Decimal expansion of rational numbers

Step 1: (q) has only the prime factors (2) and (5). Step 2: In this case, the decimal expansion of the rational number terminates. Step 3: If (q) is of the form (2^m5^n), the decimal terminates.

Step 1: (12=2^2\times3). Step 2: The denominator contains (3), so the decimal does not terminate, but because it is rational, it recurs. Step 3: A non-terminating decimal of a rational fraction is not non-recurring.

Step 1: (0.08=\frac{8}{100}). Step 2: Reducing by (4) gives (\frac{2}{25}). Step 3: If there are two digits after the decimal point, use denominator (100).

Step 1: (200=2^3\times5^2). Step 2: The larger exponent is (3), and (\frac{7}{200}=0.035), so it has exactly three places. Step 3: When exact places are asked, verify by writing the decimal.

Step 1: (22=2\times11). Step 2: The factor (11) prevents termination, and since the number is rational, the decimal recurs. Step 3: Any factor other than (2) and (5) stops termination.

Step 1: (1.25=\frac{125}{100}). Step 2: Reducing by (25) gives (\frac{5}{4}). Step 3: A terminating decimal greater than (1) can also be converted into a rational fraction.

Step 1: In (0.727272\ldots), the block (72) repeats. Step 2: Therefore, it is a non-terminating recurring decimal. Step 3: A recurring decimal must have a fixed block repeating continuously.

Step 1: (45=3^2\times5). Step 2: The factor (3) is present, so the decimal does not terminate and recurs. Step 3: A denominator having (3) along with (5) does not give a terminating decimal.

Step 1: (75=3\times5^2). Step 2: The factor (3) remains in the denominator, so the decimal will not terminate and will recur. Step 3: If the reduced denominator is not of the form (2^m5^n), it does not terminate.

Step 1: The denominator in the first option has only (2) and (5). Step 2: Hence, (\frac{23}{2^4\times5}) has a terminating decimal. Step 3: In factorised denominators, quickly spot any extra prime factor.

Step 1: (16=2^4), so the decimal terminates. Step 2: (\frac{1}{16}=\frac{625}{10000}=0.0625). Step 3: Fractions with denominator (16) may have four decimal places.

Step 1: (\frac{14}{35}=\frac{2}{5}). Step 2: The reduced denominator is (5), so the decimal terminates. Step 3: Do not be misled by the factor (7) in the original denominator; reduce first.

Step 1: (66=2\times3\times11). Step 2: The denominator contains (3) and (11), so the decimal does not terminate and recurs. Step 3: A non-terminating decimal of a rational number is recurring.

Step 1: In a recurring decimal, a fixed block of digits repeats. Step 2: Such a decimal can be converted into a fraction (\frac{p}{q}), so it is rational. Step 3: In assertion-reason questions, check whether the reason supports the assertion.

Step 1: (8=2^3), so the decimal terminates. Step 2: (\frac{7}{8}=\frac{875}{1000}=0.875). Step 3: Do not put a bar on a terminating decimal.

Step 1: (0.\overline{6}=0.666\ldots). Step 2: This is the decimal expansion of (\frac{2}{3}). Step 3: Understand the difference between (0.6) and (0.\overline{6}).

Step 1: A rational number has either a terminating decimal or a non-terminating recurring decimal. Step 2: So if it does not terminate, some digit or block will repeat. Step 3: Do not call a rational number non-terminating non-recurring.

Step 1: (1250=2\times5^4). Step 2: The larger exponent is (4), so the decimal terminates in four places. Step 3: For denominators like (1250), prime factorisation is the easy route.

Step 1: Multiply (40) by (25) to make (1000). Step 2: (\frac{31}{40}=\frac{775}{1000}=0.775). Step 3: Converting the denominator into a power of (10) is a quick method.

Step 1: In a non-terminating non-recurring decimal, digits continue without a fixed repeating block. Step 2: The second option states that there is no fixed repeat, so it is non-recurring. Step 3: To separate recurring and non-recurring decimals, check the repetition pattern.