Decimal representation

The governing concept is the criterion for a terminating decimal. First simplify the fraction: the greatest common divisor of 126 and 210 is 42, so 126/210 = 3/5. A rational number in lowest terms has a terminating decimal expansion precisely when the prime factors of its denominator are only 2 and/or 5. The denominator here is 5, so the condition is satisfied. Indeed, 3/5 = 0.6, which ends after one decimal place. Therefore option D is correct. It is not an integer because its value is between 0 and 1, and it is neither recurring nor irrational. Simplification must come first because cancellation can remove other prime factors.

The decimal 3.04004000400004... continues indefinitely, so it is not terminating. Its blocks do not repeat with a fixed period: the number of zeros between successive 4s keeps increasing. A rational number has a decimal expansion that is either terminating or non-terminating but eventually recurring. Since this expansion is non-terminating and not eventually periodic, it represents an irrational number. Thus option C is correct. Option A is impossible because digits continue forever; option B would require a repeating cycle; and option D is impossible because the decimal part is not zero. This conclusion uses the standard rational-decimal classification.

The governing concept is the denominator test for terminating decimals. A fraction in lowest terms has a terminating decimal exactly when the prime factorisation of its denominator contains only 2s, only 5s, or both. Here q = 2^9, so the condition is satisfied. To express the denominator as a power of 10, multiply by 5^9: 2^9 × 5^9 = 10^9. Therefore the fraction can be written with at most nine digits after the decimal point. This maximum occurs when the numerator does not cancel any factor from the denominator; for example, 1/2^9 = 0.001953125 has nine decimal places. Thus option C is correct. Eight is insufficient for the general maximum, while ten adds an unnecessary place. The word maximum is important because some numerators could cause cancellation and produce fewer places.

The governing concept is the equivalence of terminating and repeating decimal representations. The bar over 9 indicates 0.124999..., not a single final 9. Since 0.999... = 1, we have 0.124999... = 0.125000..., so 0.124̅9 equals 0.125 exactly. Also, 1/8 = 0.125 because dividing 1 by 8 gives 0.125, or because 8 × 0.125 = 1. Therefore all three quantities are equal and option C is correct. Option A incorrectly treats 0.124999... as slightly smaller; options B and D impose an ordering even though the values are identical. The apparent difference is only a difference in notation.

The governing concept is that a denominator of the form 2^m × 5^n produces a terminating decimal with at most max(m,n) decimal places. Here q = 2⁸ × 5¹¹. The powers must be balanced to create a power of 10. Multiply by 2³: 2⁸ × 5¹¹ × 2³ = 2¹¹ × 5¹¹ = 10¹¹. Consequently, the fraction can be written with denominator 10¹¹ and has at most eleven digits after the decimal point. The maximum occurs when the numerator causes no cancellation that shortens the result. Thus option B is correct. Option A ignores the larger exponent, option C adds exponents unnecessarily, and option D multiplies them without mathematical basis.

The governing concept is the decimal-expansion test for a rational number in lowest terms. First simplify the fraction: the greatest common divisor of 198 and 330 is 66, so 198/330=3/5. A rational number has a terminating decimal expansion exactly when, after reduction, the denominator has no prime factors other than 2 and 5. The denominator here is 5, so the condition is satisfied. Indeed, 3/5=0.6, which ends after one decimal place. Therefore option B is correct. Option A would occur if a factor such as 3 remained in the reduced denominator. Option C describes a non-terminating non-recurring decimal, generally associated with an irrational number, and option D is wrong because 3/5 is not an integer.

The governing idea is place-value alignment in decimal subtraction. Write 12.006 as 12.00600 by adding two zeros to the right; this does not change its value. Now subtract: 12.00600 − 7.21975. Subtracting in columns, the result is 4.78625. A quick check is that 7.21975 + 4.78625 = 12.00600, confirming the calculation. Thus option A is correct. Option B differs by 0.01000, while option C is close to the reversed or incorrectly borrowed computation and option D results from an error in one of the decimal columns.

The governing concept is that multiplying both dividend and divisor by the same nonzero power of 10 does not change their quotient. To remove the decimal point from the divisor, multiply both numbers by 1000: 0.945×1000=945 and 0.015×1000=15. Thus 0.945÷0.015=945÷15. Since 15×63=945, the quotient is 63, so option B is correct. A place-value check gives the same result: 0.015×60=0.900 and 0.015×3=0.045, whose sum is 0.945. Option A is ten times too small, option C is ten times too large, and option D results from shifting the decimal point incorrectly. The essential rule is to scale both numbers equally, not just the divisor.

