Decimal representation

Multiplying both numbers by (10000) gives (7040\div16=440). Removing decimals makes division easier.

The governing idea is place-value alignment in decimal subtraction. Write 12.006 as 12.00600 by adding two zeros to the right; this does not change its value. Now subtract: 12.00600 − 7.21975. Subtracting in columns, the result is 4.78625. A quick check is that 7.21975 + 4.78625 = 12.00600, confirming the calculation. Thus option A is correct. Option B differs by 0.01000, while option C is close to the reversed or incorrectly borrowed computation and option D results from an error in one of the decimal columns.

Writing equal decimal places gives (0.281250000+0.003906250+0.000078125=0.285234375). Align decimal points while adding.

In 13.24999\ldots, infinitely many 9s occur after 13.24. An infinite tail of 9s increases the preceding place value by 1; for example, 0.00999\ldots = 0.01. Hence, 13.24999\ldots = 13.25. Option 13.249 is only a truncated value and is not equal to the given decimal. Exam tip: Replace an infinite string of 9s by adding 1 to the digit immediately before it.

( \frac{31}{64}=0.484375 ) and ( \frac{17}{35}=0.485714\ldots ), so (0.4846) lies between them. Convert the boundaries into decimals.

The zero gaps after successive 1s are 1, 2, 3, ..., so no fixed digit block repeats. Hence it is non-terminating and non-recurring, therefore irrational. Exam tip: look for a repeating block, not merely repeated digits.

A rational number has a decimal expansion that either terminates or continues with a fixed repeating block. In 4.06006000600006…, the groups do not settle into a single repeating cycle: the numbers of zeros between successive 6s increase, so the pattern keeps changing. The decimal is therefore non-terminating and non-repeating, which is the characteristic decimal form of an irrational number. Hence option C is correct. It is not terminating, so A is impossible; it has no fixed recurring block, so B is wrong; and it is not an integer because nonzero digits occur after the decimal point, so D is also incorrect.

(x=3.90625\div100000=0.0000390625). On dividing by (100000), the decimal moves five places left.

In 32.008009, the digits after the decimal point occupy the tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths and millionths places, respectively. The digit 9 is in the sixth place, the millionths place. Therefore, its place value is \(9 \times \frac{1}{1000000}=\frac{9}{1000000}\). Option C, \(\frac{9}{100000}\), represents the fifth-place value, not the sixth-place value. Exam tip: Count decimal places from left to right, beginning with tenths.

Write all the numbers up to four decimal places for comparison: \(7.0070, 7.0707, 7.0077, 7.7000\). Comparing digits after the decimal point from left to right gives \(7.0070<7.0077<7.0707<7.7000\). Hence, option A is correct. In option D, \(7.0707\) and \(7.0077\) are placed in the wrong order because, at the tenths place, \(0<7\). Exam tip: Add trailing zeros when needed to make the decimal places equal before comparing decimals.

(16000\times625=10000000), so ( \frac{173}{16000}=0.0108125 ). Convert the denominator into a power of (10) to get the exact decimal.

To find the multiple, divide the larger decimal by the smaller one: \(0.015625 \div 0.00015625 = 100\). Therefore, \(0.015625\) is 100 times \(0.00015625\). The ratio is also evident because the decimal point shifts two places to the right. Exam tip: For a ‘how many times’ question, divide the larger quantity by the smaller quantity.

The bar is placed only over 81. Thus, the first three digits after the decimal point, 000, do not repeat, and 81 repeats thereafter. Hence, \(0.000\overline{81}=0.000818181\ldots\). Option B has an extra zero, while option C incorrectly makes 810 the repeating block. Exam tip: only the digits under the bar repeat; digits before the bar are written once.

\(\frac{33}{154}=\frac{3}{14}\), and \(14=2\times7\). Its reduced denominator has a prime factor other than 2 or 5, so the decimal is non-terminating recurring. Exam tip: simplify the fraction before checking denominator factors.

The fraction must be simplified before its decimal type is determined. Both 99 and 363 are divisible by 33, so 99/363 = 3/11. The denominator in lowest form is consequently 11.

For a rational fraction in simplest form, a terminating decimal is possible only when the denominator has prime factors 2 and/or 5. Since 11 is a different prime factor, the decimal does not end. Dividing gives 3/11 = 0.2727..., where the digits 27 repeat forever. Therefore the expansion is non-terminating recurring, which is option C. This also shows why checking the denominator after cancellation is essential; the original denominator alone may hide the simplest structure.

In 17.374999\ldots, infinitely many 9s occur after 4. Since \(0.009999\ldots=0.010\), we get \(17.374999\ldots=17.375\). Option 17.374 is only an approximation, not an equal value. Exam tip: when infinitely many 9s follow a digit, increase that digit by 1 and remove the repeating 9s.

After the decimal point, the digits 0, 8, 0 and 5 occupy the tenths, hundredths, thousandths and ten-thousandths places, respectively. Therefore, the place value of 5 is \(\frac{5}{10000}\). Exam tip: each successive place to the right of the decimal represents one-tenth of the preceding place.

( \frac{45}{64}=0.703125 ) and (0.703\overline{1}=0.703111\ldots). Therefore, (0.703125) is greater.

The correct order is \(b>a>c\), since \(0.690=0.6900\), \(0.609=0.6090\), and \(c=0.6009\). All three have 6 in the tenths place. In the hundredths place, \(b\) has 9, so it is the greatest. For \(a\) and \(c\), the hundredths digit is 0 in both, but the thousandths digit is 9 for \(a\) and 0 for \(c\); hence \(a>c\). The order \(b>c>a\) incorrectly places \(c\) above \(a\). Exam tip: add trailing zeros to compare decimals to the same number of places.

( \frac{31}{250}=0.124 ) and ( \frac{17}{640}=0.0265625 ), so the sum is (0.1505625). First convert the fractions into decimals.