Decimal representation

Correct answer: B. (0.6865)

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

Correct answer: B. \(\frac{17}{360}\)

Explanation: 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.

Correct answer: D. (0.02425)

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

Correct answer: C. ( \frac{1}{640} )

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

Correct answer: A. 6

Explanation: 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.

Correct answer: C. Irrational

Explanation: 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.

Correct answer: C. 0.249̅ = 0.25 = 1/4

Explanation: 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.

Correct answer: C. Irrational

Explanation: 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.

Correct answer: C. 0.374̅9 = 0.375 = 3/8

Explanation: 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.

Correct answer: D. 12

Explanation: 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.