4 + 3 = 7; keep denominator as 9.
Final result: 7/9, which is already in simplest form (only common factor is 1).

New problem: 3/10 + 2/6.
Multiples of 10: 10, 20, 30, 40.
Multiples of 6: 6, 12, 18, 24; extend to find common multiple which is 30.
Convert:
- For 3/10: Multiply numerator and denominator by 3 for equivalent 9/30.
- For 2/6: Multiply both by 5 to convert to 10/30.
Add the new fractions: 9 + 10 = 19; keep denominator as 30.
Final result: 19/30, also in simplest form (only common factor is 1).

Steps to add fractions:
- Find a common denominator (use LCM).
- Rename fractions with that common denominator.
- Add numerators and keep the common denominator.
- Simplify the result if possible.
Additional resources and links provided for further understanding of multiples and LCM.

A common denominator is essential for adding fractions; both denominators need to be identical.
The least common multiple (LCM) of the denominators is used to establish the common denominator.
For fractions with denominators of 9 and 3, the LCM is 9, as multiples of 9 include 9, 18, 27, 36, and multiples of 3 include 3, 6, 9.
In the example of 4/9, no conversion is necessary since the denominator is already 9.
To convert 1/3 to a fraction with a denominator of 9, multiply both the numerator and denominator by 3, resulting in 3/9.
After renaming fractions, add the numerators: 4 + 3 = 7, while maintaining the denominator of 9.
The sum, 7/9, is in its simplest form with no common factors other than 1.

In a new problem involving 3/10 and 2/6, establish a common denominator by finding the LCM.
Multiples of 10 include 10, 20, 30, and 40, while multiples of 6 are 6, 12, 18, and 24; the LCM here is 30.
For 3/10, convert it to 9/30 by multiplying both numerator and denominator by 3.
For 2/6, converting involves multiplying both parts by 5 to achieve 10/30.
Add the converted fractions by summing the numerators: 9 + 10 = 19, while keeping the denominator at 30.
The final result of 19/30 is also in the simplest form, as the only common factor is 1.

Steps for adding fractions with unlike denominators include:
- Determine a common denominator through the least common multiple (LCM).
- Rename each fraction using the identified common denominator.
- Add numerators while keeping the denominator the same.
- Simplify the final fraction if possible to its lowest terms.
Additional resources are available for further understanding of multiples and LCM concepts.