Decimal representation

Although \(0.\overline{3}\) is non-terminating, it repeats and \(0.\overline{3}=\frac{1}{3}\), so it is rational. \(\sqrt{2}\) is non-repeating. Exam tip: every repeating decimal represents a rational number.

\(\frac{3}{15}=\frac{1}{5}=0.2\), so its decimal expansion terminates. Always reduce a fraction first: only the prime factors of the simplified denominator determine the decimal type.

A rational number has a terminating decimal expansion only when, after simplifying the fraction, the denominator has no prime factors other than 2 and 5. If the simplified denominator contains another prime factor, the decimal digits continue indefinitely and repeat in a pattern. Thus the fraction must always be simplified before deciding its decimal type.

Simplify \(44/66\) by dividing numerator and denominator by 22: \(44/66=2/3\). The denominator 3 is neither 2 nor 5, so the decimal cannot terminate. In fact, \(2/3=0.666\ldots\), where 6 repeats endlessly. Therefore the expansion is non-terminating recurring, which is option B. It is not non-terminating non-recurring because every rational number has either a terminating or a repeating decimal expansion.

In \(0.036036036\ldots\), the digit block \(036\) repeats continuously. Therefore, the bar must be placed over the complete recurring block: \(0.\overline{036}\). In \(0.0\overline{36}\), only \(36\) repeats, giving \(0.0363636\ldots\), which is different from the given decimal. Exam tip: identify the smallest complete repeating block before placing the bar.

We can write \(0.09\) as \(0.0900\ldots\). The first three decimal digits are the same, but at the fourth digit \(0.0909\ldots\) has 9 whereas \(0.0900\ldots\) has 0. Therefore, \(0.0909\ldots>0.09\). Option B is incorrect because all digits after the terminating decimal \(0.09\) are 0. Exam tip: When comparing decimals, append zeros to a terminating decimal and compare digits from left to right.

The governing algebraic principle is maintaining equality while isolating the unknown. In 1000x = 7.25, the coefficient of x is 1000, so divide both sides by 1000: x = 7.25/1000. Dividing by 1000 moves the decimal point three places to the left, producing 0.00725. Hence option D is correct. Substitution confirms the result: 1000 × 0.00725 = 7.25. Option A results from multiplying 7.25 by 1000 instead of dividing. Option B moves the decimal only two places and therefore represents division by 100, while option C moves it only one place and represents division by 10. The zeros are essential for correct place value.

To convert a decimal into a percentage, multiply it by 100: \(0.875 \times 100 = 87.5\). Therefore, the correct answer is 87.5%. The option 8.75% results from multiplying by 10, while 875% results from multiplying by 1000. Exam tip: Move the decimal point two places to the right and add the percent sign when converting a decimal to a percentage.

(16\times625=10000), so ( \frac{11}{16}=\frac{6875}{10000}=0.6875 ). Convert the denominator into a power of (10).

After the decimal point, the places are tenths, hundredths, thousandths, and ten-thousandths. In 12.3045, 5 is in the fourth place after the decimal point, so its place value is \(\frac{5}{10000}\), or 0.0005. As an exam tip, count the digits after the decimal point to determine the denominator as the corresponding power of 10.

All numbers in the first option have some part repeating again and again. This is the key feature of non-terminating recurring decimals.

There are four digits after the decimal point in 0.0048, so it can be written as \(\frac{48}{10000}\). Dividing the numerator and denominator by 16 gives \(\frac{48}{10000}=\frac{3}{625}\). Therefore, option A is correct. In option B, the denominator 1000 is used, but four decimal places require a denominator of 10000. Exam tip: Count the digits after the decimal point, write the corresponding power of 10 as the denominator, and then simplify the fraction.

The governing concept is comparison by decimal place value. To compare fairly, write every number with four decimal places: 0.2000, 0.0202, 0.0220, and 0.2020. Reading from left to right, 0.0202 is smallest because its tenths digit is 0 and its hundredths digit is 2. Next comes 0.0220, whose thousandths digit is 2. The numbers 0.2000 and 0.2020 are larger, and 0.2020 is the greatest. Thus the ascending order is 0.0202 < 0.022 < 0.2 < 0.202, which is option A. Appending zeros does not change values; it only aligns place values for comparison.

