To find the decimal equivalent of the fraction [tex]\(\frac{4}{15}\)[/tex], we need to perform the division of 4 by 15.
Here, let's break down the steps:
1. Set up the Division:
[tex]\[
\frac{4}{15} = 0.266666666666\ldots
\][/tex]
2. Understanding Repeating Decimals:
- When we divide 4 by 15, we get a recurring decimal. This means that after a certain point, the digits start to repeat indefinitely.
- In this case, the repeating part is "6". Hence, we write this as [tex]\(0.2\overline{6}\)[/tex], where the bar over 6 indicates that this digit repeats indefinitely.
3. Checking the Options:
- Option 1: 0.26: This does not match our calculated value because it rounds off after two decimal places.
- Option 2: [tex]\(0.2 \overline{6}\)[/tex]: This matches our calculated value exactly, indicating the decimal repeats after the first non-recurring digit.
- Option 3: [tex]\(\overline{2.6}\)[/tex]: This implies that the entire sequence '2.6' repeats indefinitely, which is not correct.
- Option 4: [tex]\(\overline{0.26}\)[/tex]: This implies that the digit sequence '26' repeats, which is not correct according to our calculation.
Given these steps and comparisons, the correct decimal representation of [tex]\(\frac{4}{15}\)[/tex] is [tex]\(0.2\overline{6}\)[/tex], which corresponds to the second option:
[tex]\[
0.2 \overline{6}
\][/tex]
