Decimal expansion of rational numbers

A decimal expansion is non-terminating and non-repeating when it continues forever without settling into a fixed repeating block. In 0.2020020002..., the gaps between successive 2s keep changing: there are increasing strings of zeroes, so no finite block repeats periodically. Therefore it is an irrational decimal, and option A is correct. In contrast, 0.2222... is repeating and equals 2/9, so it is rational. The decimals 0.75 and 1.5000 terminate; they equal 3/4 and 3/2 respectively, so they are also rational. The important distinction is that an infinite decimal need not be irrational: an infinite repeating decimal is rational. Thus one must check both conditions—no ending and no repeating pattern—rather than merely noticing the ellipsis.

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 theorem states that a rational number a/b in lowest form has a terminating decimal expansion if and only if the prime factors of b are only 2 and/or 5. Here the denominator is 2³ × 5 × 11, and the factor 11 remains because the fraction is already in lowest form. Therefore the decimal cannot terminate. Since a/b is rational, its decimal expansion must eventually repeat, so it is non-terminating recurring. Option A is correct. Option B would be possible only if no prime factor other than 2 or 5 remained. Option C describes irrational numbers, not a rational fraction, and option D is not guaranteed merely from the denominator’s factorization.

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.

The governing concept is conversion of a terminating decimal into an equivalent fraction. Since 0.375 has three digits after the decimal point, write it as 375/1000. Simplifying by the common factor 125 gives 375 ÷ 125 = 3 and 1000 ÷ 125 = 8, so 0.375 = 3/8. Therefore the point representing 0.375 is also the point representing 3/8, and option A is correct. Option C, 5/8, equals 0.625; option D, 8/3, is greater than 2; and option B, 37/5, equals 7.4. These values clearly do not represent the given point. Keeping place value correct before reducing prevents errors.

The governing concept is converting a terminating decimal into a rational fraction and reducing it to lowest terms. Because −0.625 has three digits after the decimal point, write it as −625/1000. The numerator and denominator have common factor 125, so divide both by 125: −625÷125=−5 and 1000÷125=8. Thus −0.625=−5/8, making option A correct. The negative sign must remain because the original number lies to the left of zero on the number line. For comparison, −3/8=−0.375, −7/8=−0.875, and −1/8=−0.125, so options B, C, and D do not represent the given decimal.