Decimal expansion of rational numbers

Step 1: (\frac{96}{450}) simplifies by (6) to (\frac{16}{75}). Step 2: Since (75=3\times5^2), factor (3) remains in the denominator, so the decimal will not terminate. Step 3: Exam tip: If the reduced denominator has a factor other than (2) and (5), the decimal is recurring.

Step 1: (\frac{144}{320}) simplifies by (16) to (\frac{9}{20}). Step 2: Since (20=2^2\times5), the larger exponent is (2). Step 3: Exam tip: Decide decimal places from the denominator in lowest form.

Step 1: The denominator has only (2) and (5), so the decimal terminates. Step 2: The exponents are (5) and (3), and the larger exponent is (5). Step 3: Exam tip: For a terminating decimal, decimal places come from the larger exponent of (2) and (5).

Step 1: (221=13\times17) and (650=2\times5^2\times13). Step 2: After cancelling (13), we get (\frac{17}{50}), so the decimal actually terminates. Step 3: Exam tip: This calculation shows the correct decision is terminating, so the right choice should be (A).

There are infinitely many rational numbers with terminating decimal expansions. For example, 1/2 = 0.5, 1/4 = 0.25, 3/10 = 0.3, and every positive integer has a terminating decimal representation. More generally, numbers such as 1/10, 1/100, 1/1000, and so on are all distinct members. Therefore, the set is infinite, so option C is correct.

Step 1: In lowest form, the denominator is (8=2^3). Step 2: It contains only the prime factor (2), so the decimal expansion terminates. Step 3: In exams, always factorise the denominator first.

Step 1: (45=3^2\times5). Step 2: The denominator also has (3), so it is not made only of (2) and (5). Step 3: If any other prime remains in the reduced denominator, the decimal expansion is non-terminating recurring.

Step 1: (\frac{39}{312}=\frac{1}{8}). Step 2: The reduced denominator is (8=2^3), which contains only (2). Step 3: Do not judge from the original denominator; reduce the fraction first.

Step 1: (200=2^3\times5^2). Step 2: The larger exponent is (3), so the decimal terminates within (3) places. Step 3: For the number of decimal places, use the larger exponent of (2) and (5).

Step 1: (125=5^3). Step 2: The denominator has only (5) with exponent (3). Step 3: Such a fraction terminates after (3) decimal places.

Step 1: (\frac{14}{35}=\frac{2}{5}). Step 2: The reduced denominator is (5), so the decimal terminates. Step 3: The reduced form decides the decimal type.

Step 1: (28=2^2\times7). Step 2: The reduced denominator contains (7), which is not (2) or (5). Step 3: If another prime factor remains, the decimal is non-terminating recurring.

Step 1: (0.375=\frac{375}{1000}=\frac{3}{8}). Step 2: Since (8=2^3), the denominator has only (2). Step 3: Convert a terminating decimal to a fraction and check the denominator factors.

Step 1: In (0.\overline{6}), the digit (6) repeats. Step 2: A non-terminating repeating decimal represents a rational number. Step 3: Do not confuse recurring decimals with irrational numbers.

Step 1: The given decimal does not end. Step 2: It also has no fixed repeating block. Step 3: A non-terminating non-recurring decimal is associated with an irrational number.

Step 1: The number of decimal places is decided by the larger exponent of (2) and (5). Step 2: The larger exponent must be (4), so (n=4). Step 3: In such questions, identify the larger exponent directly.

Step 1: (\frac{64}{4000}=\frac{2}{125}). Step 2: Since (125=5^3), the decimal terminates after (3) places. Step 3: Assuming (4) places from (4000) without reducing is a common mistake.

Step 1: The denominator has exponent (4) for (2) and exponent (1) for (5). Step 2: The larger exponent is (4), so the decimal terminates after (4) places. Step 3: Do not add the exponents; take the larger one.

Step 1: (\frac{6}{15}=\frac{2}{5}). Step 2: The reduced denominator is (5), so the decimal terminates. Step 3: Even if the original denominator shows (3), decide only after reducing.

Step 1: For a terminating decimal, the reduced denominator must contain only (2) and (5). Step 2: (q=2^a5^b) exactly shows this form. Step 3: The numerator does not change the type once the fraction is in lowest form.

Step 1: In the reduced denominator, (3) is a prime other than (2) and (5). Step 2: So the decimal will not terminate but will repeat. Step 3: A non-terminating decimal of a rational number is recurring.

Step 1: (160=2^5\times5). Step 2: The larger exponent is (5), so the decimal terminates after (5) places. Step 3: Factorising and noting the larger exponent is the safest method.

Step 1: (625=5^4). Step 2: The denominator has only (5) with exponent (4). Step 3: Therefore the decimal terminates after (4) places.

Step 1: (40=2^3\times5). Step 2: The larger exponent is (3), so the decimal terminates after (3) places. Step 3: This is also confirmed by (\frac{1}{40}=0.025).

Step 1: (2500=2^2\times5^4). Step 2: The larger exponent is (4). Step 3: Hence the decimal terminates within (4) places.