Decimal representation

In a decimal number, each digit has a value determined by its place. In 2.045, the digit 2 is in the units place, 0 is in the tenths place, 4 is in the hundredths place, and 5 is in the thousandths place. Hence the place-value expansion is 2 + 0/10 + 4/100 + 5/1000, which is equal to 2 + 4/100 + 5/1000. Therefore option A is correct. Option B incorrectly treats 4 as tenths and 5 as hundredths. Option C combines 4 and 5 as 45 hundredths, losing the fact that 4 is hundredths and 5 is thousandths. Option D changes the whole-number part from 2 to 20.

A is correct. Since \(0.0999\ldots=0.1\), \(0.4999\ldots=0.4+0.1=0.5\). Infinite 9s do not leave a gap below the next decimal. Exam tip: remember \(0.999\ldots=1\).

Write the numbers to three decimal places for comparison: \(0.56=0.560\) and \(0.57=0.570\). Since \(0.560<0.565<0.570\), \(0.565\) lies between them. \(0.570\) is equal to \(0.57\), so it is not strictly between the two numbers. Exam tip: When comparing decimals, add zeros at the end if needed to make the number of decimal places equal.

For a rational number in lowest terms, the decimal expansion terminates precisely when the denominator has no prime factors other than 2 and 5. Here 21/125 is already in lowest terms and 125 = 5³. Hence its decimal expansion terminates; in fact, 21/125 = 0.168. Therefore, option C is correct, whereas the first two choices require a different denominator structure.

Since 3.125 has three digits after the decimal point, it can be written as \(\frac{3125}{1000}\). Dividing the numerator and denominator by 125 gives \(\frac{3125}{1000}=\frac{25}{8}\), so option A is correct. Exam tip: use a denominator of 1 followed by as many zeros as there are decimal places, then simplify the fraction. Option C incorrectly uses 100 instead of 1000 as the denominator.

In 0.0808..., the complete block 08 repeats continuously: 08, 08, 08, ... . Therefore, the bar must be placed over both digits, giving \(0.\overline{08}\). In contrast, \(0.0\overline{8}\) means 0.0888..., which is a different decimal. Exam tip: first identify the smallest repeating block after the decimal point before placing the bar.

( \frac{29}{250}=\frac{116}{1000}=0.116 ). Multiply denominator (250) by (4) to make (1000).

In 4.006, 6 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 6 is \(6 \times \frac{1}{1000}=\frac{6}{1000}\). Option B, \(\frac{6}{100}\), is incorrect because it represents the hundredths place, which is the second decimal place. Exam tip: Count the digits after the decimal point and write that many zeros in the denominator after 1.

The number 0.00064 has five digits after the decimal point, so first write it as 64/100000. Now simplify by dividing numerator and denominator by their greatest common divisor. Since 64 = 8 × 8 and 100000 is divisible by 8, division by 8 gives 8/12500. The numerator 8 and denominator 12500 have no common factor greater than 1, so this is the simplified fraction. Therefore option B is correct. Option A is not equivalent because its denominator has only three zeros. Options C and D may look like reductions, but 4/6250 = 8/12500 and 16/2500 = 8/1250; neither is the requested simplest equivalent form, and D is not even equal to the original decimal.

For comparison, 0.37 can be written as 0.370 because adding zeros at the end of a decimal does not change its value. Comparing 0.370 and 0.307, the first decimal digit, 3, is the same, but at the next place 7 is greater than 0. Therefore, \(x>y\). The relation \(x=y\) is incorrect because 0.370 and 0.307 are not equal. Exam tip: First write decimals with the same number of decimal places before comparing them.

For \(\frac{7}{40}\), \(40=2^3\times5\). A reduced fraction terminates only when its denominator has 2 and/or 5. The other denominators contain 3. Exam tip: factor the denominator first.

After the decimal point, the digits occur as 312, 312, 312, ... repeatedly. Therefore, the smallest complete recurring block is 312. The sequences 31 and 12 are only parts of the pattern; they do not repeat as complete blocks. Exam tip: Group the digits after the decimal point to identify the shortest block that repeats.

Since 0.4375 has four digits after the decimal point, it can be written as \(\frac{4375}{10000}\). Dividing the numerator and denominator by 625 gives \(\frac{4375}{10000}=\frac{7}{16}\). Therefore, option A is correct. Exam tip: after converting a decimal to a fraction, divide the numerator and denominator by their greatest common factor to obtain the simplified form.

(0.025\times100=2.5), so it gives (2.5%). To convert a decimal into a percentage, multiply by (100).

The governing concept is place value when multiplying by a power of ten. Since 100=10², multiplication by 100 shifts the decimal point two places to the right. Starting with 5.08, the first shift gives 50.8 and the second gives 508; therefore 5.08×100=508. Option C is correct. Option A is smaller and corresponds to moving the decimal point left, as in division by 10. Option B represents multiplication by only 10, because it shifts the point one place. Option D shifts it three places and would correspond to multiplication by 1000. The same result can be checked by writing 5.08=508/100, so (508/100)×100=508.

The governing concept is decimal place value under division by a power of ten. Because 1000=10³, dividing by 1000 moves the decimal point three places to the left. Starting from 37.5, the first shift gives 3.75, the second gives 0.375, and the third gives 0.0375. Hence 37.5÷1000=0.0375, so option A is correct. Option B represents division by 100, involving only two leftward shifts. Option C represents division by 10, involving one shift. Option D moves the decimal point in the opposite direction and is larger than the original positive number; division by 1000 cannot produce such a result. The calculation can also be checked as 37.5/1000=375/10000=0.0375.

Write 0.48 as 0.480 and align the decimal points: \(0.480+0.075=0.555\). Therefore, the correct answer is 0.555. An option such as 0.5075 results from adding digits without correctly using place value. Exam tip: align decimal points first, and add trailing zeros where needed.

Align the decimal points and write 3.2 as 3.200. Then \(3.200-0.875=2.325\), so 2.325 is correct. A result such as 2.425 can arise from an error while borrowing during subtraction. Exam tip: always place decimal points directly below each other before subtracting decimals.

Zeros at the end of the decimal part do not change a number’s value. Therefore, removing the final zero from 9.090 gives 9.09. In 9.009, the zero is within the decimal part, so its value is different. Exam tip: Only trailing zeros in a decimal can be removed without changing its value.

(0.1520) is greater than (0.1500), (0.1250), and (0.1205). Write all numbers up to equal decimal places for comparison.