Decimal representation

List the natural numbers whose units digit is 7: 7, 17, 27, 37, 47, and so on. Applying the strict condition x<40 keeps 7, 17, 27 and 37, but removes 47 and every larger value. Therefore the correct roster form is Q={7,17,27,37}, which is option A. Option B contains 47, C contains 40 even though its last digit is 0, and D omits 37.

Because n is a natural number and n≤5, take n=1, 2, 3, 4, and 5. Substitution in x=2n−1 gives x=1, 3, 5, 7, and 9 respectively. Thus the roster form is A={1,3,5,7,9}, so option A is correct. Option C uses 2n instead of 2n−1, while B and D use an incorrect starting value or shift.

Since 0.75 has two digits after the decimal point, it can be written as \(\frac{75}{100}\). Dividing the numerator and denominator by 25 gives \(\frac{75}{100}=\frac{3}{4}\), so option A is correct. \(\frac{75}{10}\) is not the correct conversion because two decimal places require a denominator of 100. Exam tip: Remove the decimal point, use 1 followed by as many zeros as the number of decimal places, and then simplify the fraction.

There is one digit after the decimal point, so write 0.2 as a fraction with denominator 10: \(0.2=\frac{2}{10}=\frac{1}{5}\). Therefore, \(\frac{1}{5}\) is correct. The fraction \(\frac{2}{5}\) equals 0.4, not 0.2. Exam tip: the number of digits after the decimal determines the number of zeros in the denominator, such as 10, 100, or 1000.

\(0.625=\frac{625}{1000}=\frac{5}{8}\), so it is rational and has a terminating decimal form. Option D, \(0.\overline{27}\), is a repeating decimal; it is rational but not terminating. The decimals in options B and C are non-terminating and non-repeating. Exam tip: A rational number has a terminating decimal when the denominator in its simplest fractional form contains only the prime factors 2 and 5.

A rational number can be written as p/q with integers p and q and q not equal to zero. Its decimal expansion either terminates, such as 0.25 = 1/4, or continues with a repeating pattern, such as 0.666... = 2/3. A non-terminating, non-repeating decimal has no finite ending and no recurring block of digits; it cannot be expressed as a ratio of two integers, so it is irrational. Therefore option C is correct. A terminating decimal is rational, and a repeating decimal is also rational. Zero is rational because 0 = 0/1, so option D is not correct.

8\frac{2}{5}=\frac{4}{10}=0.49. Equivalently, dividing 2 by 5 gives 0.4, so option A is correct. The nearby distractor 0.25 represents 8\frac{1}{4}9, not 8\frac{2}{5}9. Exam tip: convert the denominator to 10, 100, or 1000 whenever possible to find a decimal form quickly.

\(\frac{7}{2}\) means 7 divided by 2. The quotient is 3 with remainder 1, and \(\frac{1}{2}=0.5\); therefore, \(\frac{7}{2}=3+0.5=3.5\). Hence, option B is correct. Exam tip: A rational number whose denominator, in lowest form, has only factors 2 and/or 5 has a terminating decimal representation.

Since \(1 \div 4 = 0.25\), the negative sign in the numerator remains in the decimal form: \(\frac{-1}{4}=-0.25\). Option A is incorrect because it omits the negative sign. Exam tip: a negative rational number has a negative decimal representation.

The denominator 100 has two zeros, so dividing 11 by 100 shifts the decimal point two places to the left: \(\frac{11}{100}=0.11\). The distractor 0.011 represents \(\frac{11}{1000}\), not \(\frac{11}{100}\). Exam tip: division by 10, 100, and 1000 shifts the decimal point one, two, and three places to the left, respectively.

\(2.75=\frac{275}{100}\). Dividing the numerator and denominator by 25 gives \(\frac{275}{100}=\frac{11}{4}\), so option A is correct. Option B equals \(0.28\), option C is approximately \(2.27\), and option D is approximately \(0.36\). Exam tip: a decimal with two digits after the decimal point can first be written over 100 and then simplified.

