Sure! Let's go through the process of adding the fractions [tex]\(\frac{5}{8}\)[/tex] and [tex]\(\frac{5}{12}\)[/tex] step-by-step and simplify it if needed.
### Step 1: Find a Common Denominator
To add fractions, we first need a common denominator. The denominators are [tex]\(8\)[/tex] and [tex]\(12\)[/tex].
The least common multiple (LCM) of [tex]\(8\)[/tex] and [tex]\(12\)[/tex] can be found.
Factors of [tex]\(8\)[/tex]: [tex]\(8 = 2^3\)[/tex]
Factors of [tex]\(12\)[/tex]: [tex]\(12 = 2^2 \times 3\)[/tex]
The LCM will be the highest power of each prime number that appears in the factorization of the denominators:
LCM = [tex]\(2^3 \times 3 = 8 \times 3 = 24\)[/tex]
So, the common denominator is [tex]\(24\)[/tex].
### Step 2: Convert Each Fraction to Equivalent Fractions with the Common Denominator
Convert [tex]\(\frac{5}{8}\)[/tex] and [tex]\(\frac{5}{12}\)[/tex] to fractions with a denominator of [tex]\(24\)[/tex]:
For [tex]\(\frac{5}{8}\)[/tex]:
[tex]\[
\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}
\][/tex]
For [tex]\(\frac{5}{12}\)[/tex]:
[tex]\[
\frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{10}{24}
\][/tex]
### Step 3: Add the Fractions
Now that we have the fractions with a common denominator, we can add them:
[tex]\[
\frac{15}{24} + \frac{10}{24} = \frac{15 + 10}{24} = \frac{25}{24}
\][/tex]
So the sum of [tex]\(\frac{5}{8}\)[/tex] and [tex]\(\frac{5}{12}\)[/tex] is [tex]\(\frac{25}{24}\)[/tex].
Since [tex]\(\frac{25}{24}\)[/tex] cannot be simplified further and it is already in its simplest form, our final answer is:
[tex]\[
\frac{25}{24}
\][/tex]
Thus, the sum of [tex]\(\frac{5}{8}\)[/tex] and [tex]\(\frac{5}{12}\)[/tex] is [tex]\(\frac{25}{24}\)[/tex].
