Decimal representation

It has no fixed repeating pattern and it does not terminate. Such a decimal is non-terminating non-recurring.

A terminating decimal has a finite number of digits after the decimal point. A rational number can have terminating or non-terminating recurring decimal expansion.

\(0.272727\ldots\) is recurring, and \(0.272727\ldots=\frac{27}{99}=\frac{3}{11}\), so it is rational. \(\sqrt{2}\) and \(\pi\) are irrational. Exam tip: every recurring decimal represents a rational number.

Dividing 2 by 3 gives 0.666…, where the digit 6 repeats indefinitely. Therefore, \(\frac{2}{3}=0.666\ldots\). Option B is incorrect because 0.6 is only a one-decimal-place approximation, not the exact value. Exam tip: Remember that \(\frac{1}{3}=0.333\ldots\); doubling it gives \(\frac{2}{3}=0.666\ldots\).

Direct answer: Option B, 0.625. A fraction means division, so calculate \(5\div8\). A convenient method is to make the denominator 1000: multiply numerator and denominator by 125, because \(8\times125=1000\). Thus \(\frac58=\frac{5\times125}{8\times125}=\frac{625}{1000}=0.625\). Equivalently, long division gives 5.000 ÷ 8: 8 goes into 50 six times, leaving 2; bring down 0 to get 20, which gives 2, leaving 4; bring down 0 to get 40, which gives 5, leaving 0. Option B is correct. Option A, 0.58, incorrectly joins the digits 5 and 8 instead of dividing. Option C, 0.875, equals \(7/8\), not \(5/8\). Option D, 0.525, is another unrelated decimal and does not equal the fraction. Since the denominator 8 divides 1000 exactly after multiplication, the decimal terminates. Memory cue: \(1/8=0.125\), so \(5/8=5\times0.125=0.625\).

(8) is the second digit after the decimal point, so it is in the hundredths place. Count digits after the decimal in order to identify place value.

The correct answer is 0.50 because adding a zero at the right end of a decimal does not change its value. Hence, 0.5 = 0.50. Option 0.05 is one-tenth of 0.5, so it is not equal to it. Exam tip: Zeros added or removed at the end of a decimal do not change its value.

To compare decimals, compare digits from left to right, giving equal place value to each number by adding zeros when necessary. Write the numbers as 0.7500, 0.7050, 0.5700, and 0.7501. The first three digits of 0.7500 and 0.7501 are equal, but at the fourth decimal place, 1 is greater than 0. Therefore 0.7501 is greater than 0.7500, so choice D is correct.

Notice that 0.75 and 0.7500 have exactly the same value; adding zeros at the end does not change a decimal. However, 0.7501 has an additional positive amount, one ten-thousandth, so it is slightly larger. The number 0.705 is smaller because its hundredths digit is 0, and 0.57 is smaller still because its tenths digit is 5 rather than 7.

The direct answer is A: 2.09. To compare decimals, first compare the whole-number parts. All numbers have whole part 2, so compare the digits after the decimal from left to right. Write them with equal places: 2.090, 2.900, 2.099, and 2.190. At the tenths place, 2.09 has 0, while 2.9, 2.099, and 2.19 have 9, 0, and 1 respectively; more carefully, 2.090 is less than 2.099 because the hundredths are both 9 and the thousandths 0 is less than 9. Thus A is smallest. B is 2.900, clearly larger. C is 2.099, slightly larger than 2.090. D is 2.190, also larger. A missing decimal place may be filled with zero, but zero must be placed in the correct position.

\(0.\overline{3}\) is non-terminating but recurring, and \(0.\overline{3}=\frac{1}{3}\); hence it is rational. \(\sqrt{2}\) and \(\pi\) are non-terminating, non-recurring irrational numbers. Exam tip: every recurring decimal is rational.

In (0.04), (4) is in the hundredths place, so it is ( \frac{4}{100} ). For two decimal places, use denominator (100).

In 2.35, 2 is the whole-number part and 35 represents thirty-five hundredths, so the mixed fraction is \(2+\frac{35}{100}=2\frac{35}{100}\). Option B has denominator 10 and gives 5.5, while option C reverses the whole-number and fractional parts. Exam tip: when there are two digits after the decimal point, write the decimal part over 100.

(7) is in the third place after the decimal, so it is in the thousandths place. Zeros are also important while counting places.

(25=5^2), so the denominator has only the factor (5). Such a fraction has a terminating decimal expansion.

