Decimal expansion of rational numbers

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.

Step 1: (45=3^2\times5). Step 2: Because (3) is present in the denominator, the decimal will not terminate. Step 3: Since it is rational, the decimal will be non-terminating recurring.

Step 1: A non-terminating recurring decimal occurs when the reduced denominator has a prime factor other than (2) and (5). Step 2: Such a denominator cannot be made into a power of (10). Step 3: So always check the prime factors of the denominator.

Step 1: Four decimal places mean ten-thousandths. Step 2: So the number can be written as (\frac{n}{10000}). Step 3: In exams, reduce the fraction afterward.

Step 1: (0.875=\frac{875}{1000}). Step 2: Reducing gives (\frac{875}{1000}=\frac{7}{8}). Step 3: Count the decimal places and use the corresponding power of (10) as denominator.

Step 1: The repeating block is (45), so (0.\overline{45}=\frac{45}{99}). Step 2: (\frac{45}{99}=\frac{5}{11}). Step 3: Write as many (9)s in the denominator as the number of repeating digits.