Decimal representation

\(\frac{3}{28}\) is already in lowest terms, and \(28=2^2\times 7\). The decimal expansion of a rational number terminates only when the prime factors of its denominator are limited to \(2\) and \(5\). Since the denominator also contains the factor \(7\), the decimal expansion is non-terminating but recurring. The number is rational, so it is neither irrational nor an integer. Exam tip: After reducing a fraction, inspect its denominator; only factors \(2\) and \(5\) give a terminating decimal.

The fraction \(\frac{17}{125}\) is already in lowest terms, and \(125=5^3\). A fraction in lowest terms has a terminating decimal expansion when its denominator contains only the prime factors 2 and/or 5. In fact, \(\frac{17}{125}=\frac{136}{1000}=0.136\), so the decimal terminates. Option B is incorrect because the decimal is not infinite, and option C is incorrect because the fraction is rational. Exam tip: Check the prime factors of the denominator after reducing the fraction; only 2s and 5s give a terminating decimal.

Since \(1.69=1.3^2\), we have \(\sqrt{1.69}=1.3\). Similarly, \(0.09=0.3^2\), so \(\sqrt{0.09}=0.3\). Therefore, \(1.3+0.3=1.6\), making option A correct. Option D is only the value of the first square root and does not include the second term. Exam tip: express each decimal as the square of a simpler decimal number before finding its square root.

For a fraction in lowest terms, the decimal terminates only when the denominator has no prime factors other than 2 and 5. The fraction 8/13 is already in lowest terms, and 13 is a prime factor different from 2 and 5. Therefore its decimal expansion continues indefinitely but repeats in a cycle. Option B is correct; the fraction is defined and is not an integer.

The governing concept is the denominator test for rational decimals. A fraction in lowest terms has a terminating decimal expansion exactly when the prime factors of its denominator are only 2 and/or 5. Here, 200 = 2³ × 5², and 29 shares no factor with 200, so 29/200 is already in lowest terms. It can also be converted directly to denominator 1000: 29/200 = 145/1000 = 0.145. Since the decimal ends after three places, option A is correct. It is not irrational or non-terminating non-repeating; those descriptions apply to numbers such as √2. The expression is defined because its denominator is not zero.

A reduced fraction has a terminating decimal only if every prime factor of its denominator is 2 or 5. In 11/17, the denominator 17 is prime and is neither 2 nor 5. Long division therefore continues and eventually repeats a remainder, producing a non-terminating recurring decimal. Hence option B is correct; the fraction is defined but neither terminating nor an integer.

The governing criterion is that a rational number p/q in lowest terms has a terminating decimal expansion exactly when the prime factors of q are only 2 and/or 5. The fraction 33/160 is already in lowest terms because 33 and 160 have no common factor. Also, 160 = 2⁵ × 5, so the criterion is satisfied. To verify directly, multiply numerator and denominator by 625: 33/160 = 20625/100000 = 0.20625. The decimal ends, so option A is correct. It is not irrational, because it is a ratio of integers. It is not undefined because the denominator is non-zero, and it is not non-terminating non-repeating because its decimal expansion stops.

For a rational number written in lowest terms, the decimal expansion terminates only when the denominator has no prime factors other than 2 and 5. The fraction 7/66 is already in lowest terms, and 66 = 2 × 3 × 11. Since the denominator contains 3 and 11, its decimal expansion cannot terminate; every rational number nevertheless has either a terminating or a repeating decimal expansion. Therefore 7/66 is non-terminating but repeating, so option B is correct. The decimal begins 0.1060606..., showing a recurring pattern. Option A ignores the factors 3 and 11, option C confuses repeating rational decimals with irrational decimals, and option D is impossible because the fraction lies between 0 and 1.

A rational fraction in lowest terms has a terminating decimal expansion when the denominator contains only the prime factors 2 and 5. Here, 41 is prime and does not divide 500, so 41/500 is already reduced. Factorising the denominator gives 500 = 2² × 5³. Thus the criterion is satisfied and the decimal terminates. For a direct calculation, multiply by 2² to obtain 41/500 = 164/2000 = 0.082. Hence option A is correct. Options B and C describe a non-terminating non-repeating decimal, which cannot represent this rational fraction, while option D is impossible because the denominator is not zero.

The decimal-expansion rule must be applied after reducing the fraction. The greatest common divisor of 105 and 126 is 21, so 105/126 = 5/6. The denominator 6 factors as 2 × 3. Because the reduced denominator contains 3, not only 2 and 5, the decimal cannot terminate. Since 5/6 is rational, its decimal must be repeating; indeed, 5/6 = 0.83333..., where 3 repeats indefinitely. Hence option B is correct. Option A would be valid only if the reduced denominator had factors exclusively 2 and 5. Option C describes irrational decimals, and option D is impossible because the original denominator is non-zero.

The governing rule is that a rational number in lowest terms has a terminating decimal only when its denominator has no prime factors other than 2 and 5. First reduce the fraction: \(35/154=5/22\), because both numerator and denominator are divisible by 7. The denominator 22 factors as \(2\times11\), and the factor 11 remains. Therefore its decimal expansion does not terminate; because the number is rational, its decimal digits repeat periodically. In fact, \(5/22=0.2272727\ldots\). Hence option B is correct. It is not irrational, not terminating, and certainly not an integer.

