Decimal representation

A rational number has a terminating decimal only when its reduced denominator contains only 2 and/or 5. Since 3 is present, the decimal repeats endlessly. Exam tip: always reduce the fraction first.

The direct answer is B: non-terminating recurring. First reduce the fraction: \(35/42\) has common factor 7, so \(35/42=5/6\). A fraction in simplest form has a terminating decimal only when its denominator has no prime factors except 2 and 5. Here, \(6=2\times3\), and the factor 3 remains. Therefore its decimal cannot end; division gives \(5\div6=0.8333\ldots\), where 3 repeats forever. Option A, terminating, is wrong because the denominator contains 3. Option B is correct because the decimal continues and repeats. Option C, non-terminating non-recurring, is wrong because every rational fraction has either a terminating or recurring decimal. Option D, integer only, is wrong because \(5/6\) is not a whole number. Remember: after reducing, only 2s and 5s in the denominator mean terminating; any other prime factor means recurring.

(160\times625=100000), so ( \frac{43}{160}=\frac{26875}{100000}=0.26875 ). Making the denominator a power of (10) gives an exact answer.

(0.0064=\frac{64}{10000}=\frac{4}{625}). Count decimal places to write the denominator and then simplify the fraction.

( \frac{27}{90}=\frac{3}{10} ), and the denominator has only factors (2) and (5). Therefore, its decimal expansion will be terminating.

The governing concept is recurring-decimal notation: a bar is written only over the digit or consecutive block that repeats endlessly. In 0.0131313..., the first digit after the decimal point is 0, while the block 13 repeats: 0.0 13 13 13.... Thus the non-repeating part is 0 and the repeating part is 13, so the intended notation is 0.0 overline{13}, represented by option B. Option A incorrectly includes the initial 0 in the recurring block, implying that 013 repeats. Option C removes the non-repeating 0 and changes the number. Option D marks only 3, although 1 and 3 repeat together. Therefore B is the only unambiguous correct choice.

(2.34=2.340), so it is the greatest and (2.034) is the smallest. Write all decimals up to equal places for comparison.

In the equation \(100x=45.6\), divide both sides by 100 to isolate \(x\): \(x=\frac{45.6}{100}=0.456\). Dividing by 100 shifts the decimal point two places to the left. Option B, \(4.56\), would result from dividing by 10, not by 100. Exam tip: when dividing by 10, 100, or 1000, move the decimal point 1, 2, or 3 places to the left, respectively.

(125\times8=1000), so ( \frac{23}{125}=\frac{184}{1000}=0.184 ). Making the denominator a power of (10) is a fast method.

A terminating decimal can be converted to a fraction by using a power of ten as the denominator, followed by reduction to lowest terms. Since 0.5625 has four digits after the decimal point, 0.5625 = 5625/10000. The greatest common divisor of 5625 and 10000 is 625. Dividing numerator and denominator by 625 gives 5625 ÷ 625 = 9 and 10000 ÷ 625 = 16. Therefore 0.5625 = 9/16, so option A is correct. Option C, 45/80, has the same numerical value but is not simplified because both terms are divisible by 5. Option D uses 1000 instead of the required denominator for four decimal places, and option B represents a different value.

The governing theorem states that a rational number p/q in lowest terms has a terminating decimal expansion only when the prime factors of q are exclusively 2 and 5. The fraction 17/120 is already reduced because 17 shares no factor with 120. Factor the denominator: 120 = 2^3 × 3 × 5. Since the factor 3 remains, the decimal expansion cannot terminate. A rational number must have either a terminating or a non-terminating recurring decimal expansion, so it cannot be non-recurring. Indeed, long division gives 17/120 = 0.141666..., confirming option B. Option A overlooks the factor 3, while C is associated with irrational decimals.

In 4.0705, the digit 7 is in the second place to the right of the decimal point. This is the hundredths place, so its place value is \(7 \times \frac{1}{100}=\frac{7}{100}\). Option A represents the tenths place, not the hundredths place. Exam tip: Count the decimal places from left to right as tenths, hundredths, thousandths and ten-thousandths.

In 3.208, 2 is in the tenths place and 8 is in the thousandths place. The hundredths digit is 0, so its expanded form is \(3+\frac{2}{10}+\frac{0}{100}+\frac{8}{1000}\), which is equivalent to \(3+\frac{2}{10}+\frac{8}{1000}\). Option A is incorrect because it places 8 in the hundredths place. In an exam, check the place values after the decimal point in order: tenths, hundredths and thousandths.

