Decimal representation

To find the multiplier, divide the larger decimal by the smaller one: \(0.0078125 \div 0.000078125 = 100\). Therefore, \(0.0078125\) is 100 times \(0.000078125\). Exam tip: for “how many times,” divide the compared larger quantity by the reference quantity.

The bar is placed only over 54, so the block 54 repeats indefinitely. The first three zeros are non-repeating; hence the expansion is \(0.000545454\ldots\). Option C incorrectly treats 540 as the repeating block. Exam tip: first identify exactly which digits are covered by the bar in a recurring decimal.

The number of zeros between successive 1s is 1, 2, 3, 4, …, so no fixed repeating block exists. Hence the decimal is non-terminating and non-repeating, making it irrational. Exam tip: check repetition, not just the digits used.

First reduce the fraction before deciding the type of decimal expansion. The numerator and denominator have a common factor 44: \\(\frac{132}{484}=\frac{3}{11}\\). The simplified denominator is 11, which is a prime factor other than 2 or 5. A rational number whose simplified denominator contains such a factor has a non-terminating recurring decimal expansion.

For confirmation, division gives \\(\frac{3}{11}=0.272727\ldots\\), where the block 27 repeats endlessly. Thus option C is correct. It is important not to judge the original denominator 484 without simplification, because common factors may disappear. Here, however, the remaining factor 11 clearly prevents termination and produces repetition.

Here, 23.74999\ldots = 23.74 + 0.00999\ldots. Since 0.00999\ldots = 0.01, the value is 23.74 + 0.01 = 23.75. The number 23.749 has only finitely many digits and does not include the effect of the infinitely repeating 9s. Exam tip: An infinite string of 9s after a decimal place can be written by increasing the preceding digit by 1.

In 8.090600, the digits after the decimal point represent tenths, hundredths, thousandths and ten-thousandths in order. The digit 6 is in the fourth place after the decimal point, so its place value is \(6 \times \frac{1}{10000}=\frac{6}{10000}\). Therefore, option C is correct. Exam tip: count decimal places from left to right; the fourth place always has denominator 10,000.

Writing the numbers up to four decimal places gives \(b=0.7800\), \(a=0.7080\), and \(c=0.7008\). Thus, \(0.7800>0.7080>0.7008\), so the correct order is \(b>a>c\). Between \(a\) and \(c\), the thousandths digit of 0.7080 is 8, while that of 0.7008 is 0; hence \(a>c\). Exam tip: Append zeros to decimals when needed so that all numbers have the same number of decimal places before comparing them.

( \frac{47}{500}=0.094 ) and ( \frac{23}{1280}=0.01796875 ), so the sum is (0.11196875). First convert the fractions into decimals.

( \frac{59}{256}=0.23046875 ), so (0.23046875-0.10546875=0.125). Convert the fraction into decimal and subtract.

The governing concept is decimal division using equal scaling. Multiply both dividend and divisor by 1000 so that the divisor becomes an integer: 1.134×1000=1134 and 0.018×1000=18. Therefore 1.134÷0.018=1134÷18. Now 18×60=1080 and the remaining 54 equals 18×3, so 1134÷18=60+3=63. Hence option B is correct. The result can also be checked directly because 0.018×63=0.018×60+0.018×3=1.080+0.054=1.134. Option A is ten times too small, option C is ten times too large, and option D is one hundred times too small. Equal multiplication of both numbers preserves the quotient and prevents incorrect decimal shifting.

In \(0.0747474\ldots\), the first digit after the decimal point is \(0\), and then the block \(74\) repeats: \(0,74,74,74,\ldots\). Hence, the bar is placed only over \(74\), giving \(0.0\overline{74}\). In \(0.\overline{074}\), the entire block \(074\) would repeat, producing a different decimal. Exam tip: identify the shortest repeating block before placing the bar.

In a recurring decimal, a fixed block of \(p\) digits repeats. Here, the gaps of zeros between 1s are 1, 2, 3, … and keep increasing, so no fixed period exists; hence the number is irrational. Exam tip: using only 0 and 1 does not make a number rational.

To make (2^{11}) into (10^{11}), we multiply by (5^{11}). So there will be at most (11) decimal places.

The denominator is \\(2^6\times5^9\\). A terminating decimal is formed by making the denominator a power of 10, so the powers of 2 and 5 must be made equal. There are six factors of 2 and nine factors of 5; therefore, multiply by \\(2^3\\) to obtain \\(2^9\times5^9=10^9\\). This requires at most nine decimal places.

The same conclusion follows from the standard rule: the maximum number of places is the larger exponent in the simplified denominator. Here, \\(\max(6,9)=9\\), so option B is correct. The value 15 would incorrectly add the exponents. Although a special numerator may cancel factors and produce fewer places, the maximum possible number remains nine.

The number of zeros between (6)'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. Here, \(0.0064 \times 100 = 0.64\), so the percentage is \(0.64\%\). Option C results from shifting the decimal point one place too far. Exam tip: when multiplying a decimal by 100, move the decimal point two places to the right.

On multiplying by (10000000), the decimal point moves seven places to the right. Therefore, the answer is (4882.8125).

The governing concept is the equivalence of terminating and recurring decimal representations. The bar over 9 means that 9 continues forever, so the number is 0.3749999… . This recurring decimal has exactly the same real-number value as 0.375; the apparent difference disappears in the limiting value. Also, converting the fraction gives 3/8 = 3 ÷ 8 = 0.375. Hence all three expressions represent the same number, and option C is correct. Option A is wrong because it treats the recurring decimal as genuinely smaller. Options B and D also impose false inequalities. A recurring sequence of 9s can represent the next terminating decimal exactly.

( \frac{47}{128}=0.3671875 ), so (0.3675) is greater than (0.3671875) and less than (0.368). Convert boundary values into decimals.

Reduce the fraction by its common factor 121: 242/605 = 2/5. The denominator 5 has no prime factor other than 5, so the rational number has a terminating decimal expansion. In fact, 2/5 = 0.4. Hence option A is correct. A recurring or non-recurring infinite decimal would require a different denominator structure, while 0.4 is not an integer.