Decimal representation

\(\frac{2}{11}=0.1818\ldots\), where the block 18 repeats. Hence, it is rational despite having a non-terminating decimal expansion. \(\sqrt{2}\) is non-terminating and non-repeating. Exam tip: repeating decimals are rational.

It has no fixed repeating block and it does not terminate. Such a decimal is non-terminating non-recurring.

( \frac{2}{5}=0.4 ) and ( \frac{1}{2}=0.5 ), so (0.405) lies between them. Convert the boundaries into decimals.

(3.600) is greatest at the tenths place and (3.006) is smallest. In descending order, write from greatest to smallest.

(800\times125=100000), so ( \frac{53}{800}=\frac{6625}{100000}=0.06625 ). Convert the denominator into a power of (10) to find the decimal.

(0.00375=\frac{375}{100000}=\frac{3}{800}). For five decimal places, use denominator (100000) and simplify the fraction.

The tenths and hundredths digits are equal, so the thousandths digit must satisfy (b<8). Hence the greatest value is (7).

In a decimal expansion, the recurring part is the smallest group of digits that repeats endlessly in exactly the same order. The number before the decimal point is the whole-number part, not part of the repeating block. Also, a repeated block may begin with zero, so that zero must not be dropped when identifying the pattern.

In \(6.020202\ldots\), the digits after the decimal point are 0, 2, 0, 2, 0, 2, and so on. They can be grouped as \(02\mid02\mid02\ldots\). Thus the repeating block is (02), not (20), because the first digit of the decimal part is 0 and the order is important. The digit 6 does not repeat, and (202) is not the smallest repeating group. Therefore option B is correct.

The governing concept is conversion and simplification of a terminating decimal. Since 0.00096 has five digits after the decimal point, write it first as 96/100000. The greatest common divisor of 96 and 100000 is 32. Dividing numerator and denominator by 32 gives 96 ÷ 32 = 3 and 100000 ÷ 32 = 3125. Hence 0.00096 = 3/3125, which is already in lowest terms because 3 does not divide 3125. Therefore option C is correct. Option A is not equal to the decimal because 96/10000 = 0.0096. Option B equals 0.00096 only if the numerator and denominator are checked carefully? In fact 12/12500 = 0.00096, but it is not simplified because both terms are divisible by 4. Option D is much larger. Thus C is the required simplified fraction.

The governing concept is solving a linear equation involving decimal place value. To isolate x in 1000x = 0.875, divide both sides by 1000: x = 0.875/1000. Since 1000 = 10³, division by 1000 shifts the decimal point three places to the left, giving x = 0.000875. The fraction method confirms this: 0.875 = 875/1000, so x = 875/(1000 × 1000) = 875/1,000,000 = 0.000875. Substitution verifies the answer because 1000 × 0.000875 = 0.875. Option A forgets to divide, while options B and C shift the decimal by only one or two places. Therefore option D is the unique correct answer.

The governing concept is converting equivalent forms before performing subtraction. Convert the fraction 9/16 to a decimal: 9 ÷ 16 = 0.5625. Now subtract the given decimal: 0.5625 − 0.3125 = 0.2500, which is 0.25. Therefore, option B is correct. Option D is only the decimal value of 9/16 before subtraction, while option C is the number being subtracted. Option A can result from an arithmetic subtraction error. The calculation can also be checked in fractions: 0.3125 = 5/16, so 9/16 − 5/16 = 4/16 = 1/4 = 0.25. Both methods confirm the same answer.

The governing concept is equivalence between terminating and recurring decimal representations. The notation 0.59̅ means that only the 9 repeats: 0.59999… . A recurring string of 9s makes this number equal to the next terminating decimal, so 0.59999… = 0.60000… = 0.6. Also, 3/5 can be converted to a decimal by dividing 3 by 5, giving 0.6 exactly. Hence all three expressions represent the same real number, and option C is correct. Option A incorrectly treats the recurring 9s as making the value smaller; options B and D incorrectly introduce a strict inequality. Equal decimal representations can look different while denoting the same number.

The governing concept is the conversion of a terminating decimal into a fraction with a power of 10 as denominator, followed by reduction to lowest terms. The decimal 0.0003125 has seven digits after the decimal point, so it is 3125/10,000,000. Now divide numerator and denominator by 3125, their greatest common divisor: 3125 ÷ 3125 = 1 and 10,000,000 ÷ 3125 = 3200. Hence 0.0003125 = 1/3200, making option B correct. Option D is the unreduced fraction and is also written with an incorrect denominator for seven decimal places; 1/320 and 1/32000 have values ten times larger and ten times smaller, respectively. The simplified fraction must have numerator 1 and denominator 3200.

The governing theorem states that a rational number in lowest terms has a terminating decimal expansion exactly when the prime factors of its denominator are only 2 and/or 5. First simplify the given fraction: 84/210 can be divided by 42, giving 2/5. The denominator 5 is a permitted prime factor, so the decimal terminates; in fact, 2/5 = 0.4. Therefore option B, terminating, is correct. The original denominator 210 contains other factors, but those factors cancel during simplification and must not be used for the final classification. A non-terminating recurring expansion would require a remaining denominator factor other than 2 or 5, while a non-recurring decimal is irrational, and 2/5 is not an integer.

The key concept is interpreting repeating-decimal bar notation correctly. In 0.41̅25, the bar is intended to cover the block 25, while the digits 41 before the bar are the non-repeating part. Therefore write 41 first after the decimal point, then repeat 25 continuously: 0.41252525… . The first digits are 4, 1, 2, 5, 2, 5, 2, 5, and so on. Thus option A is correct. Option B changes the order of the repeating block, option C inserts an extra 4 and does not preserve the stated notation, and option D incorrectly treats the repeating decimal as terminating. A bar over a block means that the entire block repeats indefinitely, not that it appears only once.

