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.

\(0.125\) is a terminating decimal, so \(\frac{125}{1000}=\frac{1}{8}\). Hence it is rational, but not an integer. Exam tip: every terminating decimal is rational; digits after the decimal do not make a number irrational.

The governing concept is place value in a terminating decimal. The number 0.5 has one digit after the decimal point, so it means five tenths and can be written directly as 5/10. This fraction simplifies by dividing numerator and denominator by 5: 5/10 = 1/2. Because the question asks what is obtained when the decimal is converted and includes 5/10 as an option, option B is the intended correct answer; 1/2 is its simplest equivalent form. Option A equals 0.2, option C equals 0.05, and option D equals 2, so none represents 0.5. The general method is to remove the decimal point and use 10, 100, or another power of ten according to the number of decimal places, then simplify.

The digits in (2.75) end so it is terminating. Exam tip: a terminating decimal has a finite number of digits.

In (0.777\ldots), (7) repeats so it is non-terminating recurring. Exam tip: identify the repeating digit or group.

The governing concept is conversion of a recurring decimal into a fraction. Let x = 0.666..., where the digit 6 repeats indefinitely. Multiplying by 10 gives 10x = 6.666... . Subtracting the original equation from this one removes the repeating decimal: 10x - x = 6.666... - 0.666..., so 9x = 6 and x = 6/9 = 2/3. Therefore option B is correct. The shortcut gives the same result: one repeating digit is written over 9, so 0.̅6 = 6/9, which simplifies to 2/3. The other options have different values and do not represent the repeating decimal.