Decimal representation

(3) is in tenths, (4) is in thousandths, and (2) is in ten-thousandths. A zero place need not be written in expansion.

After the decimal point, the places are tenths, hundredths, thousandths, ten-thousandths and hundred-thousandths. In 0.50709, 9 is in the fifth place, so its place value is \(\frac{9}{100000}\). Option A represents a thousandths-place value, so it is not correct. Exam tip: the fifth digit after the decimal point has a denominator of \(10^5\).

After the first (23), the block (13) repeats again and again. The recurring part is the smallest repeating block.

In 3.0454545..., the first digit after the decimal point is 0 and it occurs only once. The digits 45 then repeat indefinitely: 3.0 45 45 45... Therefore, the bar must cover only 45, giving 3.0\overline{45}. Option A is correct. Options B, C, and D place the bar over digits that do not represent the actual repeating block.

In lowest terms, a denominator made only of 2s and 5s gives a terminating decimal. For example, \(40=2^3\times5\), so a fraction with denominator 40 terminates. Exam tip: always reduce the fraction first.

A rational number has a terminating decimal expansion when, after it is written in lowest terms, the denominator has no prime factors other than 2 and 5. These are the factors that can be matched with powers of 10, because every power of 10 is made from 2 and 5. Therefore, the important step is to inspect the denominator after simplification, not merely to look at the numerator or the written fraction.

Here, \(320=2^6\times5\), and 27 and 320 have no common factor, so the fraction is already in simplest form. Its denominator contains only 2 and 5. In fact, multiplying numerator and denominator by 5 gives \(\frac{27}{320}=\frac{135}{1600}=0.084375\), which ends. Hence option C, terminating, follows. It is not recurring because no prime factor other than 2 or 5 remains in the denominator.

(0.6250) is greater than (0.6205), (0.6025), and (0.0625). Write all decimals up to equal places for comparison.

(1.03=1.0300), so (1.0303) is slightly greater. Arrange decimals after writing equal places.

(0.7185) is greater than (0.7180) and less than (0.7190). Make decimal places equal while finding a number between two decimals.

The key concept is converting a terminating decimal into a fraction and then reducing it. The decimal 0.00072 has five digits after the decimal point, so its initial fraction is 72/100000. Now simplify by dividing numerator and denominator by their greatest common divisor, 8: 72 ÷ 8 = 9 and 100000 ÷ 8 = 12500. Hence 0.00072 = 9/12500, so option C is correct. Option A uses an incorrect denominator and does not represent the given place value. Option B is equivalent to 9/12500 but is not in simplest form because both terms are divisible by 2. Option D also uses the wrong power of 10. The final fraction has no common factor remaining.

Since 0.875 has three digits after the decimal point, it can be written as \(\frac{875}{1000}\). Dividing the numerator and denominator by 125 gives \(\frac{875}{1000}=\frac{7}{8}\). Therefore, option A is correct. Option B is incorrect because \(\frac{5}{8}=0.625\), not 0.875. Exam tip: For a terminating decimal, use a denominator of \(10^n\), where n is the number of decimal places, and then simplify the fraction.

A rational number has a terminating decimal when its denominator in lowest form has only 2 and/or 5 as prime factors. Here \(40=2^3\times5\), so \(\frac{13}{40}=0.325\). The denominator need not be a power of 10. Exam tip: factorise the denominator first.

In \(0.272727\ldots\), the block 27 repeats. If \(x=0.272727\ldots\), then \(100x-x=27\), giving \(x=\frac{3}{11}\). Hence it is non-terminating but rational. Exam tip: repeating decimals are rational, unlike non-repeating ones such as \(\sqrt{2}\).

In the decimal, the three-digit block 006 repeats continuously: 006, 006, 006, \(\ldots\). Therefore, the bar must be placed over the complete repeating block 006, giving \(0.\overline{006}\). Option A incorrectly treats only 06 as repeating, which does not reproduce the given sequence of digits. Exam tip: Before using bar notation, identify the shortest block of digits that repeats continuously.

The governing concept is the identification of the repeating block in a recurring decimal. Write the digits after the decimal point in order: 1, 0, 2, 0, 2, 0, 2, and so on. The first digit 1 occurs only once, so it is the non-repeating or initial part. After that, the pair 02 repeats: 5.1 02 02 02... Therefore, the recurring part is 02 and option B is correct. Option C, 20, reverses the order of the repeating digits and would describe a different decimal. Option A is only the initial pair and does not continue periodically. Option D includes the initial non-repeating digit, so it is not the repeating block itself. Sequence order is essential here.

It does not terminate and has no fixed recurring part. Such a decimal is called non-terminating non-recurring.

The denominator has (7), a factor other than (2) and (5). So the rational number will have a non-terminating recurring decimal.

In \(\frac{5}{21}\), the denominator is \(21=3\times7\). A reduced denominator containing a prime other than 2 or 5 gives a non-terminating recurring decimal. Exam tip: factorise the denominator first.

A rational number terminates only when, in lowest form, its denominator has prime factors 2 and/or 5 only. \(\frac{2}{11}=0.1818\ldots\) is recurring. Exam tip: factorise the denominator to test the decimal form.

The governing concept is comparison of rational numbers by expressing them in a common form. Convert the fraction to a decimal: 9/16 = 0.5625, because 16 × 0.5625 = 9. Write 0.57 as 0.5700 so that both values have the same number of decimal places. Comparing 0.5625 and 0.5700 shows that 0.5700 is larger; equivalently, 5700 ten-thousandths are greater than 5625 ten-thousandths. Therefore, 0.57 is greater and option B is correct. Option A is smaller, option C is false because the two values differ, and option D is incorrect because fractions and decimals can always be compared after conversion to equivalent forms.