The direct answer is option B: 45. In the decimal \(0.1454545\ldots\), the first digit after the decimal point is 1, and after that the digits continue as 45, 45, 45, and so on. The recurring part means the smallest block that repeats forever. Therefore the recurring block is 45. Option A, 14, includes the non-repeating initial digit and is not the repeating block. Option B, 45, is correct because the pair 45 appears again and again. Option C, 54, is not the order in which the digits repeat; the decimal shows 45, not 54. Option D, 145, contains both the initial non-repeating digit and the repeating digits, so it is larger than the smallest recurring block. A common mistake is to include the digits before repetition begins. Here, the initial 1 is non-recurring, while 45 is recurring.

A rational number has a terminating decimal only when the denominator in lowest terms has no prime factors other than 2 and 5. For example, \(40=2^3\times5\). Options B and C are only special cases, while a factor such as 3 gives a non-terminating recurring decimal. Exam tip: reduce first.

(1.045) is greater than (1.040) and less than (1.050). Make decimal places equal to find a number between them.

A digit’s place value depends on its position relative to the decimal point. The first 7 in 0.07007 is not in the tenths place; it is the second digit after the decimal point. The second position after the decimal represents hundredths, so one unit there is \\(\frac{1}{100}\\). Therefore, the 7 contributes \\(7\times\frac{1}{100}=\frac{7}{100}\\), which is option B.

Reading from left to right after the decimal point, the places are tenths, hundredths, thousandths, ten-thousandths, and so on. In 0.07007, the first 7 is in the hundredths place, while the second 7 is in the ten-thousandths place. Thus the given answer is correct. Confusing the first 7 with the second 7 would incorrectly give \\(\frac{7}{10000}\\).

The governing concept is the conversion of a terminating decimal into a fraction. Since 6.25 has two digits after the decimal point, write it over 100: 6.25 = 625/100. The numerator and denominator have the common factor 25, so divide both by 25: 625 ÷ 25 = 25 and 100 ÷ 25 = 4. Thus 6.25 = 25/4. Because the numerator 25 is greater than the denominator 4, the result is an improper fraction, making option A correct. Option B, 625/10, equals 62.5 and is therefore not equivalent. Option C equals 0.24, while option D equals 6.2. These value checks confirm that only option A represents 6.25 correctly in simplest improper form.

The governing concept is place value in a terminating decimal. There are five digits after the decimal point, so 0.00625 can first be written as 625/100000. The numerator and denominator have 625 as a common divisor. Reducing gives 625 ÷ 625 = 1 and 100000 ÷ 625 = 160, so the simplified fraction is 1/160. Therefore, option A is correct. A useful verification is that 1 ÷ 160 = 0.00625. Option B equals 0.0625 and is ten times too large. Option C equals 0.625 and is not simplified, while option D has a completely different value. The two zeros immediately after the decimal must be retained when establishing place value.

In (0.0304), the place of digits has changed, so it is smaller. Only adding or removing zeros at the end does not change value.

(0.064\times100=6.4), so it becomes (6.4%). To convert a decimal into a percentage, multiply by (100).

The governing place-value rule is that dividing by 100 moves the decimal point two places to the left, because 100 is 10^2. Starting with 8.406, moving one place left gives 0.8406, and moving a second place left gives 0.08406. Therefore, 8.406 ÷ 100 = 0.08406, so option C is correct. The result can be checked by reversing the operation: 0.08406 × 100 = 8.406. Option A moves the decimal in the wrong direction and represents multiplication by 100. Option B corresponds to division by 10, while option D corresponds to division by 1000. The digits remain ordered; only their place values change.

Align the decimal points and write 0.375 as 0.3750. Then, 0.3750 + 0.0625 = 0.4375, so option A is correct. An answer such as 0.3875 can result from adding digits without respecting their place values. Exam tip: always align decimal points first and add trailing zeros when needed.

Write 5.5 as 5.500 so that the decimal places are aligned. Then \(5.500-2.875=2.625\), so 2.625 is correct. The value 2.725 can result from subtracting digits without correctly maintaining their place values. Exam tip: align decimal points first and add zeros where needed before subtracting decimal numbers.

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