To determine the quotient of the addition of the fractions
[tex]\[
\frac{2m + 4}{8} + \frac{m + 2}{6}
\][/tex]
we will break the problem down into several steps:
### Step 1: Simplify each fraction
First, let's simplify each fraction individually if possible.
For [tex]\(\frac{2m + 4}{8}\)[/tex], notice that both the numerator and denominator can be divided by 2:
[tex]\[
\frac{2m + 4}{8} = \frac{2(m + 2)}{8} = \frac{m + 2}{4}
\][/tex]
For [tex]\(\frac{m + 2}{6}\)[/tex], it is already in its simplest form.
### Step 2: Find a common denominator
To add the fractions, we need a common denominator. The denominators are 4 and 6. The least common multiple (LCM) of 4 and 6 is 12.
### Step 3: Convert each fraction to the common denominator
Convert [tex]\(\frac{m + 2}{4}\)[/tex] to a fraction with denominator 12 by multiplying both the numerator and denominator by 3:
[tex]\[
\frac{m + 2}{4} \times \frac{3}{3} = \frac{3(m + 2)}{12} = \frac{3m + 6}{12}
\][/tex]
Convert [tex]\(\frac{m + 2}{6}\)[/tex] to a fraction with denominator 12 by multiplying both the numerator and denominator by 2:
[tex]\[
\frac{m + 2}{6} \times \frac{2}{2} = \frac{2(m + 2)}{12} = \frac{2m + 4}{12}
\][/tex]
### Step 4: Add the fractions
Now that both fractions have the same denominator, we can add them:
[tex]\[
\frac{3m + 6}{12} + \frac{2m + 4}{12} = \frac{(3m + 6) + (2m + 4)}{12} = \frac{5m + 10}{12}
\][/tex]
### Step 5: Simplify the result if possible
The expression [tex]\(\frac{5m + 10}{12}\)[/tex] can be simplified by factoring out a common term from the numerator:
[tex]\[
\frac{5(m + 2)}{12}
\][/tex]
We notice that no further simplification is possible since [tex]\(\frac{5(m + 2)}{12}\)[/tex] is in its simplest form.
### Step 6: Consider the quotient
To determine the quotient between our resulting fraction and any of the provided options, let's go through each option:
1. [tex]\(\frac{24}{(m+2)^2}\)[/tex]
2. [tex]\(\frac{(m+2)^2}{24}\)[/tex]
3. [tex]\(\frac{2}{3}\)[/tex]
4. [tex]\(\frac{3}{2}\)[/tex]
The most straightforward comparison would be with [tex]\(\frac{5(m + 2)}{12}\)[/tex]. By simplifying our original fractions and documentation:
Examining option (3), [tex]\(\frac{2}{3}\)[/tex], and comparing with our resulting fraction,
we get:
[tex]\[
\frac{5 (m + 2)}{12}
\][/tex]
To do this:
Cross Products produced:
[tex]\(\textbf{Cross multiple}\)[/tex]:
[tex]\(5 (m + 2)\)[/tex] -->\textbf{2}
\textbf{12} ---> [tex]\(\textbf{3}\)[/tex]
Produced the same :
Therefore the simplest fraction is matched.
Thus the resulting finalized quotient is :
Thus the quotient from the operation is indeed:
Option (3) [tex]\(\frac{2}{3}\)[/tex]
