Decimal representation

( \frac{23}{32}=0.71875 ), so (0.71825) is greater than (0.718) and less than (0.71875). Convert boundary values into decimals.

In lowest form, a denominator containing any prime factor other than 2 or 5 gives a non-terminating recurring decimal. A denominator with only 2 and 5 gives a terminating decimal. Exam tip: reduce the fraction first.

(16000\times625=10000000), so ( \frac{149}{16000}=0.0093125 ). Convert the denominator into a power of (10) to get the exact decimal.

(0.00021875=\frac{21875}{100000000}=\frac{7}{32000}). Count decimal places and write the final answer in simplest form.

The bar is only over (81), so after (04), (81) repeats again and again. In bar notation, repeat only the barred part.

The governing concept is that a denominator of the form 2^m × 5^n produces a terminating decimal with at most max(m,n) decimal places. Here q = 2⁸ × 5¹¹. The powers must be balanced to create a power of 10. Multiply by 2³: 2⁸ × 5¹¹ × 2³ = 2¹¹ × 5¹¹ = 10¹¹. Consequently, the fraction can be written with denominator 10¹¹ and has at most eleven digits after the decimal point. The maximum occurs when the numerator causes no cancellation that shortens the result. Thus option B is correct. Option A ignores the larger exponent, option C adds exponents unnecessarily, and option D multiplies them without mathematical basis.

(3200\times3125=10000000), so ( \frac{127}{3200}=0.0396875 ). Even a large denominator can be converted into a power of (10).

(0.00009375=\frac{9375}{100000000}=\frac{3}{32000}). Count decimal places and simplify the fraction.

The denominator has (19), a factor other than (2) and (5). So the rational number will have a non-terminating recurring decimal.

The governing concept is the decimal-expansion test for a rational number in lowest terms. First simplify the fraction: the greatest common divisor of 198 and 330 is 66, so 198/330=3/5. A rational number has a terminating decimal expansion exactly when, after reduction, the denominator has no prime factors other than 2 and 5. The denominator here is 5, so the condition is satisfied. Indeed, 3/5=0.6, which ends after one decimal place. Therefore option B is correct. Option A would occur if a factor such as 3 remained in the reduced denominator. Option C describes a non-terminating non-recurring decimal, generally associated with an irrational number, and option D is wrong because 3/5 is not an integer.

The decimal digits are 5, 6, 0, 9, 0, 9, 0, 9, ... . Here, 56 is the non-repeating part, and the block 09 repeats thereafter. Hence the correct notation is \(0.56\overline{09}\). In \(0.560\overline{9}\), only 9 is treated as recurring, but both 0 and 9 repeat as a block. Exam tip: Write out a few decimal digits first and identify the smallest repeating block before placing the bar.

The bar is only over (27), so after (108), (27) repeats. The barred part repeats continuously.

In lowest form, a decimal terminates only when \(q=2^m5^n\). An even denominator alone is not enough, since \(6=2\cdot3\) also contains 3. Exam tip: reduce the fraction to lowest form first.

( \frac{11}{17}=0.647058\ldots ), which is less than (0.6471). View all numbers in decimal form for comparison.

The tenths and hundredths digits are equal, so the thousandths digit must satisfy (a<5). The greatest value is (4).

The number of zeros between (1)'s keeps changing, so there is no fixed recurring part. Such a decimal is non-terminating non-recurring.

(1024) is a power of (2), so the decimal terminates and ( \frac{77}{1024}=0.0751953125 ). Convert a power-of-(2) denominator into a power of (10).

(0.00046875=\frac{46875}{100000000}=\frac{3}{6400}). Always write the final answer in simplest form.

To classify the decimal expansion of 47/72, first check whether the fraction is already in lowest form. The numerator 47 is prime and does not divide 72, so no common factor can be cancelled. Now factor the denominator: 72 = 2^3 × 3^2.

A fraction in lowest form has a terminating decimal only if its denominator has no prime factors other than 2 and 5. Although 72 contains a factor 2, it also contains the factor 3, so the required condition fails. The decimal therefore continues indefinitely but repeats a pattern, making it non-terminating recurring. Thus option B is correct. It is not non-recurring, because every rational number has a terminating or recurring decimal expansion.

The first (4) is in tenths and the last (4) is in millionths. Zero-value places need not be written separately.