The direct answer is option A: terminating decimal. First reduce the fraction if necessary. The fraction 3/40 is already in simplest form because 3 and 40 have no common factor greater than 1. Now factor the denominator: 40=2^3×5. A rational number in simplest form has a terminating decimal expansion exactly when its denominator has no prime factors other than 2 and 5. Here the denominator contains only 2 and 5, so the decimal terminates. We can also calculate it: 3/40=3×25/(40×25)=75/1000=0.075. Thus the decimal ends after three decimal places. Option A is correct. Option B, non-terminating recurring, is wrong because that type occurs when the simplified denominator contains another prime factor, such as 3 or 7; for example 1/3=0.333... . Option C, non-terminating non-recurring, is wrong because such decimals represent irrational numbers, while every fraction of integers is rational and has either a terminating or recurring decimal. Option D, integer only, is wrong because 3/40 is less than 1 and equals 0.075, not a whole integer. The exam rule is: simplify first, factor the denominator, and check whether only 2s and 5s remain. If yes, the decimal terminates.

(6=2\times3) includes (3), so the decimal will not terminate. Since it is rational, it will be non-terminating recurring.

(7) is neither (2) nor (5), so the decimal will not terminate. Since it is rational, it will be non-terminating recurring.

Since 0.375 has three digits after the decimal point, it can be written as \(\frac{375}{1000}\). Dividing the numerator and denominator by 125 gives \(\frac{375}{1000}=\frac{3}{8}\), which is the simplest form. In option B, the decimal digits have been handled incorrectly, while options C and D are not equal to 0.375. Exam tip: use a denominator of 1 followed by as many zeros as there are decimal places, then reduce the fraction.

The governing concept is place value in decimal representation. A denominator of 100 means that the numerator is measured in hundredths, so two digits must be placed to the right of the decimal point. Therefore 11/100 = 0.11. Option B is correct. The value 1.1 is eleven tenths, or 110/100, so it is ten times too large. The value 0.011 represents eleven thousandths, or 11/1000, so it is ten times too small. The number 11.00 is simply 11, which is much larger than 0.11. Another useful check is multiplication: 0.11 × 100 = 11, confirming the fraction exactly. Zeros may be added at the left when needed, but changing the number of decimal places changes the denominator and therefore the value.

(2) is in the first place after the decimal, so its place value is ( \frac{2}{10} ). The first decimal place is called tenths.

A zero at the extreme right of a decimal does not change its value. Hence, \(5.060=5.06\). In 5.006, the digit 6 is in the thousandths place, whereas in 5.060 it is in the hundredths place. Exam tip: Only trailing zeros may be removed; zeros between digits can affect place value.

Direct answer: Option B, 0.26. To convert \(\frac{13}{50}\) into a decimal, make the denominator 100. Multiply numerator and denominator by 2: \(\frac{13}{50}=\frac{13\times2}{50\times2}=\frac{26}{100}=0.26\). The hundredths form directly shows 26 hundredths, or 0.26. Option B is correct. Option A, 0.13, would be \(13/100\), not \(13/50\); it forgets to double the numerator when changing 50 to 100. Option C, 0.52, is twice the correct value and would correspond to \(26/50\), not \(13/50\). Option D, 1.35, is greater than 1, whereas \(13/50\) is less than 1 because the numerator is smaller than the denominator; it is therefore immediately unreasonable. As a check, 50 × 0.26 = 13, confirming the fraction. Exam cue: when the denominator is 50, multiply top and bottom by 2 to obtain denominator 100.

In 0.0909..., the two-digit block 09 repeats continuously. Therefore, the bar must be placed over the complete repeating block: \(0.\overline{09}\). Option \(0.0\overline{9}\) shows only 9 as recurring, so it does not represent the given repeating pattern correctly. Exam tip: First identify the shortest block of digits that repeats.

An irrational number has a non-terminating non-recurring decimal expansion. A non-terminating decimal without fixed repetition can be irrational.

The direct answer is C: 3 digits. In 0.875, the decimal point separates the whole-number part from the fractional part. Read the digits to its right in order: 8 is the first, 7 is the second, and 5 is the third. Therefore there are three digits after the decimal point. Option A says 1, but it counts only 8 and misses 7 and 5. Option B says 2, but it misses one digit. Option C counts 8, 7, and 5 correctly. Option D says 4, but there is no fourth digit shown. The rule is to count every written digit after the decimal point, including zeros when they appear, such as the three places in 0.205.