Decimal representation

( \frac{21}{64}=0.328125 ), so (0.328125-0.203125=0.125). Convert the fraction into decimal and subtract.

The governing concept is preserving a quotient while clearing decimal points. Multiply both dividend and divisor by 1000, which does not change their ratio: 0.567 ÷ 0.021 = 567 ÷ 21. Now divide 567 by 21. Since 21 × 27 = 567, the quotient is 27. Therefore option B is correct. A direct check gives 0.021 × 27 = 0.567, confirming the result. The answer 2.7 is ten times too small, 270 is ten times too large, and 0.27 is smaller by a factor of 100. The important point is that both numbers must be multiplied by the same power of 10; moving the decimal in only one number would change the value of the quotient.

In the decimal expansion \(0.0646464\ldots\), the first digit after the decimal point, \(0\), occurs only once. After it, the block \(64\) repeats continuously: \(64,64,64,\ldots\). Hence, the bar is placed only over \(64\), giving \(0.0\overline{64}\). \(0.\overline{64}\) is incorrect because it makes \(64\) start immediately after the decimal point. Exam tip: Identify the exact block that repeats continuously before placing the bar.

In lowest form, a denominator containing a prime other than 2 or 5 cannot produce a terminating decimal. Since \(\frac{p}{q}\) is rational, its decimal repeats. Exam tip: factorise the denominator first.

The governing rule says that a rational number has a terminating decimal when the denominator in lowest terms contains only factors 2 and 5. Here q = 2⁸, so the decimal certainly terminates. To express the denominator as a power of 10, multiply numerator and denominator by 5⁸: 2⁸ × 5⁸ = 10⁸. Thus p/q = (p × 5⁸)/10⁸, which has at most eight digits after the decimal point. The maximum is reached when cancellation in the numerator does not remove any of these places, so option C is correct. It cannot be 6 or 7 because the denominator may require all eight places, and 10 is unnecessarily large. The phrase “in simplest form” ensures that no hidden denominator factor remains.

A terminating decimal is obtained by changing the denominator into a power of 10. The denominator here is \\(2^6\times5^3\\). Since there are six factors of 2 and only three factors of 5, multiply by \\(5^3\\) to create three additional pairs. Then the denominator becomes \\(2^6\times5^6=10^6\\), so at most six decimal places are needed.

The general rule is to take the larger exponent of 2 and 5 in the simplified denominator. Thus \\(\max(6,3)=6\\), making option B correct. The answer is not 9, because the exponents are not added; unmatched factors are supplied only to form equal pairs of 2 and 5. A particular numerator might shorten the decimal, but six is the maximum.

The number of zeros between (7)'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: \(0.0024 \times 100 = 0.24\). Therefore, the correct answer is \(0.24\%\). Remember that multiplying by 100 shifts the decimal point two places to the right; hence, 0.024% is ten times too small and 2.4% is ten times too large.

The governing concept is place-value movement when multiplying a decimal by a power of 10. Since 1,000,000 = 10⁶, multiplication moves the decimal point six places to the right. Starting with 0.00390625, the shifts are 0.0390625, 0.390625, 3.90625, 39.0625, 390.625, and finally 3906.25. Therefore option C is correct. The same result can be checked by writing 0.00390625 = 390625/100000000 and multiplying by 1000000, which gives 3906.25. Option A corresponds to moving the point four places, option B to five places, and option D to seven places. Counting all six zeros prevents these place-value errors.

\(\frac{21}{88}\) is already in lowest form, and \(88=2^3\times11\). Since the denominator contains 11, its decimal expansion is non-terminating recurring. The other fractions reduce to denominators with only 2s and 5s. Exam tip: simplify first.

( \frac{11}{16}=0.6875 ), so (0.6865) is greater than (0.686) and less than (0.6875). Convert boundary values into decimals.

For \(\frac{17}{360}\), \(360=2^3\times3^2\times5\). Since the denominator has a factor other than 2 or 5, its decimal expansion is non-terminating recurring. \(\frac{21}{150}=\frac{7}{50}\) terminates. Exam tip: reduce first.

(4000\times25=100000), so ( \frac{97}{4000}=\frac{2425}{100000}=0.02425 ). Convert the denominator into a power of (10) to find the decimal.

(0.0015625=\frac{15625}{10000000}=\frac{1}{640}). Count decimal places and write the fraction in simplest form.

The tenths digit (4) and hundredths digit (5) are the same in both decimals, so compare the thousandths digits. For 0.45a < 0.456, we need a < 6. Thus, a can be 0, 1, 2, 3, 4, or 5, giving 6 possible values. If a = 6, the two decimals are equal, so it is not allowed. Exam tip: when initial decimal digits are equal, the first differing digit from the left determines the comparison.

The governing concept is recurring-decimal notation. A bar is placed only above the complete block of digits that repeats indefinitely; non-repeating digits before that block remain outside the bar. In 0.00252525..., the first two digits after the decimal point are 00, and the repeating part is 25: 0.00 25 25 25 ... . Consequently, the correct notation is 0.00 overline{25}, which is option B. Option A incorrectly includes the initial non-repeating zeros in the recurring block. Option C marks only 5, although the repeating cycle has two digits, 25. Option D treats 025 as the repeating block, which does not match the displayed decimal expansion. Therefore B is the only unambiguous answer.

(2500\times4=10000), so ( \frac{109}{2500}=\frac{436}{10000}=0.0436 ). Making the denominator a power of (10) is a fast method.

(0.00078125=\frac{78125}{100000000}=\frac{1}{1280}). Counting decimal places and simplifying the fraction is important.

The denominator has (17), a factor other than (2) and (5). So the rational number will have a non-terminating recurring decimal.

The governing concept is the criterion for a terminating decimal. First simplify the fraction: the greatest common divisor of 126 and 210 is 42, so 126/210 = 3/5. A rational number in lowest terms has a terminating decimal expansion precisely when the prime factors of its denominator are only 2 and/or 5. The denominator here is 5, so the condition is satisfied. Indeed, 3/5 = 0.6, which ends after one decimal place. Therefore option D is correct. It is not an integer because its value is between 0 and 1, and it is neither recurring nor irrational. Simplification must come first because cancellation can remove other prime factors.