To add the fractions [tex]\(\frac{4}{9}\)[/tex] and [tex]\(\frac{7}{12}\)[/tex], follow these steps:
1. Find the Least Common Denominator (LCD):
The denominators of the fractions are 9 and 12. The least common multiple (LCM) of 9 and 12 can be calculated as follows:
[tex]\[
\text{LCM}(9, 12) = 36
\][/tex]
Therefore, the least common denominator (LCD) is 36.
2. Adjust the Numerators based on the Common Denominator:
- For [tex]\(\frac{4}{9}\)[/tex]:
[tex]\[
\text{Adjusted numerator} = 4 \times \left(\frac{36}{9}\right) = 4 \times 4 = 16
\][/tex]
So, [tex]\(\frac{4}{9}\)[/tex] becomes [tex]\(\frac{16}{36}\)[/tex].
- For [tex]\(\frac{7}{12}\)[/tex]:
[tex]\[
\text{Adjusted numerator} = 7 \times \left(\frac{36}{12}\right) = 7 \times 3 = 21
\][/tex]
So, [tex]\(\frac{7}{12}\)[/tex] becomes [tex]\(\frac{21}{36}\)[/tex].
3. Sum the Adjusted Numerators:
Now, add the numerators while keeping the denominator the same:
[tex]\[
\frac{16}{36} + \frac{21}{36} = \frac{16 + 21}{36} = \frac{37}{36}
\][/tex]
4. Simplify the Resulting Fraction:
- The numerator is 37, and the denominator is 36.
- The greatest common divisor (GCD) of 37 and 36 is 1 (since 37 is a prime number and does not share any factors with 36 besides 1).
Since the GCD is 1, the fraction [tex]\(\frac{37}{36}\)[/tex] is already in its simplest form.
Therefore, the answer is:
[tex]\[
\frac{4}{9} + \frac{7}{12} = \frac{37}{36}
\][/tex]
