Decimal representation

(2500\times4=10000), so ( \frac{71}{2500}=\frac{284}{10000}=0.0284 ). Make the denominator a power of (10).

To find the ratio, divide the larger number by the smaller number: \(0.075 \div 0.00075 = 100\). Therefore, \(0.075\) is 100 times \(0.00075\). The 10-times option is incorrect because the ratio is determined by the quotient, not merely by counting the visible decimal places. Exam tip: Express both decimals with a common place value; \(0.075 = 75000\times10^{-6}\) and \(0.00075 = 750\times10^{-6}\), giving a ratio of 100.

The bar is over 45 only, so after the initial two zeros, 45 repeats continuously: \(0.00\overline{45}=0.00454545\ldots\). In option B, the repeating block begins one decimal place later, so it represents a different number. Exam tip: Repeat only the digits under the bar; digits before the bar are written only once.

\(\frac{7}{12}=\frac{7}{2^2\times3}=0.58\overline{3}\), so repetition begins only after 58, not immediately after the decimal point. Thus it disproves Riya’s claim. Exam tip: with 2 or 5 plus another prime factor, a non-repeating beginning may occur.

First reduce the fraction before deciding its decimal expansion. The numerator 63 and denominator 231 have a common factor of 21, so 63/231 becomes 3/11. For a fraction in simplest form, the decimal terminates only if its denominator has no prime factors other than 2 and 5. Since the denominator here is 11, the decimal cannot terminate.

Dividing 3 by 11 gives 0.272727 and the digits 27 continue repeating forever. Thus the decimal expansion is non-terminating recurring, not non-terminating non-recurring. The correct choice is option C. The reduction step is important because the denominator must be inspected only after all common factors have been cancelled; in this case, the simplified denominator still contains 11.

The correct answer is 5.75. Since 0.00999\ldots = 0.01, we get 5.74999\ldots = 5.74 + 0.00999\ldots = 5.75. The number 5.749 simply stops after three decimal places, whereas the given decimal has infinitely repeating 9s. Exam tip: when infinitely many 9s follow a digit, the decimal can be written by increasing the preceding place by 1.

After the decimal point, the places represent tenths, hundredths, thousandths and ten-thousandths, respectively. In 3.090800, 8 is the fourth digit after the decimal point, so its place value is \(\frac{8}{10000}\), or 0.0008. \(\frac{8}{1000}\) would represent the value of a digit in the third decimal place. Exam tip: Count decimal places from immediately after the decimal point.

( \frac{3}{16}=0.1875 ) and (0.18\overline{7}=0.18777\ldots). Therefore, the recurring number is greater.

Write all the numbers to four decimal places: \(b=0.3600\), \(a=0.3060\), and \(c=0.3006\). Thus, \(b\) is the greatest. Also, \(a=0.3060\) is greater than \(c=0.3006\) because at the hundredths place, \(a\) has 6 while \(c\) has 0. Therefore, the descending order is \(b>a>c\). Option \(b>c>a\) is wrong because \(0.3006<0.3060\). Exam tip: Add trailing zeros to make the decimal places equal before comparing decimals.

( \frac{11}{125}=0.088 ) and ( \frac{9}{80}=0.1125 ), so the sum is (0.2005). First convert the fractions into decimals.

The governing concept is converting equivalent forms before performing subtraction. Convert the fraction to a decimal by dividing 13 by 32. Since 1/32 = 0.03125, multiplying by 13 gives 13/32 = 0.40625. Now subtract the given decimal: 0.40625 − 0.15625 = 0.25000, which is 0.25. Therefore option A is correct. A useful check is to write both numbers with five decimal places; the subtraction is 40625 − 15625 in units of 1/100000, giving 25000/100000 = 0.25. Option B repeats the subtracted number, option C is only the converted fraction before subtraction, and option D is unrelated to the required difference.

The governing concept is decimal division by making the divisor a whole number while multiplying both numbers by the same power of 10. The divisor 0.016 has three decimal places, so multiply both dividend and divisor by 1000: 0.432 ÷ 0.016 = 432 ÷ 16. Since 16 × 27 = 432, the quotient is 27. A direct check gives 0.016 × 27 = 0.432, confirming the result. Therefore option B is correct. Option A, 2.7, is ten times too small; option C, 270, is ten times too large; and option D, 0.27, results from moving the decimal point incorrectly. Multiplying both terms by the same nonzero number does not change their quotient, which is why the conversion is valid.

In \(0.0767676\ldots\), the first digit after the decimal point, \(0\), occurs only once. After that, the block \(76\) repeats: \(0.0\,76\,76\,76\ldots\). Therefore, the bar is placed only over \(76\), giving \(0.0\overline{76}\). Option A incorrectly treats \(076\) as the repeating block. Exam tip: Before placing a bar, identify the shortest block of digits that repeats continuously.

A decimal terminates only when the denominator \(q\), in lowest form, has no prime factors other than 2 and 5; hence \(q=2^m5^n\). Any other prime factor gives a non-terminating recurring decimal. Exam tip: reduce the fraction first.

A rational number in lowest terms has a terminating decimal only when the prime factors of its denominator are 2 and/or 5. Here q = 5^8. To express the denominator as a power of 10, multiply numerator and denominator by 2^8: 5^8 × 2^8 = 10^8. Thus the decimal can have at most eight places. It may have fewer places if the numerator causes cancellation after conversion, but eight is the maximum possible. Therefore option C is correct. Options A and B do not supply enough factors of 2 to form 10^8, while option D overestimates the required power. The conclusion follows directly from the terminating-decimal theorem.

A fraction with denominator \\(2^5\times5^2\\) can be converted into a denominator that is a power of 10. To do this, the smaller power, \\(5^2\\), must be matched with two more factors of 5, while the two factors of 2 are already available. The resulting denominator is \\(2^5\times5^5=10^5\\). Hence the decimal can require at most five places.

The maximum is determined by the larger exponent, not by adding the exponents. Thus it is 5, which is option B. For example, multiplying numerator and denominator by \\(5^3\\) gives a denominator of \\(10^5\\). Some fractions may have fewer places because cancellation or trailing zeros can occur, but five is the greatest possible number under the stated denominator condition.

The number of zeros between (5)'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: \(0.0016\times100=0.16\). Therefore, the percentage is \(0.16\%\). Option A results from shifting the decimal by only one place, whereas multiplication by 100 shifts it two places to the right. Exam tip: For decimal-to-percentage conversion, move the decimal point two places to the right and add the percent sign.

The governing concept is multiplication of a terminating decimal by a power of 10. Since 1,000,000 = 10⁶, the decimal point in 0.0078125 moves six places to the right. Counting the shifts gives 0.078125 after one, 0.78125 after two, 7.8125 after three, 78.125 after four, 781.25 after five, and 7812.5 after six shifts. Therefore 0.0078125 × 1,000,000 = 7812.5, so option C is correct. Option A corresponds to moving the point only four places, option B to five places, and option D to seven places or an extra shift. The power-of-ten rule and the explicit place-value count independently confirm the answer.

In lowest terms, if \(q\) has a prime factor other than 2 or 5, it cannot divide any \(10^n\), so the decimal is non-terminating recurring. A denominator containing only 2 and 5 gives a terminating decimal. Exam tip: reduce the fraction first.