To perform the addition of the fractions [tex]\(\frac{5}{12} + \frac{1}{4} + \frac{1}{18}\)[/tex], we need to follow a series of steps to obtain the result. Here is a detailed, step-by-step solution:
### Step 1: Find the Least Common Denominator (LCD)
The least common denominator (LCD) is the smallest number that all denominators can divide without leaving a remainder. For the given fractions [tex]\(\frac{5}{12}\)[/tex], [tex]\(\frac{1}{4}\)[/tex], and [tex]\(\frac{1}{18}\)[/tex], we need to find the LCD of 12, 4, and 18.
The prime factorization is:
- 12 = [tex]\(2^2 \cdot 3\)[/tex]
- 4 = [tex]\(2^2\)[/tex]
- 18 = [tex]\(2 \cdot 3^2\)[/tex]
The LCD is the highest power of each prime number involved:
- [tex]\(2^2 = 4\)[/tex]
- [tex]\(3^2 = 9\)[/tex]
Thus, the LCD is [tex]\(4 \times 9 = 36\)[/tex].
### Step 2: Convert Each Fraction to an Equivalent Fraction with the LCD as the Denominator
We will rewrite each fraction to have 36 as the denominator:
- Convert [tex]\(\frac{5}{12}\)[/tex]:
[tex]\[
\frac{5}{12} = \frac{5 \times 3}{12 \times 3} = \frac{15}{36}
\][/tex]
- Convert [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\frac{1}{4} = \frac{1 \times 9}{4 \times 9} = \frac{9}{36}
\][/tex]
- Convert [tex]\(\frac{1}{18}\)[/tex]:
[tex]\[
\frac{1}{18} = \frac{1 \times 2}{18 \times 2} = \frac{2}{36}
\][/tex]
### Step 3: Add the Equivalent Fractions
Now that all fractions have a common denominator, we can simply add the numerators:
[tex]\[
\frac{15}{36} + \frac{9}{36} + \frac{2}{36} = \frac{15 + 9 + 2}{36} = \frac{26}{36}
\][/tex]
### Step 4: Simplify the Resulting Fraction
To simplify [tex]\(\frac{26}{36}\)[/tex], we find the greatest common divisor (GCD) of 26 and 36. The GCD is 2:
[tex]\[
\frac{26}{36} = \frac{26 \div 2}{36 \div 2} = \frac{13}{18}
\][/tex]
Therefore, the sum of the fractions [tex]\(\frac{5}{12} + \frac{1}{4} + \frac{1}{18}\)[/tex] simplifies to [tex]\(\frac{13}{18}\)[/tex].
Additionally, in decimal form,
[tex]\[
\frac{13}{18} \approx 0.7222\ldots
\][/tex]
And that is the complete and simplified result:
[tex]\(\boxed{\frac{13}{18}}\)[/tex]
