Decimal representation

Multiplying both by (10000) gives (2970\div9=330). In division, both numbers can be multiplied by the same number.

Write 7.003 as 7.00300 so that both numbers have the same number of decimal places. Then \(7.00300-4.05875=2.94425\), so 2.94425 is correct. The option 2.95425 results from an error in subtraction at the hundredths place. Exam tip: Before subtracting decimals, align the decimal points and add trailing zeros if needed.

Writing equal decimal places gives (0.0781250+0.0156250+0.0003125=0.0940625). Align decimal points while adding.

In 4.12999\ldots, the digit 9 continues indefinitely after the decimal point. Since 0.00999\ldots = 0.01, we get 4.12999\ldots = 4.12 + 0.01 = 4.13. Option 4.129 merely truncates the decimal and is therefore not equal to the given number. Exam tip: A decimal ending in infinitely recurring 9s can be written by adding 1 to the preceding decimal place.

( \frac{17}{32}=0.53125 ) and ( \frac{8}{15}=0.5333\ldots ), so (0.5328) lies between them. Convert the boundaries into decimals.

Let \(x=0.272727\ldots\). Then \(100x=27.272727\ldots\); subtracting gives \(99x=27\), so \(x=\frac{27}{99}=\frac{3}{11}\). A repeating decimal is rational, so option A is incorrect. Exam tip: identify the repeating block before converting it to a fraction.

The key concept is the distinction between terminating, recurring, and non-recurring decimal expansions. The decimal 5.02002000200002... continues indefinitely, so it is non-terminating. Its blocks do not repeat with one fixed period: the numbers of zeros between successive 2s increase, so no finite string of digits can repeat forever. A rational number has a decimal expansion that either terminates or eventually repeats periodically. Since this expansion is non-terminating and non-periodic, it represents an irrational number; option C is correct. It cannot be a terminating rational because digits continue, cannot be an eventually recurring rational because there is no fixed cycle, and cannot be an integer because a nonzero decimal tail remains.

The governing concept is solving a linear equation by applying the inverse operation. Since 10000x means 10000 multiplied by x, divide both sides by 10000: x = 1.1875 ÷ 10000. Dividing by 10,000 moves the decimal point four places to the left: 1.1875 → 0.11875 → 0.011875 → 0.0011875 → 0.00011875. Hence option C is correct. A quick check confirms it: 0.00011875 × 10000 = 1.1875. Option A results from shifting in the wrong direction, option B has only two decimal-place shifts, and option D has only one shift. The equation therefore requires the value shown in option C.

In 18.07006, the digits after the decimal point occupy the tenths, hundredths, thousandths, ten-thousandths and hundred-thousandths places, respectively. Therefore, 6 is in the fifth decimal place, so its place value is \(\frac{6}{100000}\). Exam tip: the denominator increases by a factor of 10 for each place to the right of the decimal point.

Write all numbers up to four decimal places: \(8.0080, 8.0808, 8.0088, 8.8000\). The first decimal digits are \(0,0,0,8\), so \(8.8000\) is the greatest. Among the remaining numbers, \(8.0080<8.0088<8.0808\). Hence, the ascending order is \(8.008<8.0088<8.0808<8.800\). Option D incorrectly places \(8.0808\) before \(8.0088\). Exam tip: append trailing zeros to make the decimal places equal before comparing decimals.

(5000\times2=10000), so ( \frac{89}{5000}=\frac{178}{10000}=0.0178 ). Make the denominator a power of (10).

Divide the larger decimal by the smaller one: \(0.0625 \div 0.000625 = 100\). Therefore, \(0.0625\) is 100 times \(0.000625\). Option C is incorrect because multiplying \(0.000625\) by 1000 gives \(0.625\). Exam tip: For a ‘how many times’ comparison, divide the larger quantity by the smaller quantity.

In \(0.00\overline{72}\), the bar is over 72 only. Hence, the first two digits after the decimal point are 0 and 0, followed by the repeated block 72: \(0.00727272\ldots\). In option B, 72 starts one place later, while option D incorrectly treats the repetition as terminating. Exam tip: Repeat only the digits covered by the bar; digits before the bar do not repeat.

Since \(0.\overline{27}=\frac{27}{99}=\frac{3}{11}\), it is rational. A non-terminating decimal is not necessarily irrational; every repeating decimal is rational. Exam tip: identify the repeating block first.

First reduce the fraction before deciding its decimal type. The numerator and denominator of 72/198 have a common factor of 18, so 72/198 = 4/11. A fraction in lowest form has a terminating decimal only when its denominator has no prime factors other than 2 and 5. If another prime factor remains, the decimal continues and repeats.

The denominator 11 is a prime factor different from 2 and 5. In fact, 4/11 = 0.3636..., so the digits 36 repeat without ending. It is therefore a non-terminating recurring decimal. Hence option C is correct. The unreduced form should not be used alone, because common factors can remove some denominator factors during simplification.

In 6.24999..., the digit 9 continues infinitely after the decimal point. Since 0.00999... = 0.01, we get 6.24999... = 6.24 + 0.01 = 6.25. The option 6.249 is only a finite decimal and does not include the infinitely repeating 9s. Exam tip: When infinitely many 9s follow a digit, increase that digit by 1 and remove the repeating 9s.

In 4.080700, the digits after the decimal point represent tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths and millionths, respectively. The digit 7 is in the fourth decimal place, so its place value is \(7 \times \frac{1}{10000}=\frac{7}{10000}\). Therefore, option C is correct. Exam tip: each place to the right of the decimal point has one-tenth the value of the preceding place; \(\frac{7}{1000}\) would correspond to the third decimal place.

The governing concept is converting equivalent forms before comparing numbers. First, 5/16 can be converted to a decimal by dividing 5 by 16, giving 0.3125. The notation 0.31̅2 means that only the digit 2 repeats: 0.3122222..., not 0.3125. Compare the latter with 0.3125 digit by digit: both begin 0.312, but at the fourth decimal place, 2 is less than 5. Thus 0.31̅2 < 0.3125, while 0.3125 = 5/16. Therefore option A is correct. Option C reverses the comparison, option B incorrectly equates the repeating decimal with 5/16, and option D ignores the different fourth and later digits.

Write the decimals to the same number of decimal places: a = 0.4070, b = 0.4700, and c = 0.4007. At the tenths place, all have 4; at the hundredths place, b has 7, so b is the greatest. For a and c, the hundredths digits are both 0, but at the thousandths place a has 7 while c has 0. Hence a > c. Therefore, the descending order is b > a > c. Exam tip: Add zeros to the right of decimals when needed before comparing place values.

( \frac{17}{125}=0.136 ) and ( \frac{11}{80}=0.1375 ), so the sum is (0.2735). First convert the fractions into decimals.