Decimal representation

For a rational number written in simplest form, the decimal expansion terminates only if the denominator has prime factors 2 and/or 5. If any other prime factor remains in the denominator, the decimal expansion is non-terminating recurring. This rule applies only after checking that the fraction is already in lowest terms.

The fraction is \(37/48\). Since 37 is prime and does not divide 48, the fraction is already simplified. Factor the denominator: \(48=2^4\times3\). Although it contains a power of 2, it also contains the factor 3. Because 3 is not allowed for a terminating decimal, \(37/48\) has an endless repeating decimal expansion. Therefore option B, non-terminating recurring, is correct. It is not non-recurring because the fraction is rational.

(0.\overline{27}) is non-terminating recurring, so it is rational but not terminating. Recurring decimals are always rational.

A real number has a terminating decimal expansion or a non-terminating recurring expansion exactly when it is rational. The displayed decimal continues indefinitely, and its groups do not repeat with a fixed period: after 04, the number of zeros before the next 4 increases. Thus it is non-terminating and non-recurring, the characteristic decimal form of an irrational number. It cannot be terminating because digits continue after every stated position, and it is not a recurring rational because no fixed block repeats forever. It is also not an integer, since its decimal part is nonzero. Under the indicated pattern, the correct classification is irrational, so option C is correct.

The places after the decimal point are tenths, hundredths, thousandths, and so on. In 12.03004, 3 is the second digit after the decimal point, so its place value is \(\frac{3}{100}\). Option A treats 3 as being in the tenths place, which is incorrect. Exam tip: label the decimal places from left to right as 10, 100, 1000, and so on.

The decimal places are equal, so compare from left to right. The smallest is (4.006) and the greatest is (4.600).

The key classification rule is that every terminating decimal and every non-terminating recurring decimal is rational, while a non-terminating decimal with no fixed repeating pattern is irrational. Option A repeats 72, so it is rational. Option B terminates and equals 25/8, so it is rational. Option D has a repeating block, so it is also rational. In option C, the decimal continues without a fixed repeating block; the zeros between successive 2s increase, indicating a non-terminating non-recurring expansion. Therefore it cannot be expressed as a ratio of integers and is definitely irrational. Hence option C is correct. The phrase “without fixed repetition” is essential to distinguish it from an ordinary recurring decimal.

(1600\times625=1000000), so ( \frac{29}{1600}=0.018125 ). Even a large denominator can be converted into a power of (10).

To find the multiplicative comparison, divide the larger decimal by the smaller one: \(0.0125 \div 0.000125 = 100\). Therefore, \(0.0125\) is 100 times \(0.000125\). Multiplying by 10 gives only \(0.00125\), whereas multiplying by 1000 gives \(0.125\). Exam tip: To compare two positive decimals multiplicatively, divide the larger number by the smaller number.

The bar is over 27 only. Therefore, the first digit after the decimal point is 0, followed by the repeating block 27: \(0.0272727\ldots\). In option C, both 0 and 27 are treated as repeating, which does not match the notation. Exam tip: Repeat only the digits covered by the bar.

\(\frac{21}{84}=\frac14\), and its reduced denominator is \(4=2^2\); therefore, its decimal expansion terminates. In \(\frac{17}{60}\), factor 3 remains in the denominator. Exam tip: always reduce first.

To determine the decimal type of a fraction, first reduce the fraction to its simplest form. Here, 56 and 154 have a common factor of 14, so the fraction becomes 4/11. A rational number has a terminating decimal expansion only when the denominator in simplest form has no prime factors other than 2 and 5. The denominator 11 does not meet this condition.

The division of 4 by 11 continues without ending and produces a repeating pattern, namely 0.363636 and so on. Therefore, its decimal expansion is non-terminating recurring. The correct answer is option C. It is not non-terminating non-recurring, because every rational number has either a terminating decimal or a repeating decimal when written in decimal form.

When (9) repeats after a place, it can equal the next terminating value. So (0.124999\ldots=0.125).

After the decimal point, the positions are tenths, hundredths, thousandths, and ten-thousandths. In 7.005400, 4 is the fourth digit after the decimal point, so its place value is \(\frac{4}{10000}\). Option A is incorrect because \(\frac{4}{1000}\) represents the place value of the third decimal digit. In an exam, count the decimal places from left to right, including zeros.

( \frac{1}{15}=0.0666\ldots=0.06\overline{6} ), which is greater than (0.0625). Convert the fraction into decimal for comparison.

Write the decimals to four decimal places: \(a=0.2040\), \(b=0.2400\), and \(c=0.2004\). At the tenths place, \(b\) has 4 whereas \(a\) and \(c\) have 2, so \(b\) is greatest. For \(a\) and \(c\), the hundredths digits are equal, but at the thousandths place \(a\) has 4 and \(c\) has 0; hence \(a>c\). Therefore, the descending order is \(b>a>c\). Exam tip: Add trailing zeros when needed to compare the same number of decimal places.

( \frac{13}{125}=0.104 ) and ( \frac{7}{40}=0.175 ), so the sum is (0.279). First convert the fractions into decimals.

After the decimal point, the first digit 0 occurs only once. Then 45 repeats as 45, 45, 45. Therefore, the bar is placed only over 45: \(0.0\overline{45}\). Option \(0.\overline{45}\) is incorrect because it starts repeating 45 immediately after the decimal point. Exam tip: identify the non-repeating digits before marking the repeating block with a bar.

\(\frac{21}{84}=\frac14\), and its reduced denominator is \(4=2^2\); hence its decimal expansion terminates. The other denominators contain 3, 7, or 11. Exam tip: reduce the fraction before checking prime factors of the denominator.

The direct answer is option C, 7 decimal places. For a fraction in simplest form to have a terminating decimal, its denominator must contain only the prime factors 2 and/or 5. Here q=2^7. To turn the denominator into a power of 10, multiply numerator and denominator by 5^7: 2^7×5^7=10^7. Therefore the denominator can become 10^7, which gives at most seven digits after the decimal point. The numerator may cause cancellation only if it contains factors that reduce the denominator, so seven is the maximum, not necessarily the actual number for every p. Option A, 5, is too small; option B, 6, is also too small; option C, 7, follows directly from the exponent of 2; option D, 8, is one place too many. Exam cue: for denominator 2^m or 5^m in lowest form, the maximum terminating length is m.

\(0.9+0.09+\cdots\) is a geometric series: \(S=\frac{0.9}{1-0.1}=1\). Every finite truncation is below 1, but their limit is 1. In exams, treat \(\ldots\) as an infinite process.

The number of zeros between (2)'s increases, so there is no fixed repetition. Such a decimal is non-terminating non-recurring.

To convert a decimal into a percentage, multiply it by 100: 0.0008 × 100 = 0.08. Therefore, the correct answer is 0.08%. Choosing 0.8% shifts the decimal point one place too far. Exam tip: Move the decimal point two places to the right when converting a decimal to a percentage.

On multiplying by (100000), the decimal point moves five places to the right. Therefore, (0.004096\times100000=409.6).

( \frac{7}{8}=0.875 ), so (0.8735) is greater than (0.873) and less than (0.875). Convert boundary values into decimals.

The tenths digits are equal and the hundredths digit must satisfy (a>3). So (a=4,5,6,7,8,9), giving (6) possible values.