Decimal expansion of rational numbers

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.

Step 1: Multiply (250) by (4) to make (1000). Step 2: (\frac{9}{250}=\frac{36}{1000}=0.036). Step 3: Pay attention to the correct position of zeros in decimals.

Step 1: (33=3\times11). Step 2: The denominator has factors other than (2) and (5), so the decimal does not terminate and recurs. Step 3: Such denominators in rational fractions give recurring decimals.

Step 1: (1000=10^3=2^3\times5^3). Step 2: The denominator is made only of (2) and (5), so the decimal terminates. Step 3: A denominator that is a power of (10) always gives a terminating decimal.

Step 1: (4.125) has a finite number of digits after the decimal point. Step 2: Every terminating decimal can be written as a fraction, so it is rational. Step 3: Identifying a terminating decimal is an easy way to identify rationality.

Step 1: (\frac{36}{150}=\frac{6}{25}). Step 2: The reduced denominator is (25=5^2), so the decimal terminates. Step 3: Simplification can remove extra factors from the original denominator.

Step 1: (54=2\times3^3). Step 2: The denominator contains (3), so the decimal will not terminate and will recur because it is rational. Step 3: Any power of (3) in the denominator prevents termination.

Step 1: (2.04=\frac{204}{100}). Step 2: Reducing by (4) gives (\frac{51}{25}). Step 3: When converting a decimal to a fraction, include the whole part in the numerator.

Step 1: (250=2\times5^3). Step 2: The denominator has only (2) and (5), so (\frac{11}{250}) has a terminating decimal. Step 3: Choose the denominator made only of (2) and (5).

Step 1: (27=3^3). Step 2: The denominator contains only (3), so the decimal will not terminate. Step 3: The fraction is rational, so its non-terminating decimal will recur.

Step 1: (\frac{15}{60}=\frac{1}{4}). Step 2: The reduced denominator is (4=2^2), so the decimal terminates. Step 3: The reduced denominator, not the original one, decides the type.

Step 1: (0.0008=\frac{8}{10000}). Step 2: Reducing by (8) gives (\frac{1}{1250}). Step 3: Count decimal places carefully in very small decimals.

Step 1: The denominator is (2^5\times5^5). Step 2: It becomes (10^5), so the decimal terminates after five places. Step 3: When the exponents are equal, that exponent gives the number of decimal places.

Step 1: (90=2\times3^2\times5). Step 2: The factor (3) remains in the denominator, so the decimal will not terminate. Step 3: Since it is rational, it will be non-terminating recurring.

Step 1: (5000=2^3\times5^4). Step 2: The larger exponent is (4), so the decimal terminates within four places. Step 3: Even denominators with zeros should be written in prime factors.

Step 1: In (0.2\overline{3}), after (2), the digit (3) repeats. Step 2: The decimal does not terminate and has a repeating digit, so it is recurring. Step 3: The bar is placed only over the repeating part.

Step 1: (\frac{1}{11}=0.\overline{09}). Step 2: Multiplying by (6) gives (\frac{6}{11}=0.\overline{54}). Step 3: If two digits repeat, put the bar over the whole block.

Step 1: (0.625=\frac{625}{1000}). Step 2: Reducing by (125) gives (\frac{5}{8}). Step 3: Converting the terminating decimal to a fraction and reducing is the safest method.

Step 1: (39=3\times13). Step 2: The denominator has factors other than (2) and (5), so the decimal is non-terminating recurring. Step 3: In options, check denominator factors first.

Step 1: For a terminating decimal, the denominator in lowest form must be made only of (2) and (5). Step 2: If (7) remains, this condition fails and the decimal will not terminate. Step 3: The reason correctly explains the assertion, so the first option is correct.

Step 1: (0.0125=\frac{125}{10000}). Step 2: Reducing by (125) gives (\frac{1}{80}). Step 3: For small decimals, count the zeros carefully and then reduce the fraction.