To add the fractions [tex]\(\frac{1}{8}\)[/tex] and [tex]\(\frac{1}{12}\)[/tex], follow these steps:
1. Find the Least Common Multiple (LCM) of the Denominators:
- The denominators of the fractions are 8 and 12.
- The prime factorization of 8 is [tex]\(2^3\)[/tex].
- The prime factorization of 12 is [tex]\(2^2 \times 3\)[/tex].
- The LCM is found by taking the highest power of each prime number that appears in these factorizations. Therefore, the LCM of 8 and 12 is [tex]\(2^3 \times 3 = 8 \times 3 = 24\)[/tex].
2. Rewrite Each Fraction with the Common Denominator:
- Convert each fraction so that they both have the LCM (24) as the new denominator.
- For [tex]\(\frac{1}{8}\)[/tex]:
[tex]\[
\frac{1}{8} = \frac{1 \times 3}{8 \times 3} = \frac{3}{24}
\][/tex]
- For [tex]\(\frac{1}{12}\)[/tex]:
[tex]\[
\frac{1}{12} = \frac{1 \times 2}{12 \times 2} = \frac{2}{24}
\][/tex]
3. Add the Numerators Together while Keeping the Common Denominator:
- Now that both fractions have a denominator of 24, add their numerators:
[tex]\[
\frac{3}{24} + \frac{2}{24} = \frac{3 + 2}{24} = \frac{5}{24}
\][/tex]
4. Express the Result in Both Fractional and Decimal Form:
- The sum of [tex]\(\frac{1}{8}\)[/tex] and [tex]\(\frac{1}{12}\)[/tex] in fractional form is [tex]\(\frac{5}{24}\)[/tex].
- To convert [tex]\(\frac{5}{24}\)[/tex] to a decimal, simply divide the numerator by the denominator:
[tex]\[
\frac{5}{24} \approx 0.20833333333333334
\][/tex]
Therefore, the detailed solution is:
[tex]\[
\frac{1}{8} + \frac{1}{12} = \frac{5}{24} \approx 0.20833333333333334
\][/tex]
The intermediate steps can be summarized as:
1. LCM of 8 and 12 is 24.
2. Rewrite each fraction with denominator 24:
- [tex]\(\frac{1}{8} = \frac{3}{24}\)[/tex]
- [tex]\(\frac{1}{12} = \frac{2}{24}\)[/tex]
3. Add the fractions:
- [tex]\(\frac{3}{24} + \frac{2}{24} = \frac{5}{24}\)[/tex]
4. In decimal form, the sum is approximately [tex]\(0.20833333333333334\)[/tex].
The final result is:
[tex]\[
\frac{5}{24} \approx 0.20833333333333334
\][/tex]
