Decimal representation

The direct answer is option A. In 8.20002, the 8 is in the units place, the first 2 is in the tenths place, and the last 2 is in the hundred-thousandths place. Hence the expanded form is 8+2/10+2/100000. The zeros in the middle have zero place value and do not add a term. Option A uses the correct positions and reconstructs the original decimal. Option B places its first 2 in the hundredths place and its final 2 in the ten-thousandths place, so it represents another number. Option C writes 82 as the whole-number part, incorrectly joining digits across the decimal point. Option D has denominator 1000 and therefore does not represent the two separated 2s in their actual places. The memory rule is to count every place after the decimal, even when some digits are zero; the last digit here is in the fifth decimal place.

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.

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.

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.

( \frac{21}{64}=0.328125 ), so (0.328125-0.203125=0.125). Convert the fraction into decimal and subtract.

In the decimal expansion \(0.0646464\ldots\), the first digit after the decimal point, \(0\), occurs only once. After it, the block \(64\) repeats continuously: \(64,64,64,\ldots\). Hence, the bar is placed only over \(64\), giving \(0.0\overline{64}\). \(0.\overline{64}\) is incorrect because it makes \(64\) start immediately after the decimal point. Exam tip: Identify the exact block that repeats continuously before placing the bar.

In lowest form, a denominator containing a prime other than 2 or 5 cannot produce a terminating decimal. Since \(\frac{p}{q}\) is rational, its decimal repeats. Exam tip: factorise the denominator first.

A terminating decimal is obtained by changing the denominator into a power of 10. The denominator here is \\(2^6\times5^3\\). Since there are six factors of 2 and only three factors of 5, multiply by \\(5^3\\) to create three additional pairs. Then the denominator becomes \\(2^6\times5^6=10^6\\), so at most six decimal places are needed.

The general rule is to take the larger exponent of 2 and 5 in the simplified denominator. Thus \\(\max(6,3)=6\\), making option B correct. The answer is not 9, because the exponents are not added; unmatched factors are supplied only to form equal pairs of 2 and 5. A particular numerator might shorten the decimal, but six is the maximum.

The number of zeros between (7)'s increases, so there is no fixed repetition. Such a decimal is non-terminating non-recurring.

To convert a decimal into a percentage, multiply it by 100: \(0.0024 \times 100 = 0.24\). Therefore, the correct answer is \(0.24\%\). Remember that multiplying by 100 shifts the decimal point two places to the right; hence, 0.024% is ten times too small and 2.4% is ten times too large.

\(\frac{21}{88}\) is already in lowest form, and \(88=2^3\times11\). Since the denominator contains 11, its decimal expansion is non-terminating recurring. The other fractions reduce to denominators with only 2s and 5s. Exam tip: simplify first.