Decimal representation

( \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.

The governing concept is converting equivalent forms before performing subtraction. Convert the fraction 9/16 to a decimal: 9 ÷ 16 = 0.5625. Now subtract the given decimal: 0.5625 − 0.3125 = 0.2500, which is 0.25. Therefore, option B is correct. Option D is only the decimal value of 9/16 before subtraction, while option C is the number being subtracted. Option A can result from an arithmetic subtraction error. The calculation can also be checked in fractions: 0.3125 = 5/16, so 9/16 − 5/16 = 4/16 = 1/4 = 0.25. Both methods confirm the same answer.

The governing concept is decimal division. To remove the decimal points without changing the quotient, multiply both the dividend and divisor by 1000, because each has at most three decimal places. Thus, 0.625 ÷ 0.025 = 625 ÷ 25. Now 25 × 25 = 625, so 625 ÷ 25 = 25. Therefore, option B is correct. Option A, 2.5, would result from an incorrect placement of the decimal point. Option C, 250, is ten times too large, while option D, 0.25, is much too small. Multiplying both numbers by the same nonzero number preserves their ratio, which is why this method is valid.

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.

To make (2^7) into (10^7), we multiply by (5^7). So there can be at most (7) decimal places.

\(0.9+0.09+\cdots\) 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).

The governing concept is equivalence between terminating and recurring decimal representations. The notation 0.59̅ means that only the 9 repeats: 0.59999… . A recurring string of 9s makes this number equal to the next terminating decimal, so 0.59999… = 0.60000… = 0.6. Also, 3/5 can be converted to a decimal by dividing 3 by 5, giving 0.6 exactly. Hence all three expressions represent the same real number, and option C is correct. Option A incorrectly treats the recurring 9s as making the value smaller; options B and D incorrectly introduce a strict inequality. Equal decimal representations can look different while denoting the same number.

( \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.

(128\times78125=10000000), so ( \frac{37}{128}=0.2890625 ). A denominator that is a power of (2) gives a terminating decimal.

(640\times15625=10000000), so ( \frac{57}{640}=0.0890625 ). Convert the denominator into a power of (10) to find the decimal.

The governing concept is the conversion of a terminating decimal into a fraction with a power of 10 as denominator, followed by reduction to lowest terms. The decimal 0.0003125 has seven digits after the decimal point, so it is 3125/10,000,000. Now divide numerator and denominator by 3125, their greatest common divisor: 3125 ÷ 3125 = 1 and 10,000,000 ÷ 3125 = 3200. Hence 0.0003125 = 1/3200, making option B correct. Option D is the unreduced fraction and is also written with an incorrect denominator for seven decimal places; 1/320 and 1/32000 have values ten times larger and ten times smaller, respectively. The simplified fraction must have numerator 1 and denominator 3200.

The denominator has (11), a factor other than (2) and (5). So the rational number will have a non-terminating recurring decimal.

The governing theorem states that a rational number in lowest terms has a terminating decimal expansion exactly when the prime factors of its denominator are only 2 and/or 5. First simplify the given fraction: 84/210 can be divided by 42, giving 2/5. The denominator 5 is a permitted prime factor, so the decimal terminates; in fact, 2/5 = 0.4. Therefore option B, terminating, is correct. The original denominator 210 contains other factors, but those factors cancel during simplification and must not be used for the final classification. A non-terminating recurring expansion would require a remaining denominator factor other than 2 or 5, while a non-recurring decimal is irrational, and 2/5 is not an integer.