The governing classification of decimal numbers is as follows: a terminating decimal is rational, a non-terminating decimal with a fixed repeating block is rational, and a non-terminating decimal with no repeating pattern is irrational. In 8.03003000300003…, the groups of zeros between the 3s keep changing in length: one zero, then two, then three, and so on. There is therefore no fixed finite block that repeats indefinitely. The decimal does not terminate and is non-repeating, so it represents an irrational number. Option C is correct. It cannot be a terminating rational or an integer because digits continue forever, and it is not a recurring rational because no stable repeating cycle appears.

The governing concept is decimal division by making the divisor a whole number while multiplying both numbers by the same power of 10. The divisor 0.016 has three decimal places, so multiply both dividend and divisor by 1000: 0.432 ÷ 0.016 = 432 ÷ 16. Since 16 × 27 = 432, the quotient is 27. A direct check gives 0.016 × 27 = 0.432, confirming the result. Therefore option B is correct. Option A, 2.7, is ten times too small; option C, 270, is ten times too large; and option D, 0.27, results from moving the decimal point incorrectly. Multiplying both terms by the same nonzero number does not change their quotient, which is why the conversion is valid.

A rational number in lowest terms has a terminating decimal only when the prime factors of its denominator are 2 and/or 5. Here q = 5^8. To express the denominator as a power of 10, multiply numerator and denominator by 2^8: 5^8 × 2^8 = 10^8. Thus the decimal can have at most eight places. It may have fewer places if the numerator causes cancellation after conversion, but eight is the maximum possible. Therefore option C is correct. Options A and B do not supply enough factors of 2 to form 10^8, while option D overestimates the required power. The conclusion follows directly from the terminating-decimal theorem.

A rational number has a terminating decimal only when, after reducing the fraction completely, the denominator has no prime factors other than 2 and 5. If any other prime factor remains in the denominator, the decimal continues forever in a repeating pattern. This rule helps us identify the type without needing to perform a long division. It also shows why the phrase “in simplest form” is important: common factors may hide the actual denominator structure.

The greatest common divisor of 96 and 180 is 12. Therefore, \(96/180=8/15\). The denominator 15 factors as \(3\times5\), so it contains the prime factor 3 as well as 5. Hence the decimal does not terminate; because \(8/15\) is rational, its continuing digits must repeat. Thus option B, non-terminating recurring decimal, is correct. It is not non-recurring, since non-terminating non-recurring decimals are irrational numbers.

The governing idea is recurring-decimal notation: a bar is placed only over the digits that repeat indefinitely. In 0.0181818..., the first digit after the decimal point is 0 and it is nonrepeating. After that, the block 18 repeats: 0.0 18 18 18 ... Therefore the correct notation is 0.0 overline{18}, option C. Option A incorrectly includes the initial nonrepeating 0 and changes the repeating block; option B treats only 8 as repeating, although every 1 is followed by 8; and option D incorrectly makes the first 0 part of the repetition. Careful separation of the nonrecurring prefix from the recurring cycle identifies C unambiguously.

Simplify the fraction by dividing numerator and denominator by 9: 144/225 = 16/25. The denominator 25 equals 5², so its prime factors are only 5. By the terminating-decimal criterion, the expansion must terminate; indeed, 16/25 = 0.64. Thus option B is correct. It is not recurring or irrational, and the value is not an integer.

The governing rule is the terminating-decimal criterion for rational numbers. After reducing a fraction to lowest terms, its decimal expansion terminates only when the denominator has no prime factors other than 2 and 5. Here 41/60 is already in simplest form because 41 is prime and does not divide 60. Factor the denominator: 60 = 2² × 3 × 5. The factor 3 remains, so the denominator cannot be converted into a power of 10 by multiplying only by powers of 2 and 5. Therefore the decimal expansion is non-terminating and recurring, making option B correct. It is not terminating, and it is not non-recurring because every rational number has either a terminating or an eventually recurring decimal expansion.

The key concept is the distinction between terminating, recurring, and non-recurring decimal expansions. The decimal 5.02002000200002... continues indefinitely, so it is non-terminating. Its blocks do not repeat with one fixed period: the numbers of zeros between successive 2s increase, so no finite string of digits can repeat forever. A rational number has a decimal expansion that either terminates or eventually repeats periodically. Since this expansion is non-terminating and non-periodic, it represents an irrational number; option C is correct. It cannot be a terminating rational because digits continue, cannot be an eventually recurring rational because there is no fixed cycle, and cannot be an integer because a nonzero decimal tail remains.

The governing concept is converting equivalent forms before comparing numbers. First, 5/16 can be converted to a decimal by dividing 5 by 16, giving 0.3125. The notation 0.31̅2 means that only the digit 2 repeats: 0.3122222..., not 0.3125. Compare the latter with 0.3125 digit by digit: both begin 0.312, but at the fourth decimal place, 2 is less than 5. Thus 0.31̅2 < 0.3125, while 0.3125 = 5/16. Therefore option A is correct. Option C reverses the comparison, option B incorrectly equates the repeating decimal with 5/16, and option D ignores the different fourth and later digits.

The governing rule says that a rational number has a terminating decimal when the denominator in lowest terms contains only factors 2 and 5. Here q = 2⁸, so the decimal certainly terminates. To express the denominator as a power of 10, multiply numerator and denominator by 5⁸: 2⁸ × 5⁸ = 10⁸. Thus p/q = (p × 5⁸)/10⁸, which has at most eight digits after the decimal point. The maximum is reached when cancellation in the numerator does not remove any of these places, so option C is correct. It cannot be 6 or 7 because the denominator may require all eight places, and 10 is unnecessarily large. The phrase “in simplest form” ensures that no hidden denominator factor remains.