Decimal representation

The governing concept is comparison of decimal numbers by place value. To compare accurately, write the endpoints with the same number of decimal places: 1.2 = 1.20 and 1.3 = 1.30. Now test each option. The number 1.25 has the same whole-number part, 1, and its hundredths value places it after 1.20 but before 1.30; therefore 1.20 < 1.25 < 1.30. Option C is correct. The value 1.19 is below 1.20, 1.31 is above 1.30, and 1.03 is also below 1.20. The phrase strictly between excludes the endpoints themselves, although none of the listed options equals an endpoint. This comparison relies on digit positions, not on the number of written digits.

There are four digits after the decimal point in 0.0045, so it can be written as \(\frac{45}{10000}\). Dividing the numerator and denominator by 5 gives \(\frac{45}{10000}=\frac{9}{2000}\). Therefore, option B is correct. In an exam, count the digits after the decimal point to determine the denominator as the appropriate power of 10.

A recurring decimal contains a block of digits that continues to repeat in the same order indefinitely. In the decimal 0.1252525..., the first digit after the decimal point is 1. After that initial non-repeating digit, the digits are 25, 25, 25, and so on. Thus, the repeating block must be identified from the continuous repetition, not by taking every digit from the beginning.

The decimal can be viewed as 0.1 followed by the recurring sequence 25. Since 25 reappears without interruption, the recurring part is 25, which is option C. The digit 1 is only a preliminary, non-repeating digit. The choice 125 is also not correct because it does not repeat as one complete block in the displayed decimal.

(21=3\times7) has factors other than (2) and (5). So this rational number gives a non-terminating recurring decimal.

In 18.305, 5 is in the third place to the right of the decimal point. The places after the decimal are tenths, hundredths and thousandths, so the place value of 5 is \(5 \times \frac{1}{1000}=\frac{5}{1000}\). Therefore, option C is correct. Exam tip: distinguish the digit 5 from its place value.

The governing concept is ordering decimals by equalizing their decimal places. Rewrite the relevant numbers as 0.5050, 0.5500, and 0.5005. All have the same whole-number part, 0, so compare digits from left to right after the decimal point. Their tenths digits are equal at 5; at the hundredths place, 0.5005 has 0, while 0.5050 has 0 and 0.5500 has 5. Comparing the next digits gives 0.5005 < 0.5050 < 0.5500. Thus option C is correct. Option A is larger by 0.0045, and option B is much larger because its hundredths digit is 5. Option D is not among the original three and is also larger than all of them. Trailing zeros do not change a decimal's value; they only make comparison clearer.

\(0.\overline{27}\) is recurring but rational. If \(x=0.\overline{27}\), then \(100x-x=27\), so \(99x=27\) and \(x=\frac{3}{11}\). Exam tip: every repeating decimal represents a rational number.

In lowest terms, a decimal terminates only when the denominator has prime factors 2 and/or 5. A factor such as 3 causes repetition. Exam tip: reduce the fraction before checking its denominator.

The governing concept is conversion of a mixed number into decimal notation. A mixed number consists of a whole part and a fractional part, so convert 3/4 first and then add the whole number 1. Since 3 ÷ 4 = 0.75, we get 1 + 0.75 = 1.75. Therefore option B is correct. The value 0.75 represents only the fractional part and leaves out the whole unit. The number 1.25 would correspond to 1 1/4, not 1 3/4. The notation 1.34 incorrectly treats the numerator and denominator as adjacent decimal digits rather than performing division. This also illustrates that a fraction with denominator 4 has a terminating decimal because 4 can be converted to 100 by multiplying numerator and denominator by 25: 3/4 = 75/100 = 0.75.

Since 0.06 has two digits after the decimal point, it is first written as \(\frac{6}{100}\). The greatest common divisor of 6 and 100 is 2, so \(\frac{6}{100}=\frac{3}{50}\). Therefore, option C is correct. Remember that \(\frac{6}{10}\) equals 0.6, not 0.06.

A terminating decimal can be written as a fraction by using a denominator of 10, 100, 1000, and so on, according to the number of decimal places. Since 0.48 has two digits after the decimal point, it is first written as 48 over 100. The resulting fraction must then be reduced by dividing its numerator and denominator by their common factor, so that the answer is in simplest form.

Thus, \\(0.48=\\frac{48}{100}\\). Both 48 and 100 are divisible by 4, giving \\(\\frac{48\\div4}{100\\div4}=\\frac{12}{25}\\). Therefore, option A is correct. The fraction \\(48/10\\) has the wrong denominator, while \\(4/8\\) and \\(24/100\\) are not the simplest form of 0.48.

(2.125=\frac{2125}{1000}=\frac{17}{8}). A mixed decimal can also be converted into an improper fraction.

After the decimal point, the first place is tenths, the second is hundredths, and the third is thousandths. In 0.009, 9 is the third digit after the decimal point, so it is in the thousandths place. The hundredths digit in 0.009 is 0, not 9. Exam tip: Count decimal places in order: tenths, hundredths, then thousandths.

(4) is in the first place after the decimal point, so its place value is ( \frac{4}{10} ). The first decimal place is tenths.

In 8.032, the digit 3 is in the second place to the right of the decimal point. This is the hundredths place, so its place value is \(3 \times \frac{1}{100}=\frac{3}{100}\). Remember that 3 is the face value, whereas \(\frac{3}{100}\) is its place value.

After the decimal point, the places represent tenths, hundredths, thousandths and ten-thousandths. In 0.0008, 8 is in the fourth place after the decimal point, so its place value is \(\frac{8}{10000}\). Exam tip: Count decimal places from immediately after the decimal point.

In 4.73, 4 is in the ones place, 7 is in the tenths place, and 3 is in the hundredths place. Therefore, its expanded form is \(4+\frac{7}{10}+\frac{3}{100}\). In option B, the place values of 7 and 3 are interchanged. Exam tip: the first digit after the decimal has denominator 10, and the second has denominator 100.

In 0.508, 5 is in the tenths place, 0 is in the hundredths place, and 8 is in the thousandths place. Therefore, its expanded form is \(\frac{5}{10}+\frac{0}{100}+\frac{8}{1000}\). Option A omits the tenths term, while option C incorrectly places 8 in the hundredths place. Exam tip: the first, second, and third digits after the decimal point represent tenths, hundredths, and thousandths, respectively.

Adding a zero to the right end of 0.08 does not change its value, so 0.08 = 0.080. In 0.008, the digit 8 is in the third decimal place, so its value is different. Exam tip: Zeros added only at the end of a decimal do not change its value.

Zeros at the end of the decimal part do not change the value of a number. Thus, 6.500 = 6.5, so option B is correct. In 6.05 and 6.005, the zeros are in different places, so their values are not equal to 6.500. Exam tip: Only trailing zeros in a decimal can be added or removed without changing its value.