To find the decimal equivalent of the fraction [tex]\(\frac{2}{27}\)[/tex], we need to divide [tex]\(2\)[/tex] by [tex]\(27\)[/tex].
First, we start by conducting the division:
1. We divide [tex]\(2\)[/tex] by [tex]\(27\)[/tex], which gives [tex]\(0.\)[/tex] (since [tex]\(2 < 27\)[/tex], the integer part is [tex]\(0\)[/tex]).
2. We need to continue the division beyond the decimal point. This involves considering [tex]\(2.000000...\)[/tex] and dividing each digit in sequence by [tex]\(27\)[/tex].
3. [tex]\(20\)[/tex] divided by [tex]\(27\)[/tex] is [tex]\(0\)[/tex] (quotient), so we put a zero in the decimal place and bring down the next digit.
4. [tex]\(200\)[/tex] divided by [tex]\(27\)[/tex] gives roughly [tex]\(7.\)[/tex] (since [tex]\(27 \times 7 = 189\)[/tex]), leaving us with a remainder of [tex]\(11\)[/tex].
5. Bringing down the next digit, we have [tex]\(110\)[/tex]. This [tex]\(110\)[/tex] divided by [tex]\(27\)[/tex] gives roughly [tex]\(4\)[/tex] (since [tex]\(27 \times 4 = 108\)[/tex]), leaving us with a remainder of [tex]\(2\)[/tex].
6. Then, we bring down the next digit, making it [tex]\(20\)[/tex] again, essentially starting a repeating pattern.
This sequence continues, demonstrating that [tex]\(\frac{2}{27}\)[/tex] converts to a repeating decimal. The repeating part is "074".
Therefore, the decimal equivalent of [tex]\(\frac{2}{27}\)[/tex] is [tex]\(0.\overline{074}\)[/tex].
Thus, the correct answer is:
[tex]\[
0 . \overline{074}
\][/tex]
