Decimal representation

For \(\frac{13}{40}\), the denominator is \(40=2^3\times5\). A fraction in lowest terms has a terminating decimal only when its denominator has no prime factors other than 2 and 5. Hence, \(\frac{13}{40}=0.325\) is terminating. In contrast, \(\frac{7}{15}\) has 3 in its denominator, and \(\frac{5}{21}\) has 3 and 7, so their decimals are non-terminating recurring. Exam tip: First reduce the fraction to lowest terms, then factorise the denominator.

Since \(125=5^3\), the denominator of \(\frac{11}{125}\) in lowest form has only the prime factor \(5\). Therefore, its decimal expansion terminates. In fact, \(\frac{11}{125}=\frac{88}{1000}=0.088\). A non-terminating recurring decimal occurs when the denominator has a prime factor other than \(2\) or \(5\). Exam tip: first reduce the fraction, then check the prime factors of its denominator.

The correct answer is \(\frac{5}{12}\). A fraction in lowest terms has a terminating decimal only when the prime factors of its denominator are 2 and/or 5. Here, \(12=2^2\times3\), and the factor 3 is present, so its decimal expansion is non-terminating recurring. In contrast, 20, 25, and 125 have only 2 and/or 5 as prime factors. Exam tip: First reduce the fraction to lowest terms, then check the denominator's prime factors.

For \(\frac{7}{12}\), the denominator has prime factorisation \(12=2^2\times3\). In lowest form, a fraction has a non-terminating recurring decimal when its denominator contains a prime factor other than 2 or 5. Hence, \(\frac{7}{12}=0.58\overline{3}\). In contrast, 16, 40, and 125 have only 2 and/or 5 as prime factors, so their decimal expansions terminate. Exam tip: first reduce the fraction, then check the prime factors of its denominator.

\(\frac{11}{50}=0.22\), so its decimal expansion terminates. A rational number \(\frac{p}{q}\) has a terminating decimal expansion only when, in lowest terms, the prime factors of \(q\) are only 2 and/or 5. Here, \(50=2\times5^2\). In contrast, \(24\), \(18\), and \(27\) contain 3 as a prime factor, so their decimal expansions are non-terminating recurring. Exam tip: First reduce the fraction, then check the prime factors of its denominator.

For \(\frac{13}{15}\), the prime factorisation of the denominator is \(15=3\times5\). A rational number has a terminating decimal only when, in lowest terms, its denominator has no prime factors other than \(2\) and \(5\). Since the denominator here contains \(3\), its decimal expansion is non-terminating recurring. The denominators in the other options contain only factors of \(2\) and/or \(5\). Exam tip: First reduce the fraction to lowest terms, then check the prime factors of its denominator.

\(0.125\) 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.

For \(\frac{7}{40}\), the denominator is \(40=2^3\times5\), so its decimal expansion terminates. The other denominators contain 3. Exam tip: first reduce the fraction to lowest terms.

In (\frac{11}{40}), the denominator (40=2^3\times5), so the decimal is terminating. Exam tip: check whether the denominator has only (2) and (5) as prime factors.

In \(0.121212\ldots\), the block 12 repeats endlessly, so it is non-terminating recurring. In \(0.1010010001\ldots\), no fixed block repeats. Exam tip: look for a repeating digit pattern.

(4) is in the hundredths place, so its place value is (\frac{4}{100}). Exam tip: after the decimal point, the first place is tenths and the second is hundredths.

\(0.125000\ldots=\frac{125}{1000}=\frac18\), so it is rational and has a terminating decimal expansion. Infinitely many trailing zeros do not change the value. Exam tip: remove trailing zeros before classifying a decimal.

In (0.04), (4) is in the hundredths place, so (0.04=\frac{4}{100}). Exam tip: do not ignore zeros.

The governing concept is decimal place value. In a number such as 2.305, the first digit to the right of the decimal point is in the tenths place, the second is in the hundredths place, and the third is in the thousandths place. Reading 2.305 from left to right after the decimal gives 3, 0, and 5. Thus 3 is in the tenths place, 0 is in the hundredths place, and 5 is in the thousandths place. Therefore option B is correct. The digit 2 is in the ones place, so option D is incorrect. The zero contributes no quantity by itself, but it is important as a place holder because it preserves the hundredths position and distinguishes 2.305 from numbers such as 2.35 or 2.035.

A fraction with denominator 10 represents tenths. The numerator 7 means seven tenths, and seven tenths is written as 0.7 in decimal notation. The zero before the decimal point shows that the value is less than one. Thus the correct choice is C, 0.7. The other choices do not represent seven tenths: 0.07 is seven hundredths, 7.10 is greater than seven, and 70.0 is seventy.

To convert directly, divide the numerator by the denominator: 7 ÷ 10 = 0.7. Equivalently, move the decimal point in 7 one place to the left because division by 10 reduces the value tenfold. Therefore, \(\frac{7}{10}=0.7\). The denominator tells us the place value: 10 gives one digit after the decimal point, so 0.7 is the required representation.

(0.6=0.60) because a zero at the end does not change the value. Exam tip: trailing zeros after decimals may be removed.