Decimal representation

The number of zeros between successive 1s increases as 1, 2, 3, 4, …, so no fixed block repeats. Its decimal expansion is non-terminating and non-recurring; hence it is irrational. Exam tip: a non-terminating decimal is rational only when it eventually repeats.

In the decimal 0.312312312…, the three-digit block 312 repeats continuously. Therefore, the bar must be placed over the complete repeating block: \(0.\overline{312}\). In \(0.3\overline{12}\), only 12 is treated as repeating, which does not match the given decimal pattern. Exam tip: Before using bar notation, identify the shortest block that repeats continuously.

Compare decimal numbers from left to right. Both numbers have 0 in the ones place, but in the tenths place, x has 6 while y has 0. Since 6>0, 0.608>0.068; hence, (x>y). The relation (x<y) would be true only if the first differing decimal digit of x were smaller. Exam tip: Compare ones, then tenths, hundredths, and so on.

First reduce the fraction before applying the decimal-expansion test. The greatest common divisor of 45 and 150 is 15, so 45/150 = 3/10. The denominator 10 has only the prime factors 2 and 5, which is the exact condition for a rational number in lowest terms to have a terminating decimal expansion. Indeed, 3/10 = 0.3, which ends after one decimal place. Therefore option A is correct. Looking only at the original denominator 150 can be misleading because it contains 3, but that factor disappears during simplification. Option B would apply if a factor other than 2 or 5 remained in the reduced denominator; C is impossible for a rational number here.

( \frac{1}{9}=0.111\ldots ), so (0.125) is the greatest. Convert fractions into decimals for comparison.

(400\times25=10000), so ( \frac{27}{400}=\frac{675}{10000}=0.0675 ). Convert the denominator into a power of (10) to find the decimal.

The tenths digits are equal and the hundredths digit (a) must be greater than (3). So the smallest value of (a) is (4).

The number of zeros keeps changing, so there is no fixed recurring part. Such a decimal is non-terminating non-recurring.

The governing concept is decimal representation of a rational number. To make the denominator a power of 10, multiply numerator and denominator by 5, because 200 × 5 = 1000. Thus, 17/200 = (17 × 5)/(200 × 5) = 85/1000 = 0.085. Therefore, option A is correct. The digits must be placed according to thousandths: 85 thousandths is 0.085, not 0.85. Option B would represent 17/1000, while option C is ten times too large. Option D is greater than 1, whereas 17/200 is clearly less than 1 because the numerator is smaller than the denominator. The terminating decimal occurs because the denominator factors only into 2s and 5s.

(400\times25=10000), so ( \frac{31}{400}=\frac{775}{10000}=0.0775 ). Making the denominator a power of (10) is useful.

(0.21875=\frac{21875}{100000}=\frac{7}{32}). Convert the decimal to a fraction and reduce it to simplest form.

The direct answer is option D: \(\frac{17}{16}\). To convert 1.0625 into a fraction, note that there are four digits after the decimal point, so write \(1.0625=\frac{10625}{10000}\). Now simplify. Both numerator and denominator are divisible by 625: \(10625\div625=17\) and \(10000\div625=16\). Thus the simplest fraction is \(\frac{17}{16}\), and it is improper because the numerator is greater than the denominator. Option A, \(\frac{16}{17}\), is the reciprocal and is less than 1, whereas 1.0625 is greater than 1. Option B, \(\frac{10625}{1000}\), uses the wrong denominator; four decimal places require 10,000, not 1,000, and it is not the simplified value. Option C, \(\frac98\), equals 1.125, not 1.0625. Option D is correct. Always count decimal places and simplify fully.

(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.

The direct answer is option A: 1.003 < 1.03 < 1.0303 < 1.300. Write all decimals with the same number of decimal places: 1.0030, 1.0300, 1.0303, and 1.3000. The whole parts are all 1, so compare the decimal parts from left to right. At the thousandths level, 1.0030 is smallest. Next, 1.0300 is smaller than 1.0303 because they agree until the fourth decimal place, where 0 is less than 3. Finally, 1.3000 is largest because its tenths digit is 3, while the others have 0. Option A gives exactly this order. Option B incorrectly places 1.03 before 1.003. Option C is descending rather than ascending. Option D incorrectly places 1.0303 before 1.03. Remember: append zeros to compare place values; never judge only by the number of written digits.

(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 \(\sqrt{2}\).