Decimal representation

Let \(x=0.\overline{2}=0.222\ldots\). Multiplying by 10 gives \(10x=2.222\ldots\). Subtracting the first equation from the second gives \(9x=2\), so \(x=\frac{2}{9}\). Therefore, option A is correct. Exam tip: a decimal with one repeating digit, such as \(0.\overline{a}\), equals \(\frac{a}{9}\), not \(\frac{a}{10}\).

The governing concept is comparison of decimal numbers by their whole-number parts and fractional parts. A positive number that is less than 1 must have whole-number part 0 and a positive decimal part. Among the choices, 0.89 satisfies the inequality 0 < 0.89 < 1, so option B is correct. The number 1.25 is greater than 1, while 1.00 is exactly equal to 1 and therefore is not less than 1. The number 2.09 is also greater than 1. A quick method is to inspect the digit before the decimal point: a positive decimal strictly between zero and one begins with 0, and at least one digit after the decimal must be positive. This confirms that 0.89, and only 0.89 among the choices, meets both conditions.

For \(\frac{7}{12}\), \(12=2^2\times3\); factor 3 makes the decimal recur. A terminating decimal has only 2 and/or 5 in its reduced denominator. Exam tip: factor the denominator first.

(4.000=4) because only zeros appear after the decimal point. Exam tip: such zeros do not change the value.

(0.\overline{45}) is non-terminating recurring, so it is rational. Exam tip: terminating or recurring decimals are rational.

(0.09=\frac{9}{100}) because it has two decimal places. Exam tip: count decimal places carefully after (0).

The denominator of (\frac{1}{6}) has (3), so the decimal is non-terminating recurring. Exam tip: look for factors other than (2) and (5).

(8) is in the hundredths place, so its place value is (\frac{8}{100}). Exam tip: the middle (0) changes the place.

For comparison, write 0.3 as 0.30. In 0.31, the tenths digit is the same as in 0.30, but the hundredths digit is 1, which is greater than 0. Therefore, 0.31 > 0.30. The value of 0.300 is equal to 0.3, not greater. Exam tip: add zeros at the end of decimals when needed to make the number of decimal places equal.

The governing concept is place value when the denominator is a power of ten. Since 100 has two zeros, 13/100 means thirteen hundredths, which is written as 0.13. Equivalently, dividing 13 by 100 moves the decimal point two places to the left: 13.00 becomes 0.13. Therefore option A is correct. Option B, 1.3, represents 13/10 rather than 13/100. Option C, 13.00, has value 13, and option D, 0.013, represents thirteen thousandths, or 13/1000. The two digits after the decimal point in 0.13 correspond exactly to the hundredths denominator. The repeated value after the slash in option A does not change the fact that it is the only correct choice.

(0.7) is seven tenths, while (0.07) is seven hundredths. Exam tip: notice the position of zeros after the decimal point.

(0.25=\frac{1}{4}), so (6.25=6\frac{1}{4}). Exam tip: first convert the decimal part into a fraction.

(2.56) is greater than (2.5) and less than (2.7). Exam tip: compare using (2.50) and (2.70).

The claim is false. Since \(40=2^3\times5\), its denominator has only 2 and 5 as prime factors; hence \(7\div40=0.175\) terminates. Exam tip: factor the reduced denominator first.

The governing concept is the definition of decimal places: they are the digits written to the right of the decimal point. In 0.375, the digits after the decimal point are 3, 7, and 5. Counting them gives three decimal places, so option C is correct. The zero before the decimal point belongs to the whole-number part and must not be counted as a decimal place. Option A would be correct for a number such as 0.3, and option B would fit 0.37. Option D would require four digits after the decimal point, for example 0.3750. A trailing zero can indicate an additional written decimal place even when it does not change the numerical value, but the given number is written as 0.375 and visibly contains exactly three digits after the point.

(5.08) is less than (5.09) because (8<9) at the hundredths place. Exam tip: compare after making equal decimal places.

The governing concept is recurring-decimal notation. A bar placed over a group of digits means that the entire group, not merely its last digit, repeats endlessly immediately after the decimal point. Therefore 0.̅12 means 0.12121212…, so option A is correct. Option B represents the terminating decimal 0.12, followed by zeros, and contains no repeated block 12. Option C begins with 11, so its decimal pattern is different. Option D begins with 012 and likewise does not match the indicated notation. Reading the complete block under the bar is essential. The same value can also be written as 12/99, which simplifies to 4/33, confirming that it is a recurring rational decimal.

In \(0.\overline{12}=0.121212\ldots\), the block 12 repeats endlessly, so it is non-terminating recurring. 0.625 and 2.75 terminate, while option C has no fixed repeating block. Exam tip: identify the shortest repeating block.

In (\frac{2}{11}), the denominator has (11), so the decimal will not terminate. Exam tip: check for denominator factors other than (2) and (5).

(4) is at the third place after the decimal point, so it is in the thousandths place. Exam tip: include zeros while counting places.