Decimal representation

( \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.

(625\times16=10000), so ( \frac{43}{625}=\frac{688}{10000}=0.0688 ). Making the denominator a power of (10) is the safest method.

(0.003125=\frac{3125}{1000000}=\frac{1}{320}). Count decimal places to write the denominator and then simplify.

The denominator has (7), a factor other than (2) and (5). So the rational number will have a non-terminating recurring decimal.

A rational number has a terminating decimal expansion when, after simplification, the denominator has no prime factors other than 2 and 5. First reduce the fraction by dividing 91 and 140 by their common factor 7: \\(\frac{91}{140}=\frac{13}{20}\\). The denominator 20 factors as \\(2^2\times5\\), so it contains only the allowed prime factors.

Indeed, \\(\frac{13}{20}=\frac{65}{100}=0.65\\), which ends after two decimal places. Therefore the expansion is terminating, and option A is correct. It is not non-terminating recurring because no factor such as 3, 7, or 11 remains in the simplified denominator. Simplifying before judging the decimal type is essential.

In the decimal expansion, after 12 the block 03 repeats as 03, 03, 03, … . Thus, 12 is the non-repeating part and the bar must be placed only over 03: \(0.12\overline{03}\). Option A incorrectly treats 1203 as the repeating block. Exam tip: Write the digits in order first, then identify the smallest block that repeats continuously.

The bar is only over (18), so after (3), (18) repeats. In bar notation, the barred part repeats continuously.

(17\div22=0.772727\ldots) and (27) repeats. In long division, watch the repeating remainder.

( \frac{5}{12}=0.41666\ldots ), which is less than (0.417). Think of all numbers in decimal form for comparison.

The tenths and hundredths digits are equal, so the thousandths digit must satisfy (a>6). Hence the smallest value is (7).

The number of zeros keeps changing, so there is no fixed recurring part. Such a decimal is non-terminating non-recurring.

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.

(64\times15625=1000000), so ( \frac{7}{64}=0.109375 ). A denominator that is a power of (2) can be converted to a power of (10).

( \frac{18}{45}=\frac{2}{5} ), and (5) has only factor (5). It is necessary to simplify the fraction first.

The first (5) is in tenths and the last (5) is in ten-thousandths. Zero-value places need not be written separately.