Decimal representation

In the decimal, the three-digit block 006 repeats continuously: 006, 006, 006, \(\ldots\). Therefore, the bar must be placed over the complete repeating block 006, giving \(0.\overline{006}\). Option A incorrectly treats only 06 as repeating, which does not reproduce the given sequence of digits. Exam tip: Before using bar notation, identify the shortest block of digits that repeats continuously.

The governing concept is the identification of the repeating block in a recurring decimal. Write the digits after the decimal point in order: 1, 0, 2, 0, 2, 0, 2, and so on. The first digit 1 occurs only once, so it is the non-repeating or initial part. After that, the pair 02 repeats: 5.1 02 02 02... Therefore, the recurring part is 02 and option B is correct. Option C, 20, reverses the order of the repeating digits and would describe a different decimal. Option A is only the initial pair and does not continue periodically. Option D includes the initial non-repeating digit, so it is not the repeating block itself. Sequence order is essential here.

It does not terminate and has no fixed recurring part. Such a decimal is called non-terminating non-recurring.

The denominator has (7), a factor other than (2) and (5). So the rational number will have a non-terminating recurring decimal.

In \(\frac{5}{21}\), the denominator is \(21=3\times7\). A reduced denominator containing a prime other than 2 or 5 gives a non-terminating recurring decimal. Exam tip: factorise the denominator first.

A rational number terminates only when, in lowest form, its denominator has prime factors 2 and/or 5 only. \(\frac{2}{11}=0.1818\ldots\) is recurring. Exam tip: factorise the denominator to test the decimal form.

The governing concept is comparison of rational numbers by expressing them in a common form. Convert the fraction to a decimal: 9/16 = 0.5625, because 16 × 0.5625 = 9. Write 0.57 as 0.5700 so that both values have the same number of decimal places. Comparing 0.5625 and 0.5700 shows that 0.5700 is larger; equivalently, 5700 ten-thousandths are greater than 5625 ten-thousandths. Therefore, 0.57 is greater and option B is correct. Option A is smaller, option C is false because the two values differ, and option D is incorrect because fractions and decimals can always be compared after conversion to equivalent forms.

To convert a decimal into a percentage, multiply it by 100. Here, \(0.075\times100=7.5\), so the correct answer is \(7.5\%\). In option B, the decimal point has been shifted incorrectly. Exam tip: When converting a decimal to a percentage, move the decimal point two places to the right.

The governing rule is place-value movement when dividing a decimal by a power of 10. Since 1000 = 10^3, division by 1000 moves the decimal point three places to the left. Starting with 12.06, move it once to 1.206, twice to 0.1206, and three times to 0.01206. Thus, 12.06 ÷ 1000 = 0.01206, making option C correct. Option A shows only one place of movement, while option B shows two places. Option D moves the decimal in the wrong direction and represents multiplication by 1000 rather than division. Zeros must be retained as placeholders so that the place values remain correct.

Multiplying both numbers by (1000) gives (480\div6=80). Multiplying both parts of a division by the same number does not change the quotient.

Align the decimal points and write 2.75 as 2.7500. Then \(2.7500+0.0875=2.8375\), so 2.8375 is correct. The value 2.7625 results from incorrect addition of decimal-place digits. Exam tip: Add trailing zeros first, when needed, so that the place values are aligned.

The direct answer is option B: 3.0625. Align the decimal points and add a zero to the shorter decimal: \(4.125=4.1250\). Now subtract place by place: \(4.1250-1.0625=3.0625\). Checking by parts gives \(4-1=3\) and \(0.1250-0.0625=0.0625\), so the result is 3.0625. Option A, 3.1875, results from an incorrect subtraction or borrowing step. Option B is correct because the aligned subtraction gives exactly this value. Option C, 3.1250, ignores the extra difference of 0.0625. Option D, 2.9375, is too small and could come from subtracting the fractional parts incorrectly. A useful check is that 1.0625 is a little more than 1, so subtracting it from 4.125 should give a little less than 3.125; 3.0625 fits this check, while some other options do not. Write trailing zeros freely: adding zero after a decimal does not change a number.