To convert \(\frac{7}{8}\) into a decimal, divide 7 by 8. Alternatively, multiplying the numerator and denominator by 125 gives \(\frac{7}{8}=\frac{875}{1000}=0.875\). Therefore, option A is correct. The value 0.78 has incorrect place values, and 0.708 is also not equal to \(\frac{7}{8}\). Exam tip: Convert a fraction to a denominator of 10, 100, or 1000 whenever possible.

\(9 \div 4 = 2.25\), so the decimal form of \(\frac{9}{4}\) is 2.25. Since the denominator 4 is a power of 2, the fraction has a terminating decimal expansion. Option D, 1.25, is the value of \(\frac{5}{4}\), not \(\frac{9}{4}\). Exam tip: divide the numerator by the denominator to convert a fraction into decimal form.

The bar over 12 means that the two-digit block repeats indefinitely, so the decimal is 0.12121212… This is a recurring decimal, and every recurring decimal can be expressed as a ratio of two integers. For an algebraic verification, let x = 0.121212…. Multiplying by 100 gives 100x = 12.121212…. Subtracting the first equation from the second removes the repeating decimal part: 99x = 12. Therefore x = 12/99 = 4/33. Since 4/33 is a quotient of integers with a nonzero denominator, the number is rational. Every rational number is real, so option B is correct. It is not an integer because its value is between 0 and 1.

A rational number has a decimal expansion that terminates or repeats periodically. In option B, 0.272727… contains the repeating block 27, so it is rational. Let x = 0.272727…. Multiplying by 100 gives 100x = 27.272727…. Subtracting x from this equation gives 99x = 27, so x = 27/99 = 3/11. This is a ratio of integers, and therefore it is a rational real number. Option A is described as non-repeating and infinite, which is the characteristic form of an irrational decimal. √10 is irrational because 10 is not a perfect square, and π is also irrational. Thus only option B satisfies the stated condition.

The governing concept is conversion of a terminating decimal into a rational fraction. Since 3.125 has three decimal places, write it as 3125/1000. Dividing numerator and denominator by their greatest common divisor, 125, gives 3125 ÷ 125 = 25 and 1000 ÷ 125 = 8. Thus option B, 25/8, is correct; the other options do not equal 3.125.

Converting \(4.5\) into a fraction gives \(4.5=\frac{45}{10}\). Dividing the numerator and denominator by 5 gives \(\frac{45}{10}=\frac{9}{2}\), so option A is correct. Exam tip: when a decimal has one digit after the decimal point, write it over 10 first and then simplify the fraction.

Multiply both the numerator and denominator by 5 to make the denominator 100: \(\frac{13}{20}=\frac{13\times5}{20\times5}=\frac{65}{100}=0.65\). Therefore, option C is correct. Option 0.20 merely resembles the denominator 20 and is not the value of the fraction, while 0.13 is also not equal to \(\frac{13}{20}\). Exam tip: Converting the denominator to 10, 100, or 1000 helps write decimal forms quickly.

To convert \(\frac{17}{25}\) into a decimal, multiply both numerator and denominator by 4: \(\frac{17}{25}=\frac{68}{100}=0.68\). Therefore, option B is correct. Option D represents the decimal value of the denominator 25, not the value of the complete fraction. Exam tip: If the denominator can be changed to 10, 100, or 1000, the decimal form can be found quickly.

The notation 0.\overline{45} means 0.454545…, where the two-digit block 45 repeats indefinitely. A repeating decimal is rational because it can be written as a fraction of integers. Let x = 0.454545…; then 100x = 45.454545…. Subtracting gives 99x = 45, so x = 45/99 = 5/11. Therefore the number is rational, and every rational number is also a real number. It is not irrational or undefined. It is also not an integer: 5/11 is between 0 and 1, whereas an integer has no nonzero fractional part. Hence option C is the complete and precise classification.