Decimal representation

For \(\frac{7}{40}\), the denominator is \(40=2^3\times5\), so its decimal expansion terminates. The other denominators contain 3. Exam tip: first reduce the fraction to lowest terms.

In (\frac{11}{40}), the denominator (40=2^3\times5), so the decimal is terminating. Exam tip: check whether the denominator has only (2) and (5) as prime factors.

In \(0.121212\ldots\), the block 12 repeats endlessly, so it is non-terminating recurring. In \(0.1010010001\ldots\), no fixed block repeats. Exam tip: look for a repeating digit pattern.

(4) is in the hundredths place, so its place value is (\frac{4}{100}). Exam tip: after the decimal point, the first place is tenths and the second is hundredths.

\(0.125000\ldots=\frac{125}{1000}=\frac18\), so it is rational and has a terminating decimal expansion. Infinitely many trailing zeros do not change the value. Exam tip: remove trailing zeros before classifying a decimal.

In (0.04), (4) is in the hundredths place, so (0.04=\frac{4}{100}). Exam tip: do not ignore zeros.

The governing concept is decimal place value. In a number such as 2.305, the first digit to the right of the decimal point is in the tenths place, the second is in the hundredths place, and the third is in the thousandths place. Reading 2.305 from left to right after the decimal gives 3, 0, and 5. Thus 3 is in the tenths place, 0 is in the hundredths place, and 5 is in the thousandths place. Therefore option B is correct. The digit 2 is in the ones place, so option D is incorrect. The zero contributes no quantity by itself, but it is important as a place holder because it preserves the hundredths position and distinguishes 2.305 from numbers such as 2.35 or 2.035.

A fraction with denominator 10 represents tenths. The numerator 7 means seven tenths, and seven tenths is written as 0.7 in decimal notation. The zero before the decimal point shows that the value is less than one. Thus the correct choice is C, 0.7. The other choices do not represent seven tenths: 0.07 is seven hundredths, 7.10 is greater than seven, and 70.0 is seventy.

To convert directly, divide the numerator by the denominator: 7 ÷ 10 = 0.7. Equivalently, move the decimal point in 7 one place to the left because division by 10 reduces the value tenfold. Therefore, \(\frac{7}{10}=0.7\). The denominator tells us the place value: 10 gives one digit after the decimal point, so 0.7 is the required representation.

(0.6=0.60) because a zero at the end does not change the value. Exam tip: trailing zeros after decimals may be removed.

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.

The number equal to 4.000 is 4, so B is correct. Decimal notation allows zeros to be written to the right of the last nonzero digit without changing the value: 4.000=4.000−0.000=4. The digit 4 is in the units place, while the decimal zeros represent zero tenths, hundredths, and thousandths. A, 0.4, means four tenths and is ten times smaller. C, 40, is ten times larger and has a different units value. D, 0.004, means four thousandths, also much smaller. Do not confuse trailing zeros after a decimal with a change in place value; zeros after the last nonzero decimal digit are harmless. Memory cue: 7.0, 7.00 and 7 have the same 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.

The direct answer is option A,
\(6\frac{1}{4}\). Separate the whole and decimal parts:
\(6.25=6+0.25\). Since
\(0.25=25/100=1/4\) after cancelling the common factor 25, we get
\(6.25=6\frac{1}{4}\). As an improper fraction, this is
\(25/4\), which confirms the mixed form. Option A is correct. Option B,
\(6\frac{1}{2}\), equals 6.5, not 6.25. Option C,
\(6\frac{3}{4}\), equals 6.75, so it is too large. Option D,
\(6\frac{25}{10}\), is not the correct mixed-number conversion; also
\(25/10=2.5\), so its value would be 8.5, and the fraction is not simplified. Memory cue: 0.25 is one quarter.

The direct answer is option B,
\(2.56\). A number lies between two numbers when it is greater than the smaller number and less than the larger number. Write the endpoints with equal decimal places:
\(2.5=2.50\) and
\(2.7=2.70\). Now compare:
\(2.50<2.56<2.70\), so 2.56 is between them. Option A,
\(2.49\), is less than 2.50, so it lies below the interval. Option B is correct. Option C,
\(2.75\), is greater than 2.70, so it lies above the interval. Option D,
\(2.07\), is also less than 2.50. Adding a zero to the right of a decimal does not change its value, so this comparison method is safe. Exam cue: align decimal places before comparing.

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.