Decimal expansion of rational numbers

Step 1: (500=2^2\times5^3). Step 2: The larger exponent is (3), so the decimal terminates after (3) places. Step 3: Thinking of making (500) into (1000) also helps.

Step 1: (\frac{81}{90}=\frac{9}{10}). Step 2: The reduced denominator is (10=2\times5). Step 3: Even if the original denominator has (3), apply the rule to the reduced denominator.

Step 1: (\frac{37}{74}=\frac{1}{2}). Step 2: The reduced denominator is (2), so the decimal terminates. Step 3: Cancel common factors before deciding.

Step 1: (\frac{121}{242}=\frac{1}{2}). Step 2: (\frac{1}{2}=0.5), so the decimal terminates after (1) place. Step 3: Do not be distracted by large numbers; reduce the fraction first.

Step 1: (11) is neither (2) nor (5). Step 2: If (11) remains in the reduced denominator, the decimal cannot terminate. Step 3: Since the number is rational, its non-terminating decimal will be recurring.

Step 1: To make (10^5), the denominator should be (2^5\times5^5). Step 2: It already has (2^3\times5^5), so it lacks (2^2). Step 3: Making the denominator a power of (10) reveals the decimal places clearly.

Step 1: (\frac{21}{56}=\frac{3}{8}). Step 2: The reduced denominator is (8=2^3), so it contains only the prime factor (2). Step 3: In exams, do not decide from the original denominator; reduce the fraction first.

Step 1: (75=3\times5^2), and (\frac{13}{75}) is already in lowest form. Step 2: The reduced denominator contains (3), which is neither (2) nor (5). Step 3: If the denominator has a prime factor other than (2) and (5), the decimal is non-terminating recurring.

Step 1: For a terminating decimal, the number of places is decided by the larger exponent of (2) and (5). Step 2: Here the exponent of (2) is (4), and the exponent of (5) is (6). Step 3: The larger exponent is (6), so the decimal terminates after (6) places.

Step 1: In (0.04\overline{7}), the digit (7) repeats. Step 2: The decimal does not terminate, but it has a fixed repeating pattern. Step 3: A non-terminating decimal with a fixed repetition represents a rational number.

Step 1: (80=2^4\times5). Step 2: The denominator has only (2) and (5), so the decimal terminates. Step 3: The larger exponent is (4), so it terminates after (4) places.

Step 1: (90=2\times3^2\times5). Step 2: The fraction is in lowest form and (3) remains in the denominator. Step 3: If a reduced denominator has a prime other than (2) and (5), the decimal is non-terminating recurring.

Step 1: (\frac{36}{144}=\frac{1}{4}). Step 2: The reduced denominator is (4=2^2), so it has only (2). Step 3: Always reduce the fraction before deciding the decimal type.

Step 1: (320=2^6\times5). Step 2: The denominator has only (2) and (5), so the decimal terminates. Step 3: The larger exponent is (6), so it terminates within (6) places.

Step 1: (6250=2\times5^5). Step 2: The denominator contains only (2) and (5). Step 3: The larger exponent is (5), so the decimal terminates after (5) places.

Step 1: (\frac{18}{42}=\frac{3}{7}). Step 2: The reduced denominator is (7), which is neither (2) nor (5). Step 3: If another prime remains in the reduced denominator, the decimal is non-terminating recurring.

Step 1: (\frac{49}{140}=\frac{7}{20}). Step 2: The reduced denominator is (20=2^2\times5), which has only (2) and (5). Step 3: Therefore the decimal expansion is terminating.

Step 1: (0.0625=\frac{625}{10000}). Step 2: Reducing it gives (\frac{1}{16}). Step 3: Write a terminating decimal over a power of (10), then reduce it.

Step 1: The repeating block is (18), so (0.\overline{18}=\frac{18}{99}). Step 2: (\frac{18}{99}=\frac{2}{11}). Step 3: The number of (9)s in the denominator equals the number of repeating digits.

Step 1: The given decimal does not end. Step 2: It has no fixed repeating block. Step 3: Such a decimal is called non-terminating non-recurring.