Which one of the following statements is true? Compare \( .98 \ldots . . .980 \) using (<, >, =) A) \( .908<.9008 \)

To compare the two numbers \(0.98\ldots\) (which represents the repeating decimal \(0.980980980\ldots\)) and \(0.9008\), we can analyze their decimal representations. 1. **Understanding \(0.98\ldots\)**: - The number \(0.98\ldots\) can be expressed as \(0.980980980\ldots\), which is a repeating decimal. - This can be interpreted as \(0.98\) followed by the repeating sequence \(0.980\). 2. **Understanding \(0.9008\)**: - The number \(0.9008\) is a finite decimal. Now, let's compare the two numbers: - The first few digits of \(0.980980980\ldots\) are \(0.980\). - The first few digits of \(0.9008\) are \(0.9008\). Now we can compare them digit by digit: - The first digit after the decimal point is the same for both: \(0\). - The second digit is also the same: \(9\). - The third digit is where they differ: \(0.980\) has \(0\) while \(0.9008\) has \(0\). - The fourth digit in \(0.9008\) is \(8\), while \(0.980980980\ldots\) continues with \(9\). Since \(0.980\) is greater than \(0.9008\) (because \(0.980 > 0.9008\)), we can conclude that: \[ 0.980980980\ldots > 0.9008 \] Thus, the correct comparison is: \[ 0.98\ldots > 0.9008 \] Now, let's evaluate the statement \(0.908 < 0.9008\): - Comparing \(0.908\) and \(0.9008\): - The first digit after the decimal point is the same: \(0\). - The second digit is also the same: \(9\). - The third digit of \(0.908\) is \(8\), while the third digit of \(0.9008\) is \(0\). Since \(0.908 > 0.9008\), the statement \(0.908 < 0.9008\) is false. In conclusion, the true statement is: \[ 0.980980980\ldots > 0.9008 \] And the statement \(0.908 < 0.9008\) is false.