Decimal representation

( \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.

A rational number has a decimal expansion that either terminates or continues with a fixed repeating block. In 4.06006000600006…, the groups do not settle into a single repeating cycle: the numbers of zeros between successive 6s increase, so the pattern keeps changing. The decimal is therefore non-terminating and non-repeating, which is the characteristic decimal form of an irrational number. Hence option C is correct. It is not terminating, so A is impossible; it has no fixed recurring block, so B is wrong; and it is not an integer because nonzero digits occur after the decimal point, so D is also incorrect.

The bar over 9 means that 9 repeats forever: 0.249̅ means 0.2499999… rather than 0.249 only. A fundamental decimal identity is 0.2499999… = 0.2500000…, because an infinite tail of 9s carries into the preceding digit. Also, 1/4 = 0.25 exactly. Therefore all three quantities have the same value, so option C is correct. Option A incorrectly treats the repeating decimal as a finite 0.249, while options B and D incorrectly place one of the equal numbers strictly between the others. The equality can also be checked by writing 0.25 − 0.249999… = 0.

A rational number has a decimal expansion that either terminates or continues with a fixed repeating block. In 6.03003000300003…, the groups of zeros between successive 3s keep increasing, so there is no fixed block that repeats forever. The decimal is non-terminating and non-recurring, which is the characteristic decimal form of an irrational number. Therefore option C is correct. Option A is unsuitable because the decimal does not end. Option B is unsuitable because no constant cycle repeats. Option D is also impossible because an integer has no nonzero digits after the decimal point.

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.

A fraction in lowest terms has a terminating decimal exactly when the denominator contains no prime factors other than 2 and 5. To express q = 2⁹ × 5¹² as a power of 10 multiplied by a remaining factor, pair nine 2s with nine 5s to form 10⁹. Five 5s remain, so the denominator can be converted to 10¹² by supplying three additional factors of 2 in the numerator’s decimal scaling. In general, the required number of places is the larger exponent, max(9, 12) = 12. Therefore option D is correct. Cancellation in p may reduce the actual number, but 12 is the maximum possible.