Decimal representation

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

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 direct answer is B: \(0.730769230769\ldots\). Divide 19 by 26. Since 19 is less than 26, the whole-number part is 0. After adding a decimal zero, 190 divided by 26 gives 7, remainder 8. Bringing down 0 gives 80 divided by 26 = 3, remainder 2; then 20 divided by 26 = 0, remainder 20; then 200 divided by 26 = 7, remainder 18; continuing produces 69230 and eventually the remainder repeats, so the block repeats. Option A, 0.7313131..., has the wrong digits. Option B matches the division. Option C, 0.192626..., does not represent 19 divided by 26. Option D, 0.769230..., is close in pattern but is actually associated with a different numerator, not 19/26. Because 26 has factor 13 after simplification, the decimal is recurring, not terminating. Check by multiplying the first digits or performing long division.

( \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 direct answer is option A. Expanded form means writing a decimal as the sum of the value contributed by each non-zero digit. In 3.70007, the 3 is in the units place, the first 7 is in the tenths place, and the last 7 is in the hundred-thousandths place. Thus 3.70007=3+7/10+7/100000. The zeros between these digits contribute zero, so they need not be written as separate terms. Option A has exactly these place values. Option B places the first 7 in hundredths and the last 7 in ten-thousandths, producing a different number. Option C changes the whole-number part to 37, which is incorrect. Option D uses 70007/1000 and gives an incorrect value. Remember to count decimal places from the decimal point: the fifth place is hundred-thousandths.

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.

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.

(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.