Decimal expansion of rational numbers

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.

Step 1: The number of places in a terminating decimal is decided by the larger exponent of (2) and (5). Step 2: Here the larger exponent is (7). Step 3: Therefore the decimal terminates after (7) places.

Step 1: (\frac{45}{360}=\frac{1}{8}). Step 2: Since (8=2^3), the decimal terminates after (3) places. Step 3: Do not get confused by the original denominator (360); check the reduced denominator.

Step 1: The denominator has exponent (2) for (2) and exponent (4) for (5). Step 2: The larger exponent is (4). Step 3: So the decimal terminates after (4) places.

Step 1: (\frac{16}{28}=\frac{4}{7}). Step 2: The reduced denominator is (7), which is neither (2) nor (5). Step 3: Hence the decimal expansion is non-terminating recurring.

Step 1: A rational number can be written as (\frac{p}{q}). Step 2: Its decimal expansion either terminates or has a fixed repetition. Step 3: A non-terminating non-recurring decimal is not rational.

Step 1: (13) is neither (2) nor (5). Step 2: If (13) remains in the reduced denominator, the decimal cannot terminate. Step 3: Since it is rational, the non-terminating decimal will be recurring.

Step 1: (250=2\times5^3). Step 2: The denominator contains only (2) and (5). Step 3: The larger exponent is (3), so the decimal terminates after (3) places.

Step 1: (2000=2^4\times5^3). Step 2: The larger exponent is (4). Step 3: Therefore the decimal expansion terminates after (4) places.

Step 1: (128=2^7). Step 2: The denominator has only (2), so the decimal terminates. Step 3: Since the exponent of (2) is (7), it terminates after (7) places.

Step 1: (3125=5^5). Step 2: The denominator has only (5), so the decimal terminates. Step 3: Since the exponent of (5) is (5), it terminates after (5) places.

Step 1: The number of places is decided by the larger exponent of (2) and (5). Step 2: The exponent of (5) is (3), so for exactly (6) places the exponent of (2) must be (6). Step 3: Match the larger exponent with the required decimal places.

Step 1: The terminating decimal rule applies to the denominator in lowest form. Step 2: The original denominator may contain factors that cancel out. Step 3: So reduce the fraction first, then factorise the denominator.

Step 1: (\frac{52}{195}=\frac{4}{15}). Step 2: The reduced denominator is (15=3\times5), which still has (3). Step 3: Therefore the decimal expansion is non-terminating recurring.

Step 1: (\frac{84}{210}=\frac{2}{5}). Step 2: The reduced denominator is (5), so the decimal terminates. Step 3: Extra factors in the original denominator do not matter after cancellation.

Step 1: (22=2\times11). Step 2: The reduced denominator contains (11), so the decimal will not terminate. Step 3: Since it is a rational fraction, it gives a non-terminating recurring decimal.