Decimal expansion of rational numbers

Step 1: (45=3^2\times5). Step 2: Because (3) is present in the denominator, the decimal will not terminate. Step 3: Since it is rational, the decimal will be non-terminating recurring.

Step 1: A non-terminating recurring decimal occurs when the reduced denominator has a prime factor other than (2) and (5). Step 2: Such a denominator cannot be made into a power of (10). Step 3: So always check the prime factors of the denominator.

Step 1: Four decimal places mean ten-thousandths. Step 2: So the number can be written as (\frac{n}{10000}). Step 3: In exams, reduce the fraction afterward.

Step 1: (0.875=\frac{875}{1000}). Step 2: Reducing gives (\frac{875}{1000}=\frac{7}{8}). Step 3: Count the decimal places and use the corresponding power of (10) as denominator.

Step 1: The repeating block is (45), so (0.\overline{45}=\frac{45}{99}). Step 2: (\frac{45}{99}=\frac{5}{11}). Step 3: Write as many (9)s in the denominator as the number of repeating digits.

Step 1: (0.1\overline{6}=0.1666\ldots). Step 2: This is equal to (\frac{1}{6}). Step 3: In a mixed recurring decimal, identify the non-repeating part and then the repeating part.

Step 1: (6.375=\frac{6375}{1000}=\frac{51}{8}). Step 2: The reduced denominator is (8=2^3). Step 3: The reduced denominator of a terminating decimal is made only of (2) and (5).

Step 1: (12.004) has three decimal places. Step 2: It stops there, so it is a terminating decimal. Step 3: Every terminating decimal can be written as a rational number.

Step 1: Terminating and non-terminating recurring decimals are rational. Step 2: A non-terminating non-recurring decimal is not rational; it is irrational. Step 3: Read the words recurring and non-recurring carefully in statement questions.

Step 1: A non-terminating recurring decimal has a fixed block repeating. Step 2: Such a decimal can be written as (\frac{p}{q}). Step 3: So treating it as irrational is a mistake.

Step 1: The denominator is (2^6). Step 2: It has only (2), so the decimal terminates. Step 3: Since the exponent of (2) is (6), it terminates after (6) places.

Step 1: (\frac{27}{216}=\frac{1}{8}). Step 2: (8=2^3). Step 3: Therefore the decimal terminates after (3) places.

Step 1: (1250\times8=10000). Step 2: (\frac{7}{1250}=\frac{56}{10000}=0.0056). Step 3: Converting the denominator into a power of (10) is a quick and safe method.

Step 1: (400=2^4\times5^2). Step 2: The larger exponent is (4), and (\frac{43}{400}=0.1075) has four places. Step 3: Therefore it terminates after (4) places.

Step 1: (\frac{63}{175}=\frac{9}{25}). Step 2: The reduced denominator is (25=5^2). Step 3: Since the denominator has only (5), the decimal terminates.

Step 1: (\frac{65}{104}=\frac{5}{8}). Step 2: The reduced denominator is (8=2^3), so the decimal terminates. Step 3: Only the denominator left after cancellation decides the result.

Step 1: (\frac{169}{338}=\frac{1}{2}). Step 2: (\frac{1}{2}=0.5), so the decimal terminates after one place. Step 3: Even with large numbers, cancel common factors first.

Step 1: (17) is a prime other than (2) and (5). Step 2: If (17) remains in the reduced denominator, the decimal will not terminate. Step 3: Such a non-terminating decimal of a rational number is recurring.

Step 1: To make (10^6), we need (2^6\times5^6). Step 2: The denominator already has (2^6\times5^2). Step 3: So it lacks (5^4), and we must multiply by (5^4).

Step 1: After reduction, the denominator is (20). Step 2: (20=2^2\times5), so it has only (2) and (5). Step 3: Therefore the decimal expansion is terminating.

Step 1: (\frac{55}{88}=\frac{5}{8}). Step 2: Since (8=2^3), the decimal terminates after (3) places. Step 3: You can also check with (\frac{5}{8}=0.625).

Step 1: (\frac{14}{63}=\frac{2}{9}). Step 2: The reduced denominator is (9=3^2). Step 3: Since (3) is present in the denominator, the decimal is non-terminating recurring.

Step 1: Here the block (142857) repeats in a fixed way. Step 2: A non-terminating decimal with fixed repetition is a rational number. Step 3: If repetition is visible, do not treat it as non-recurring.

Step 1: (0.75) is terminating, while (0.\overline{3}) and (5.2\overline{4}) are recurring. Step 2: (2.10110111011110\ldots) has no fixed repetition. Step 3: A non-terminating non-recurring decimal does not represent a rational number.

Step 1: A rational number has a terminating decimal only when the denominator in lowest form has prime factors only (2) and (5). Step 2: Here the denominator also contains (7), so the decimal will not terminate, but since the number is rational, it will repeat. Step 3: In exams, always reduce the fraction first and then check the prime factors of the denominator.