Decimal representation

( \frac{5}{8}=0.625 ), so (0.6245) is greater than (0.624) and less than (0.625). Convert boundary values into decimals.

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.

(3200\times3125=10000000), so ( \frac{83}{3200}=0.0259375 ). Convert the denominator into a power of (10) to get the exact decimal.

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.

The tenths digits are equal and the hundredths digit must satisfy (a\leq4). So (a=0,1,2,3,4), giving (5) possible values.

For a denominator of \\(2^4\times5^9\\), the powers of 2 and 5 must be balanced to form a power of 10. There are only four factors of 2 but nine factors of 5, so multiply by \\(2^5\\) to obtain \\(2^9\times5^9=10^9\\). Therefore a terminating decimal can have at most nine decimal places.

The rule is that the maximum number of places is the larger of the exponents of 2 and 5 in the simplified denominator. Here, \\(\max(4,9)=9\\), so option C is correct. The exponents should not be added to get 13, because the factors are paired to make tens, with each pair of 2 and 5 producing one factor of 10.

(1250\times8=10000), so ( \frac{73}{1250}=\frac{584}{10000}=0.0584 ). Making the denominator a power of (10) is an exact method.

(0.00015625=\frac{15625}{100000000}=\frac{1}{6400}). First count decimal places and then simplify the fraction.

The denominator has (13), a factor other than (2) and (5). So the rational number will have a non-terminating recurring decimal.

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 decimal digits are 3, 4, 0, 9, 0, 9, \(\ldots\). After 34, the block 09 repeats continuously, so the correct notation is \(0.34\overline{09}\). In \(0.\overline{3409}\), the entire block 3409 is incorrectly treated as repeating from the beginning. Exam tip: Before placing a bar, identify the smallest block of digits that repeats indefinitely.

The bar is only over (36), so after (07), (36) repeats. The barred part repeats continuously.

For \(\frac{7}{48}\), the denominator is \(48=2^4\times3\). A fraction in simplest form has a non-terminating recurring decimal when its denominator contains a prime factor other than 2 or 5. The other denominators contain only 2 and/or 5. Exam tip: simplify first, then factorise the denominator.

( \frac{17}{24}=0.70833\ldots ), which is less than (0.7084). View all numbers in decimal form for comparison.

The governing concept is comparison of decimal numbers by place value. Write the numbers as 0.6a4 and 0.674. Their tenths digits are both 6, so the comparison moves to the hundredths place. In 0.6a4, that digit is a; in 0.674, it is 7. To make the first number greater, we need a > 7. Since a is a digit, the possible values are 8 or 9, and the smallest is 8. Therefore option C is correct. If a = 6, then 0.664 is smaller than 0.674; if a = 7, the numbers become 0.674 and 0.674, which are equal rather than greater. Although 9 also works, it is not the smallest possible value.

The number of zeros between (9)'s keeps changing, so there is no fixed recurring part. Such a decimal is non-terminating non-recurring.

(256\times390625=100000000), so ( \frac{23}{256}=0.08984375 ). A denominator that is a power of (2) gives a terminating decimal.

(0.0021875=\frac{21875}{10000000}=\frac{7}{3200}). Count decimal places and simplify the final fraction.

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 first (2) is in tenths and the last (2) is in hundred-thousandths. Zero-value places are not written separately.