Decimal expansion of rational numbers

Step 1: (80=2^4\times5). Step 2: The larger exponent is (4), so the decimal terminates in four places. Step 3: Writing (\frac{7}{80}=0.0875) confirms the answer.

Step 1: The denominator has exponent (7) on (2) and exponent (2) on (5). Step 2: The larger exponent is (7), so the decimal terminates in seven places. Step 3: Comparing exponents gives the answer directly.

Step 1: The denominator is (2^3\times5^3). Step 2: Both exponents are (3), so the denominator becomes like (10^3). Step 3: Therefore, the decimal terminates in three places.

Step 1: (343=7^3). Step 2: The denominator contains (7), not (2) or (5), so the decimal does not terminate. Step 3: Since the fraction is rational, the non-terminating decimal is recurring.

Step 1: (0.0005=\frac{5}{10000}). Step 2: Dividing by (5) gives (\frac{1}{2000}). Step 3: Count zeros carefully in very small terminating decimals.

Step 1: A terminating decimal has a finite number of digits after the decimal point. Step 2: (2.375) stops after three decimal places, so it is terminating. Step 3: If digits continue endlessly after the point, it is not terminating.

Step 1: (\sqrt{2}) is not rational. Step 2: The decimal expansion of an irrational number is non-terminating and non-recurring. Step 3: Rational numbers do not behave this way; they terminate or repeat.

Step 1: (50=2\times5^2). Step 2: The denominator has only (2) and (5), so (\frac{3}{50}) gives a terminating decimal. Step 3: Decide by prime factors of the denominator, not by a rough divisibility idea.

Step 1: A denominator made of powers of (2) and (5) can be converted into a power of (10). Step 2: When the denominator becomes like (10^k), the decimal terminates. Step 3: In assertion-reason questions, check whether the reason also explains the assertion.

Step 1: The negative sign in (-\frac{17}{200}) only makes the value negative. Step 2: Since (200=2^3\times5^2), the denominator has only (2) and (5), so the decimal terminates. Step 3: To decide the decimal type, check the denominator in lowest form, not the negative sign.

Step 1: (32=2^5). Step 2: The denominator has only (2), so the decimal terminates and may go up to five places. Step 3: The highest power of (2) or (5) gives the number of decimal places.

Step 1: Dividing (4) by (9) gives the digit (4) repeatedly. Step 2: Therefore, (\frac{4}{9}=0.\overline{4}). Step 3: Put the repeating digit under the bar.

Step 1: (\frac{18}{48}=\frac{3}{8}). Step 2: Since (8=2^3), the reduced denominator has only (2), so the decimal terminates. Step 3: Always apply the rule to the reduced fraction.

Step 1: (\frac{21}{28}=\frac{3}{4}). Step 2: (\frac{3}{4}=\frac{75}{100}=0.75). Step 3: If numerator and denominator look large, reduce the fraction first.

Step 1: The denominator has exponent (2) on (2) and exponent (4) on (5). Step 2: The larger exponent is (4), so the decimal terminates within four places. Step 3: In such questions, the larger exponent gives the answer.

Step 1: (18=2\times3^2). Step 2: The denominator contains (3), so (\frac{7}{18}) will not terminate. Step 3: In options, identify the denominator that has a factor other than (2) and (5).

Step 1: (0.375=\frac{375}{1000}). Step 2: Reducing by (125) gives (\frac{3}{8}). Step 3: For three decimal places, first use denominator (1000) and then reduce.

Step 1: In (0.\overline{12}), the block (12) repeats. Step 2: A recurring decimal is rational, but it is not terminating. Step 3: A non-terminating rational decimal always has a fixed repeat.

Step 1: (625=5^4). Step 2: Multiplying by (2^4) can make the denominator (10^4). Step 3: Therefore, the decimal terminates in four places.

Step 1: The negative sign does not change the type of decimal expansion. Step 2: (28=2^2\times7), so the factor (7) makes the decimal non-terminating recurring. Step 3: Check the denominator in lowest form, not the sign.