The governing concept is recurring-decimal notation: a bar is placed over exactly the digit or block that repeats indefinitely. In 0.000727272…, the first three digits after the decimal point are 000 and do not repeat as part of the recurring cycle. After them, the two-digit block 72 repeats: 0.000 72 72 72… . Therefore the correct notation is 0.000̅72, with the bar covering both 7 and 2, so option B is correct. Option A places the bar over non-repeating digits and does not represent the actual cycle. Option C shifts the recurring block and includes an extra zero. Option D places the bar only over 2, although 7 also repeats in every cycle. Correct bar placement must identify the complete minimal repeating block.

The governing concept is the denominator test for the decimal expansion of a rational number. First reduce the fraction: the greatest common divisor of 264 and 440 is 88, so 264/440=3/5. In lowest terms, a rational number has a terminating decimal exactly when the denominator contains no prime factors other than 2 and 5. The reduced denominator is 5, so the condition is satisfied. Indeed, 3/5=0.6, which terminates after one decimal place. Therefore option B is correct. Option A would be appropriate if another prime factor, such as 3 or 7, remained in the denominator. Option C describes a non-terminating non-recurring decimal, while option D is false because 0.6 is not an integer. Simplification must be done before applying the test.

The fraction 61/90 is already in lowest terms because 61 is prime and does not divide 90. Factor the denominator: 90 = 2 × 3² × 5. A reduced rational fraction has a terminating decimal only when its denominator contains no primes other than 2 and 5. Since 3 remains as a factor, the decimal cannot terminate; because every rational number has either a terminating or an eventually repeating decimal, it must be non-terminating recurring. Thus option B is correct. Option A ignores the factor 3, option C describes an irrational decimal, and option D is not a valid classification for this fraction.

Use decimal place value by writing both numbers with the same number of decimal places: 15.009 = 15.00900. Now subtract 8.34575 from 15.00900 in aligned columns. The calculation gives 15.00900 − 8.34575 = 6.66325. A reverse check confirms the result because 8.34575 + 6.66325 = 15.00900. Therefore option A is correct. Option B differs by 0.01000 and indicates a borrowing error; option C is the complementary-looking result from mishandling the subtraction; and option D reflects an error in one decimal place. Keeping the decimal points aligned prevents these mistakes.

The governing concept is decimal division using equal scaling. Multiply both dividend and divisor by 1000 so that the divisor becomes an integer: 1.134×1000=1134 and 0.018×1000=18. Therefore 1.134÷0.018=1134÷18. Now 18×60=1080 and the remaining 54 equals 18×3, so 1134÷18=60+3=63. Hence option B is correct. The result can also be checked directly because 0.018×63=0.018×60+0.018×3=1.080+0.054=1.134. Option A is ten times too small, option C is ten times too large, and option D is one hundred times too small. Equal multiplication of both numbers preserves the quotient and prevents incorrect decimal shifting.

Reduce the fraction by its common factor 121: 242/605 = 2/5. The denominator 5 has no prime factor other than 5, so the rational number has a terminating decimal expansion. In fact, 2/5 = 0.4. Hence option A is correct. A recurring or non-recurring infinite decimal would require a different denominator structure, while 0.4 is not an integer.

We need to divide 313 by 128000. Since 128000 = 128 × 1000 and 313 ÷ 128 = 2.4453125, division by the additional factor 1000 gives 0.0024453125. The same result follows by long division or by writing 313/128000 and shifting the decimal appropriately. Thus option B is correct. Option A is ten times too large, which corresponds to placing the decimal one position too far to the right. Option C is ten times too small, with one extra zero after the decimal. Option D does not represent the given fraction at all. The denominator has only factors 2 and 5, so a terminating decimal is expected.

Answer: D, 7/16. A rational number written in lowest terms has a terminating decimal expansion only when the prime factors of its denominator are 2, 5, or both. The fraction 7/16 is already in lowest terms, and 16 = 2⁴, so it terminates: 7 ÷ 16 = 0.4375. Option A, √2, is irrational, so its decimal expansion is non-terminating and non-repeating. Option B, π, is also irrational; its decimal expansion does not terminate. Option C, √3, is irrational for the same relevant reason and does not have a terminating decimal expansion. The earlier forms involving an expression divided by itself were ambiguous because each could simplify to 1, so they have been removed. Memory cue: for a reduced fraction, inspect the denominator only—2s and 5s mean terminating; any other prime factor means non-terminating repeating.

The governing concept is the decimal representation of rational and irrational numbers. A terminating decimal or a non-terminating decimal with a repeating block is rational, because it can be expressed as a fraction of integers. Therefore, 2.47000... terminates, while 2.474747... and 2.471471471... repeat definite blocks and are also rational. In option D, the digits continue indefinitely without one fixed block repeating regularly. Hence it represents an irrational number. The ellipsis alone does not make a number irrational; the essential test is whether the decimal terminates or eventually repeats.