Decimal representation

After 45, the block 07 repeats as 07, 07, 07, ... . Therefore, the bar must be placed only over the recurring block 07: \(0.45\overline{07}\). In option B, only 7 is treated as repeating, but 0 also occurs before every repeating 7. Exam tip: group the decimal digits and identify the shortest block that repeats.

The bar is only over (18), so after (205), (18) repeats. The barred part repeats continuously.

(29\div42=0.690476190476\ldots). In long division, the recurring part appears when a remainder repeats.

( \frac{13}{17}=0.764705\ldots ), which is less than (0.7648). View all numbers in decimal form for comparison.

The tenths and hundredths digits are equal, so the thousandths digit must satisfy (a>8). So the smallest value is (9).

The number of zeros between (8)'s keeps changing, so there is no fixed recurring part. Such a decimal is non-terminating non-recurring.

(512) is a power of (2), so the decimal terminates and ( \frac{41}{512}=0.080078125 ). The power-of-(10) method also works for large denominators.

(0.00140625=\frac{140625}{100000000}=\frac{9}{6400}). Always write the final answer in simplest form.

(84=2^2\times3\times7) has (3) and (7). Such a rational number in simplest form gives a non-terminating recurring decimal.

The first (3) is in tenths and the last (3) is in millionths. Zero-value places need not be written separately.

Multiplying both numbers by (10000) gives (5390\div7=770). Removing decimals makes division easier.

Before subtracting, align the decimal points and write 9.002 as 9.00200. Then \(9.00200-5.18725=3.81475\), so the correct answer is \(3.81475\). An option such as \(3.82475\) results from an error in aligning decimal places or borrowing. Exam tip: Add zeros at the end when needed before subtracting decimals.

Writing equal decimal places gives (0.14062500+0.00781250+0.00015625=0.14859375). Align decimal points while adding.

In 8.03999\ldots, 9 repeats forever from the third decimal place onward. Since \(0.00999\ldots=0.01\), \(8.03999\ldots=8.03+0.00999\ldots=8.04\). Option 8.039 merely stops the decimal and does not account for the infinitely repeating 9s. Exam tip: When infinitely many 9s follow a digit, increase the relevant preceding digit by 1.

( \frac{29}{40}=0.725 ) and ( \frac{8}{11}=0.7272\ldots ), so (0.7265) lies between them. Convert boundaries into decimals for comparison.

A non-terminating, non-recurring expansion has no repeating block, so it is irrational. A recurring expansion after initial digits is rational. Exam tip: a bar denotes repetition.

The decimal 3.04004000400004... continues indefinitely, so it is not terminating. Its blocks do not repeat with a fixed period: the number of zeros between successive 4s keeps increasing. A rational number has a decimal expansion that is either terminating or non-terminating but eventually recurring. Since this expansion is non-terminating and not eventually periodic, it represents an irrational number. Thus option C is correct. Option A is impossible because digits continue forever; option B would require a repeating cycle; and option D is impossible because the decimal part is not zero. This conclusion uses the standard rational-decimal classification.

(x=2.34375\div100000=0.0000234375). On dividing by (100000), the decimal moves five places left.

The places after the decimal point are tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths and millionths. In 21.005007, the digit 7 is in the sixth place after the decimal point, so its place value is \(\frac{7}{1000000}\). The closest distractor, \(\frac{7}{100000}\), represents the place value of a digit in the fifth decimal place, not the sixth. Exam tip: count the decimal places from left to right carefully.

Write all the numbers up to four decimal places for comparison: \(9.0090, 9.0099, 9.0909, 9.9000\). Comparing the digits after the decimal from left to right gives \(9.0090<9.0099<9.0909<9.9000\). Therefore, option A is correct. Option D incorrectly places \(9.0909\) before \(9.0099\). Exam tip: Add zeros at the end, when needed, to make the decimal places equal before comparing decimals.