The governing concept is expressing a fraction as a terminating decimal. To make the denominator a power of ten, multiply 40 by 25, giving 1000. The numerator must also be multiplied by 25: 7/40 = (7 × 25)/(40 × 25) = 175/1000 = 0.175. Therefore option A is correct. The result is sensible because 7/40 is less than 1, so its decimal must be less than 1; this rules out 1.75 immediately. The value 0.075 equals 3/40, so it uses the wrong numerator. The value 0.740 is not the result of the fraction calculation and equals 0.74. Since 40 = 2³ × 5, its denominator contains no prime factors other than 2 and 5; consequently the decimal expansion terminates after a finite number of places. Both denominator scaling and ordinary division verify 0.175.

(80\times125=10000), so ( \frac{19}{80}=\frac{2375}{10000}=0.2375 ). Make the denominator a power of (10).

Since 0.3125 has four digits after the decimal point, it can be written as \(\frac{3125}{10000}\). Dividing the numerator and denominator by 625 gives \(\frac{3125}{10000}=\frac{5}{16}\). Option A, \(\frac{31}{100}\), and option B, \(\frac{3125}{1000}\), are not equal to the given decimal. Exam tip: use \(10^n\) as the denominator when there are n decimal places, then reduce the fraction.

The fraction 13/150 is already in simplest form because 13 shares no common factor with 150. Factor the denominator: 150 = 2 × 3 × 5². A rational number has a terminating decimal only when, after simplification, its denominator contains no prime factors other than 2 and 5. Here the factor 3 remains, so division by 150 cannot end; instead, the remainders eventually repeat and produce a recurring block. Therefore the decimal expansion is non-terminating and recurring, making option D correct. Option A would apply to a denominator made only from 2s and 5s, option B describes an irrational decimal, and option C is wrong because the fraction is not an integer.

A is correct. Since \(0.0999\ldots=0.1\), \(0.4999\ldots=0.4+0.1=0.5\). Infinite 9s do not leave a gap below the next decimal. Exam tip: remember \(0.999\ldots=1\).

Write the numbers to three decimal places for comparison: \(0.56=0.560\) and \(0.57=0.570\). Since \(0.560<0.565<0.570\), \(0.565\) lies between them. \(0.570\) is equal to \(0.57\), so it is not strictly between the two numbers. Exam tip: When comparing decimals, add zeros at the end if needed to make the number of decimal places equal.

For a rational number in lowest terms, the decimal expansion terminates precisely when the denominator has no prime factors other than 2 and 5. Here 21/125 is already in lowest terms and 125 = 5³. Hence its decimal expansion terminates; in fact, 21/125 = 0.168. Therefore, option C is correct, whereas the first two choices require a different denominator structure.

Since 3.125 has three digits after the decimal point, it can be written as \(\frac{3125}{1000}\). Dividing the numerator and denominator by 125 gives \(\frac{3125}{1000}=\frac{25}{8}\), so option A is correct. Exam tip: use a denominator of 1 followed by as many zeros as there are decimal places, then simplify the fraction. Option C incorrectly uses 100 instead of 1000 as the denominator.

In 0.0808..., the complete block 08 repeats continuously: 08, 08, 08, ... . Therefore, the bar must be placed over both digits, giving \(0.\overline{08}\). In contrast, \(0.0\overline{8}\) means 0.0888..., which is a different decimal. Exam tip: first identify the smallest repeating block after the decimal point before placing the bar.

The direct answer is option A: 0.116. A fraction means numerator divided by denominator, so we need the decimal value of 29 divided by 250. Convert the denominator into 1000, because a denominator of 10, 100, or 1000 gives a decimal immediately. Since 250 × 4 = 1000, multiply the numerator by the same 4: 29 × 4 = 116. Thus 29/250 = 116/1000 = 0.116. Option A matches this result. Option B, 0.29, would be 29/100, not 29/250. Option C, 0.250, represents 250/1000, or 1/4, and is unrelated to the numerator. Option D, 1.16, is ten times 0.116 and is too large because 29 is less than 250, so the fraction must be less than 1. Memory cue: multiply numerator and denominator by the same number, never by different numbers.

In 4.006, 6 is in the third place to the right of the decimal point. The places after the decimal are tenths, hundredths, and thousandths, so the place value of 6 is \(6 \times \frac{1}{1000}=\frac{6}{1000}\). Option B, \(\frac{6}{100}\), is incorrect because it represents the hundredths place, which is the second decimal place. Exam tip: Count the digits after the decimal point and write that many zeros in the denominator after 1.

The number 0.00064 has five digits after the decimal point, so first write it as 64/100000. Now simplify by dividing numerator and denominator by their greatest common divisor. Since 64 = 8 × 8 and 100000 is divisible by 8, division by 8 gives 8/12500. The numerator 8 and denominator 12500 have no common factor greater than 1, so this is the simplified fraction. Therefore option B is correct. Option A is not equivalent because its denominator has only three zeros. Options C and D may look like reductions, but 4/6250 = 8/12500 and 16/2500 = 8/1250; neither is the requested simplest equivalent form, and D is not even equal to the original decimal.

For comparison, 0.37 can be written as 0.370 because adding zeros at the end of a decimal does not change its value. Comparing 0.370 and 0.307, the first decimal digit, 3, is the same, but at the next place 7 is greater than 0. Therefore, \(x>y\). The relation \(x=y\) is incorrect because 0.370 and 0.307 are not equal. Exam tip: First write decimals with the same number of decimal places before comparing them.

For \(\frac{7}{40}\), \(40=2^3\times5\). A reduced fraction terminates only when its denominator has 2 and/or 5. The other denominators contain 3. Exam tip: factor the denominator first.