Decimal representation

In \(7.003\), the part to the right of the decimal point is \(.003\). Therefore, the decimal part is \(.003\). \(7\) is the integer part, whereas \(003\) are the digits after the decimal point. Exam tip: To identify the decimal part, look to the right of the decimal point and write its value with the decimal point.

A decimal number has a whole-number part to the left of the decimal point and a fractional part to the right. In 12.45, the decimal point separates 12 from 45. Therefore 12 is the whole-number part, and 45 represents forty-five hundredths, or 0.45. Hence choice C is correct.

This can also be understood by place value. The digit 1 is in the tens place and 2 is in the ones place, so together they form the whole number 12. The digits 4 and 5 lie after the decimal point and form the fractional part. The number 1245 would ignore the decimal point, while 45 alone is only the digits after it. Thus the complete number is 12+0.45, and its whole-number part is 12.

Write 0.09 as 0.090 to compare the decimals. In 0.091, the thousandths digit is 1, whereas in 0.090 it is 0. Therefore, 0.091 is greater than 0.09. Note that 0.090 is equal to 0.09 because a zero added at the end of a decimal does not change its value. Exam tip: Add zeros to make the number of decimal places equal before comparing decimals.

The governing concept is comparison of decimal numbers using place value and the number line. Rewrite the endpoints with equal decimal places: 0.3 = 0.30 and 0.4 = 0.40. A number lies between them when it is greater than 0.30 and less than 0.40. The number 0.35 satisfies 0.30 < 0.35 < 0.40, so option C is correct. The value 0.25 is less than 0.30, while 0.45 and 0.50 are greater than 0.40. On a number line, 0.35 is located halfway between 0.3 and 0.4, which gives a visual confirmation. Equalizing decimal places avoids the common mistake of comparing digits without considering their place values.

The direct answer is B: 60%. A percentage means a quantity out of 100. To change a decimal into a percentage, multiply it by 100 and attach the percent sign: \(0.6\times100=60\), so \(0.6=60\%\). Equivalently, move the decimal point two places to the right: 0.6 becomes 60. Option A, 6%, would equal 0.06, not 0.6. Option B is correct because 60 out of 100 is exactly 0.6. Option C, 0.6%, equals 0.006 and is much smaller. Option D, 600%, equals 6 and is too large. A common mistake is moving the decimal in the wrong direction. For decimal to percent, move right two places.

Writing 1.25 as hundredths gives \(1.25=\frac{125}{100}\). Dividing the numerator and denominator by 25 simplifies this to \(\frac{5}{4}\), so option A is correct. Option B equals 12.5, option C equals 0.25, and option D equals 0.8. Exam tip: when a decimal has two digits after the decimal point, first write it over 100 and then reduce the fraction.

The decimal 0.25 ends after two decimal places, so it is a terminating decimal. Hence, it is not a non-terminating recurring decimal. In contrast, 0.121212\ldots, 0.777\ldots, and 3.454545\ldots repeat a digit or block of digits indefinitely. Exam tip: A decimal with \ldots and a repeating fixed pattern is non-terminating recurring.

To the right of the decimal point, the places are tenths, hundredths, thousandths, and ten-thousandths in order. In 0.0005, 5 is the fourth digit after the decimal point, so it is in the ten-thousandths place. The thousandths place is the third place after the decimal point. Exam tip: Count the digits after the decimal point to identify the place value.

For \(\frac{7}{12}\), \(12=2^2\times3\). Since 3 occurs in the denominator, its decimal form is \(0.58\overline{3}\), which is non-terminating recurring. The other denominators contain only 2 and 5. Exam tip: first reduce the fraction to lowest terms.

Dividing \(7\) by \(12\) gives \(0.58333\ldots=0.58\overline{3}\); since 3 repeats, the decimal is non-terminating recurring. \(0.58\) is only a truncated value. Exam tip: a repeated remainder produces repeating digits.

The correct sum is \(9.99+0.01=10.00\). Here, 9 hundredths plus 1 hundredth makes 10 hundredths, producing a carry of 1 and giving 10. Option \(9.991\) is incorrect because adding \(0.01\) does not add a digit in the thousandths place. Exam tip: While adding decimals, align the decimal points so that digits with the same place value are added together.

( \frac{1}{16}=\frac{625}{10000}=0.0625 ). Try to convert the denominator into (10), (100), (1000), or (10000).

For \(\frac{7}{24}\), \(24=2^3\times3\). A reduced denominator containing a prime other than 2 or 5 gives a non-terminating recurring decimal. \(\frac{3}{40}\) terminates. Exam tip: factor the denominator.

In 2.07, 2 is in the ones place and 7 is in the hundredths place. Therefore, its expanded form is \(2+\frac{7}{100}\). Option A places 7 in the tenths place, while option D equals 2.70 and option C equals 2.007. Exam tip: the first digit after the decimal point represents tenths, and the second represents hundredths.

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.