Decimal expansion of rational numbers

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.

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.

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.

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.

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.

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.