Decimal representation

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).

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.

To identify a recurring part, inspect the digits after the decimal point and separate any non-repeating prefix from the block that continues indefinitely. The decimal is 1.2303030..., so after the initial digits 23, the digits 03 repeat: 1.23 03 03 03... . Therefore the recurring block is 03, making option C correct. Option A includes the non-repeating prefix and is not the repeating cycle. Option B is incomplete because the repeated two-digit block begins with 0; omitting that zero changes the pattern. Option D incorrectly treats the initial 230 as one repeating block, whereas the displayed continuation confirms that only 03 is repeated.

In \(0.272727\ldots\), the block 27 repeats, so it is recurring and \(27/99=3/11\); hence it is rational. \(\sqrt{2}\), \(\pi\), and \(0.1010010001\ldots\) are non-recurring. Exam tip: every recurring decimal is rational.

All numbers in the first option end after a finite number of digits. A terminating decimal has finite digits.

The direct answer is C, which can represent an irrational number: 2.01001000100001... without a fixed repeating pattern. A real number is rational if its decimal expansion terminates or eventually repeats a fixed block. A decimal that continues forever without any fixed repetition can be irrational. Option A, 0.625, terminates, and equals \(625/1000=5/8\), so it is rational. Option B, \(0.\overline{54}\), repeats the block 54 and is rational; every repeating decimal can be written as a fraction. Option C has continuing digits and no fixed repeating block, so it can be irrational. Option D, 5.75, terminates and equals \(575/100=23/4\), so it is rational. The word “can” matters: non-terminating alone is not enough; the decimal must also lack eventual repetition.

The zeros between successive 1s number 1, 2, 3, 4, …, so no fixed repeating block exists. An infinite non-repeating decimal is irrational. Exam tip: check for a repeating cycle, not just the digits used.

The denominator has (3), a factor other than (2) and (5). So the rational number will have a non-terminating recurring decimal.

In the given decimal, the bar is only over 7. Therefore, after 2, the digit 7 repeats endlessly: \(0.2\overline{7}=0.2777\ldots\). In option B, the entire block 27 repeats, while option D is a terminating decimal. Exam tip: Repeat only the digit or group of digits covered by the bar.

To find how many times 0.004 is 0.0004, divide the larger number by the smaller number: \(0.004 \div 0.0004 = 10\). Therefore, 0.004 is 10 times 0.0004, so option B is correct. Option C is incorrect because the quotient is 10, not 100. Exam tip: For “how many times” questions, divide the given quantity by the reference quantity.

(32\times3125=100000), so ( \frac{3}{32}=\frac{9375}{100000}=0.09375 ). Converting the denominator into a power of (10) is useful.

In 7.5408, the first digit after the decimal point, 5, is in the tenths place, and the second digit, 4, is in the hundredths place. Therefore, the place value of 4 is \(\frac{4}{100}\), or 0.04. Option A represents the tenths place, so it is incorrect. Exam tip: After the decimal point, the place-value sequence is tenths, hundredths, thousandths and ten-thousandths.

In 0.09009, count the places after the decimal point from left to right. The digits are 0 in the tenths place, 9 in the hundredths place, 0 in the thousandths place, 0 in the ten-thousandths place, and 9 in the hundred-thousandths place. Thus the first 9 is in the second decimal place, so its place value is \(\frac{9}{100}\). Choice B is correct.

The leading zero after the decimal does not make the first 9 a tenths digit. It occupies the tenths place, pushing the first 9 to the hundredths place. Therefore the number can be expanded as \(0+\frac{0}{10}+\frac{9}{100}+\frac{0}{1000}+\frac{0}{10000}+\frac{9}{100000}\). The later 9 has a different place value, so it must not be confused with the first one.

Convert the decimal part into a fraction: \(0.375=\frac{375}{1000}=\frac{3}{8}\). Therefore, \(2.375=2+\frac{3}{8}=2\frac{3}{8}\), so option A is correct. Option B represents \(2.625\), while option C represents \(2.875\); neither equals the given decimal. Exam tip: for three digits after the decimal point, first write the decimal part over \(1000\) and then simplify.

In ascending order, the smallest number comes first. Here, 0.2 = 2/10, 0.02 = 2/100, and 0.002 = 2/1000. With the same numerator 2, the fraction with denominator 1000 is the smallest, so 0.002 comes first. Although 0.02 is the closest option, it is 10 times greater than 0.002. Exam tip: for decimals with the same non-zero digit, shifting that digit further right makes the number smaller.

There are five digits after the decimal point, so \(0.00125=\frac{125}{100000}\). Dividing the numerator and denominator by 125 gives \(\frac{125}{100000}=\frac{1}{800}\), so option A is correct. As an exam tip, use a denominator of 1 followed by as many zeros as the number of decimal places, then reduce the fraction. Option D equals \(\frac{125}{1000}=0.125\), not 0.00125.

In \(\frac{1}{6}\), the denominator is \(6=2\times3\). Because it contains 3, its decimal is \(0.1\overline{6}\), which is non-terminating recurring. In contrast, \(\frac{3}{8}\) has only powers of 2 in its denominator. Exam tip: simplify the fraction first.

(0.695) is greater than (0.690) and less than (0.700). For a number between two limits, write both limits up to equal decimal places.

(0.0125) is smaller than (0.1250) because the place of digits changes. Only adding or removing zeros at the end does not change value.

After the decimal point, the digits occur as \(505\,505\,505\ldots\). Therefore, the recurring block is \(505\). If the block were \(50\), the decimal would be \(4.505050\ldots\), which is different from the given number. Exam tip: identify the smallest group of digits that repeats continuously without changing.