The first (6) is in tenths and the last (6) is in ten-thousandths. So the expanded form is (5+\frac{6}{10}+\frac{6}{10000}).

To compare a fraction and a decimal, convert them to the same form. The denominator 40 can be changed to 100 by multiplying by 2.5, or the division can be performed directly. Dividing 13 by 40 gives 0.325. This value can then be compared digit by digit with 0.32, which may also be written as 0.320 for equal decimal places.

We get \\(\\frac{13}{40}=0.325\\), while \\(0.32=0.320\\). Comparing 0.325 and 0.320 shows that 0.325 is larger, so \\(\\frac{13}{40}\\) is greater. Therefore, option A is correct. They are not equal because the fraction has an additional value of 0.005 beyond 0.320.

\(0.272727\ldots=\frac{27}{99}=\frac{3}{11}\), so it is rational. A non-terminating decimal is rational when a block repeats; option A ignores this rule. Exam tip: use 9, 99, or 999 in the denominator for repeating blocks.

The denominator has (3), a factor other than (2) and (5). So the rational number will have a non-terminating recurring decimal.

To find the multiple, divide the larger decimal by the smaller one: \(0.007 \div 0.0007 = 10\). Thus, \(0.007 = 10 \times 0.0007\), so the correct answer is 10 times. The value is not 7 times because shifting the decimal point one place to the right multiplies 0.0007 by 10. In the exam, divide the larger number by the smaller number to find the multiple.

After the decimal point, the digits are \(9,0,9,0,\ldots\). The digit \(9\) is followed by \(0\), and this two-digit block repeats continuously; therefore, the recurring part is \(90\). The digit \(9\) alone is not the repeating block because it is followed by \(0\) each time. Exam tip: identify the shortest block after the decimal point that repeats in the same order.

The governing concept is comparison of decimal place values. Express all numbers to four decimal places: 0.8750, 0.8570, 0.8705, and 0.8700. The tenths digit is 8 in every number, so compare the hundredths digits. The value 0.8570 has 5 and is immediately smaller than the others, which have 7. Among the remaining values, compare the thousandths digits: 0.8750 has 5, whereas 0.8705 and 0.8700 have 0. Therefore 0.8750 is the greatest, so option A is correct. Notice that 0.87 is exactly 0.8700; trailing zeros do not change value. Option C is close but remains smaller than 0.875.

( \frac{1}{3}=0.333\ldots ) and ( \frac{1}{2}=0.5 ), so (0.34) lies between them. Convert boundaries to decimals for comparison.

The direct answer is option B: 0.205. To convert 41/200 into a decimal, make the denominator 1000. Since 200 × 5 = 1000, multiply the numerator by 5 as well: 41 × 5 = 205. Therefore 41/200 = 205/1000 = 0.205. Because 41 is less than 200, the result must be less than 1, which also rules out 2.05. Option B is correct. Option A, 0.0205, is one-tenth of the correct value and has the decimal point in the wrong place. Option C, 0.41, would be 41/100, not 41/200. Option D, 2.05, is greater than 1 and cannot represent this proper fraction. Always multiply numerator and denominator by the same number; then read the digits over 1000 as thousandths.

Digits after the decimal point have place values tenths, hundredths, thousandths, ten-thousandths, and hundred-thousandths. In 7.04004, the final 4 is the fifth digit after the decimal point, so it is in the hundred-thousandths place. Its place value is therefore 4 × 1/100000 = 4/100000. Option D is correct; the other choices belong to earlier decimal positions.

\(12.5 \times 0.1 = 1.25\). Since \(0.1 = \frac{1}{10}\), multiplying a number by \(0.1\) is equivalent to dividing it by \(10\). Therefore, \(12.5 \div 10 = 1.25\). Option B leaves the number unchanged, so it is incorrect. Exam tip: When multiplying a decimal by \(0.1\), move the decimal point one place to the left.

Multiplying both by (100) gives (36\div12=3). Multiplying both parts of a division by the same number does not change the quotient.

A non-terminating decimal without fixed repetition is non-terminating non-recurring. Such decimals can represent irrational numbers.