Decimal expansion of rational numbers

Correct answer: A. Non-terminating and repeating

Explanation: Dividing 1 by 6 gives 0.166666..., so the digit 6 repeats indefinitely after the first decimal place. The decimal expansion is therefore non-terminating and repeating. This also agrees with the rational-number theorem: when a fraction is in lowest terms, its decimal expansion terminates only if the denominator has no prime factors other than 2 and 5. The denominator 6 = 2 × 3 contains the prime factor 3, so the decimal cannot terminate and must repeat. Option B is false because there is no final decimal digit, option C is false because the digits do repeat, and option D is false because 1/6 is a real rational number. Hence A is correct.

Correct answer: A. 6.1010010001...

Explanation: A decimal represents a rational number if it terminates or if its digits eventually repeat in a fixed pattern. The decimal 6.1010010001... continues without ending, and its blocks do not settle into a repeating cycle: the number of zeros between successive 1s keeps changing. Hence it is non-terminating and non-repeating, so it represents an irrational number. Option B, 6.101010..., repeats the block 10 and is rational. Option C terminates and equals 49/8, so it is rational. Option D equals 6 because trailing zeros do not change a value, and 6 is rational. Therefore only option A does not represent a rational number.

Correct answer: A. Rational number

Explanation: The governing concept is the decimal characterization of rational numbers. A decimal is rational when it terminates or eventually repeats a finite block of digits. In 2.3454545..., the digits after the initial 3 continue with the repeating block 45: 2.3\overline{45}. Therefore it is a recurring, non-terminating decimal and must represent a rational number. It could be converted into a fraction by the standard place-value subtraction method, although that calculation is not needed to classify it. It is not irrational because irrational decimals never repeat periodically. It is real, so option C is impossible, and it is not an integer because it has a nonzero fractional part. Hence option A is correct.

Correct answer: A. 15/4

Explanation: The governing concept is conversion of a terminating decimal into an equivalent fraction. Since 3.75 has two digits after the decimal point, write 3.75=375/100. Reduce this fraction by dividing numerator and denominator by their greatest common factor, 25: 375÷25=15 and 100÷25=4. Hence 3.75=15/4, and equivalent numbers represent the same point on the number line. The alternatives can be checked directly: 7/4=1.75, 3/4=0.75, and 75/3=25. None of these equals 3.75. Therefore option A is correct. The denominator 100 arises from the two decimal places, and reduction gives the simplest exact fraction rather than an approximation.

Correct answer: B. 2.2

Explanation: The governing concept is that equivalent rational numbers occupy the same point on the number line. Convert 11/5 to decimal form by division or by making the denominator 10. Multiplying numerator and denominator by 2 gives 11/5=22/10=2.2. Direct division also confirms this: 5 goes into 11 twice, leaving remainder 1; continuing with decimals gives 10 tenths, of which 5 goes twice, producing 2.2 exactly. Thus option B is correct. Option A represents 2.1, option C represents 1.2, and option D represents 5.11, so none of those is equal to 11/5. The exact conversion identifies the unique point without relying on a rough estimate.

Correct answer: A. -5/8

Explanation: Answer: A, −5/8. Convert the terminating decimal to a fraction by using 1000 as the denominator because there are three digits after the decimal point: −0.625=−625/1000. Reduce the fraction by dividing numerator and denominator by 125: −625÷125=−5 and 1000÷125=8. Thus −0.625=−5/8. The negative sign must remain because the number lies to the left of zero on the number line. B is −3/8=−0.375, C is −7/8=−0.875, and D is −1/8=−0.125; none equals −0.625. A useful check is to convert 5/8: 1/8=0.125, so 5/8=0.625, and then restore the negative sign. Do not drop the sign while simplifying.