Decimal expansion of rational numbers

The governing concept is converting a terminating decimal to lowest fractional form and then prime-factorising the reduced denominator. Since 0.0375 has four decimal places, write it as 375/10000. Divide numerator and denominator by their greatest common divisor, 125: 375 ÷ 125 = 3 and 10000 ÷ 125 = 80. Thus 0.0375 = 3/80. Now factor the denominator: 80 = 8 × 10 = 2³ × (2 × 5) = 2⁴ × 5. Therefore option B is correct. Option A misses one factor of 2, while options C and D introduce an extra factor of 5 and do not represent the prime factorisation of 80. Reduction before factorisation is important.

The governing rule is that a rational number p/q in lowest form has a terminating decimal expansion only when the prime factors of q are 2 and/or 5. Here q contains 7^r, and r > 0, so at least one factor 7 remains in the reduced denominator. No cancellation with p is possible because p/q is already in lowest form. A denominator containing another prime factor cannot be converted into a power of 10, so the division continues indefinitely. Since the number is rational, its repeating remainder pattern must eventually recur. Therefore option B, non-terminating recurring, is correct. Option A would apply only if no factor other than 2 or 5 remained; option C describes an irrational decimal; option D incorrectly makes termination depend on m and n being equal.

First reduce the fraction before applying the decimal-expansion rule. Since 72 = 2^3 × 3^2, the numerator cancels completely with the factors 2^3 × 3^2 in the denominator. Thus 72/(2^3 × 3^2 × 5^5) = 1/5^5 = 1/3125. To express this with a denominator that is a power of 10, multiply numerator and denominator by 2^5: 1/5^5 = 2^5/10^5 = 32/100000 = 0.00032. The decimal therefore ends after five digits to the right of the decimal point. Option C is correct. Options A and B use smaller exponents without justification, while option D is false because the reduced denominator contains only the prime factor 5, so the decimal is terminating.

The denominator must be examined after cancelling common factors. Factor the numerator as 14 = 2 × 7. Cancelling these factors from 2^2 × 5^3 × 7^2 leaves the reduced denominator 2 × 5^3 × 7. The criterion says that a rational number has a terminating decimal only when its reduced denominator has no prime factors other than 2 and 5. The remaining factor 7 prevents termination. Because the number is rational, its non-terminating decimal is recurring rather than non-recurring. Therefore option B is correct. Option A is wrong because the factor 7 remains; option D is also wrong for the same reason, and option C would describe a non-rational decimal rather than this rational number.

For a rational number written in lowest form, the decimal expansion terminates precisely when the denominator has no prime factors other than 2 and 5. In this question, r > 0, so 11^r is a genuine factor of the reduced denominator. Because the fraction is already in lowest form, this factor 11 cannot cancel with the numerator. Consequently the denominator cannot be changed into a power of 10 by multiplying numerator and denominator by suitable factors. The division will continue without ending. Since every rational decimal is either terminating or eventually recurring, this continuing decimal must be non-terminating recurring. Thus option B is correct. Option A and option D ignore the surviving factor 11, whereas option C is associated with irrational numbers, not a rational fraction.

Reduce the fraction first. The numerator is 22 = 2 × 11. Cancelling these factors from the denominator 2^2 × 5^4 × 11^2 leaves 2 × 5^4 × 11. A rational number has a terminating decimal only if every prime factor in its reduced denominator is 2 or 5. Although the factors 2 and 5 would support termination, the factor 11 remains after cancellation, so the denominator is not a power of 10 and the decimal cannot terminate. The number is rational, therefore its infinite decimal expansion is eventually periodic, or recurring. Hence option B is correct. Option A and option D incorrectly assume that the factors 2 and 5 alone decide the result before reduction; option C incorrectly treats a rational decimal as non-recurring.

The relevant theorem states that a rational number p/q in lowest terms has a terminating decimal if and only if the reduced denominator q is of the form 2^a × 5^b. Here q also contains 13^r, and r is positive, so a factor 13 remains in the denominator. Lowest form guarantees that this factor cannot be cancelled by the numerator. Therefore q cannot be transformed into a power of 10, and the decimal expansion does not end. Since p/q is rational, its non-terminating decimal must eventually repeat; it cannot be non-terminating non-recurring. Thus option B is correct. Options A and D would require the absence of the factor 13, while option C does not describe the decimal behaviour of a rational number.

Because 7^0 = 1, the denominator is effectively 2^5 × 5^5. These powers combine to give 10^5, since 2^5 × 5^5 = (2 × 5)^5 = 10^5. Thus the fraction has a denominator of 100000 after reduction and its decimal expansion terminates within five places. Moreover, because the fraction is in lowest form, the numerator is coprime to both 2 and 5. It therefore cannot supply a factor that would cancel the final power of 10; the fifth decimal digit cannot become an unnecessary trailing zero. Hence the expansion terminates exactly after five places, making option A correct. Option B incorrectly adds exponents, while options C and D contradict the fact that the denominator contains only 2 and 5.

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.

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 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.