Decimal representation

In the decimal, the block 06 repeats after 25: 0.25 06 06 06 \(\ldots\). Therefore, the bar must be placed only over the repeating block 06, giving \(0.25\overline{06}\). In option C, only 6 repeats, while option D incorrectly treats 506 as the repeating block. Exam tip: Write the decimal digits in groups first and identify the shortest block that repeats.

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.

(19\div26=0.730769230769\ldots). In long division, the recurring part appears when a remainder repeats.

( \frac{7}{12}=0.58333\ldots ), which is slightly less than (0.5834). Think of all numbers in decimal form for comparison.

Compare 0.5a9 and 0.549 by examining their decimal places from left to right. The digits in the tenths place are both 5, so that place does not decide the comparison. In the hundredths place, the first number has a and the second number has 4. For 0.5a9 to be less than 0.549, a must be less than 4. The greatest digit satisfying this condition is 3.

With a equal to 3, the numbers are 0.539 and 0.549, and 0.539 is indeed smaller. If a were 4, the numbers would be 0.549 and 0.549, giving equality rather than a strict less-than relation. Any digit above 4 would make the first number larger. Hence the greatest possible value is 3, which is option B.

The number of zeros between (7)'s keeps changing, so there is no fixed recurring part. Such a decimal is non-terminating non-recurring.

(128\times78125=10000000), so ( \frac{13}{128}=0.1015625 ). A denominator that is a power of (2) gives a terminating decimal.

(0.000875=\frac{875}{1000000}=\frac{7}{8000}), not ( \frac{7}{800} ). Count decimal places and simplify carefully.

A fraction in lowest terms has a terminating decimal expansion precisely when its denominator contains no prime factors other than 2 and 5. If another prime factor occurs, division continues indefinitely and produces a repeating pattern. Therefore, the denominator must be factored after confirming that no common factor remains between numerator and denominator.

For \(13/28\), the fraction is already in simplest form because 13 does not divide 28. Factor the denominator as \(28=2^2\times7\). The factor 7 is different from 2 and 5, so the decimal expansion cannot terminate. Since the fraction is rational, its infinite decimal digits must repeat rather than become non-repeating. Thus the correct classification is non-terminating recurring, option B. Option A would be correct only if the denominator had factors 2 and 5 alone.

The first (7) is in tenths and the last (7) is in hundred-thousandths. Zero-value places are not written separately.

Multiplying both by (10000) gives (2160\div8=270). In division, both numbers can be multiplied by the same number.

Write 5.004 as 5.0040 so that both numbers have four digits after the decimal point. Then \(5.0040-2.0785=2.9255\). Therefore, 2.9255 is correct. A value such as 2.9355 can result from an error while subtracting the decimal digits. Exam tip: always align decimal points vertically before subtracting decimals.

Writing equal decimal places gives (0.062500+0.031250+0.000625=0.094375). Align decimal points while adding.

Separate the repeating part: \(0.0999\ldots = 0.1\). Therefore, \(2.3999\ldots = 2.3 + 0.0999\ldots = 2.4\). The option 2.399 has only three 9s, whereas the question has infinitely many 9s. Exam tip: Replace an infinite tail of 9s by an increase of 1 in the preceding decimal place.

( \frac{13}{25}=0.52 ) and ( \frac{47}{90}=0.5222\ldots ), so (0.521) lies between them. Convert the boundaries into decimals.

The first two digits after the decimal point, 9 and 2, are the same in both numbers. Therefore, the comparison depends on the thousandths digit. For the inequality to hold, b must be greater than 7. Since b is a digit, it can only be 8 or 9, so there are 2 possible values. A common mistake is to include 7, but b = 7 makes the two decimals equal, not greater. Exam tip: when initial decimal digits are the same, the first differing digit from the left determines the comparison.

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 algebraic idea is to isolate the variable by performing the inverse operation on both sides. Since 10000 is multiplying x, divide both sides by 10000: x = 0.5625/10000. Dividing by 10,000 moves the decimal point four places to the left. Starting with 0.5625, the successive shifts give 0.05625, 0.005625, 0.0005625, and finally 0.00005625. Hence x = 0.00005625, so option C is correct. Option A ignores the division, option B shifts the decimal only three places, and option D shifts it only two places. Substitution confirms the result because 10,000 × 0.00005625 = 0.5625.

In 15.00405, the digits after the decimal point represent the tenths, hundredths, thousandths, ten-thousandths and hundred-thousandths places, respectively. The digit 4 is in the third decimal place, so its place value is \(4 \times \frac{1}{1000}=\frac{4}{1000}\). Exam tip: count decimal places from left to right; the third place is the thousandths place.

Write all the numbers up to four decimal places: \(6.0060, 6.0066, 6.0606, 6.6000\). Thus, \(6.0060<6.0066<6.0606<6.6000\), so option A is correct. In option D, \(6.0606\) and \(6.0066\) are placed in the wrong order: their hundredths digits are 6 and 0 respectively. Exam tip: Add trailing zeros when needed to make the decimal places equal before comparing decimals.