In 8.040, the zero at the end after the decimal point is a trailing zero, so it does not change the value. Hence, 8.040 = 8.04. In 8.4, the digit 4 is in the tenths place, whereas in 8.040 it is in the hundredths place, so they are not equal. Exam tip: Only trailing zeros to the right of a decimal point can be removed without changing the value.

(0.500) is greater than (0.0500) because the place of (5) has changed. Only adding or removing ending zeros does not change value.

A rational number can be written as a fraction of integers. Its decimal expansion either terminates or continues with a repeating block. In option A, 0.125 terminates, 0.777... repeats the digit 7, and 2.3434... repeats the block 34; therefore all three are rational. Hence option A is correct. In option B, 0.1010010001... has a non-repeating pattern, so it is irrational even though 0.25 and 1.5 are rational. Option C contains a non-repeating expansion of pi, which is irrational. Option D also begins a non-terminating, non-repeating pattern, so not all its numbers are rational. The presence of a decimal alone is not enough; its pattern must be checked.

A non-terminating decimal without fixed repetition can be the form of an irrational number. Terminating and recurring decimals are rational.

Write the decimals to the same number of places: \(y=0.49=0.490\). Comparing \(0.409\) and \(0.490\), the hundredths digits give \(0<9\); hence \(0.409<0.490\), so \(x<y\). The option \(x>y\) is incorrect because \(y\) has 9 in the hundredths place. Exam tip: Add zeros at the end, when needed, to make decimal places equal before comparing.

The decimal number 0.6a3 has three places after the decimal point. To compare it with 0.653, first compare the digits from left to right. The tenths digits are both 6, so they are equal. The next digits are a and 5, so the number will be smaller only when a is less than 5. Since a is a digit, the greatest possible digit less than 5 is 4.

Therefore, the greatest value of a is 4, which is option B. If a were 5, both numbers would have the same first three decimal digits, and the comparison would depend on the final digits: 0.653 would not be less than 0.653. Any digit greater than 5 would make 0.6a3 larger than 0.653. Thus 4 is the largest valid choice.

Dividing both sides of the equation by 100 gives \(x=\frac{3.75}{100}=0.0375\). When a decimal is divided by 100, its decimal point moves two places to the left. Option B moves it only one place and represents \(3.75\div10\), not \(3.75\div100\). Exam tip: For division by 10, 100, or 1000, move the decimal point 1, 2, or 3 places to the left, respectively.

Use the equality-preserving operation of dividing both sides by 1000. From 1000x = 64.8, we get x = 64.8/1000. Because 1000 is 10^3, the decimal point in 64.8 moves three places to the left: 64.8 → 6.48 → 0.648 → 0.0648. Therefore, x = 0.0648 and option B is correct. Option A moves the decimal only two places, option D moves it only one place, and option C moves it in the wrong direction and has an unrealistic magnitude. Substitution verifies the result: 1000 × 0.0648 = 64.8. This also illustrates decimal representation and place value.

Between successive 1s, the number of zeros is \(1,2,3,4,\ldots\), so no fixed digit block repeats. Hence it is non-terminating, non-recurring and irrational. Exam tip: check for a fixed repeating block, not merely repeated digits.

( \frac{1}{11}=0.0909\ldots ), which is slightly less than (0.091). Convert the fraction into decimal for comparison.

Since 13 and 30 are coprime, 30 is the denominator in lowest form. As \(30=2\times3\times5\) contains 3, the decimal is non-terminating recurring. Exam tip: check prime factors of the reduced denominator.

First reduce the fraction by dividing numerator and denominator by their greatest common divisor, 14: 28/98 = 2/7. The denominator 7 is neither 2 nor 5 and remains in lowest terms, so the decimal cannot terminate. Since every rational number has either a terminating or recurring decimal expansion, 2/7 gives the non-terminating recurring decimal 0.285714..., making option B correct.

In \(0.272727...\), the block 27 repeats, so it is rational. In fact, \(0.272727...=\frac{27}{99}=\frac{3}{11}\). An infinite decimal is not automatically irrational. Exam tip: recurring decimals represent rational numbers.