Decimal representation

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.

A non-terminating decimal without fixed repetition can be irrational. Terminating and recurring decimals are rational.

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.

A rational number has a terminating decimal only when its reduced denominator contains only 2 and/or 5. Since 3 is present, the decimal repeats endlessly. Exam tip: always reduce the fraction first.

( \frac{35}{42}=\frac{5}{6} ) and (6=2\times3). Since the denominator has (3), the decimal will be non-terminating recurring.