Decimal expansion of rational numbers

Step 1: (\frac{63}{245}) simplifies by (7) to (\frac{9}{35}). Step 2: Since (35=5\times7), factor (7) remains and the decimal will not terminate. Step 3: Exam tip: Having (5) in the denominator is not enough; only (2) and (5) should remain.

Step 1: (32=2^5), so the decimal ends after (5) places. Step 2: (32\times3125=100000), so (\frac{3}{32}=\frac{9375}{100000}=0.09375). Step 3: Exam tip: Count the total decimal places carefully while placing the decimal point.

Step 1: (0.0625=\frac{625}{10000}). Step 2: Simplifying by (625) gives (\frac{1}{16}). Step 3: Exam tip: Remembering common decimals like (0.0625), (0.125), and (0.25) is useful.

Step 1: The denominator (2^4\times5^4=(2\times5)^4=10^4). Step 2: So the decimal terminates after (4) places. Step 3: Exam tip: Equal powers of (2) and (5) directly form a power of (10).

Step 1: (2000=2^4\times5^3). Step 2: The larger exponent is (4), so the decimal terminates after (4) places. Step 3: Exam tip: Imagine converting the denominator to (10000); the number of places becomes clear.

Step 1: In (0.135135135\ldots), the block (135) repeats. Step 2: A recurring decimal is rational, but it does not terminate. Step 3: Exam tip: If a non-terminating decimal has a regular repeated block, treat it as rational.

Step 1: (\frac{18}{225}) simplifies by (9) to (\frac{2}{25}). Step 2: Since (25=5^2), the decimal terminates after (2) places. Step 3: Exam tip: The original denominator had (3^2), but it disappeared after simplification.

Step 1: (\frac{44}{242}) simplifies by (22) to (\frac{2}{11}). Step 2: The denominator (11) is not made of (2) and (5), so the decimal is non-terminating recurring. Step 3: Exam tip: Simplify every option before making the final choice.

Step 1: (0.48) stops after two decimal places. Step 2: In (0.\overline{48}), the block (48) repeats, so it is non-terminating recurring. Step 3: Exam tip: Keep the difference between stopping decimals and repeating decimals clear.

Step 1: The numerator (5) cancels one factor of (5) from (5^2) in the denominator. Step 2: The reduced denominator still has (2^3\times3\times5), so factor (3) remains. Step 3: Exam tip: If (3) remains after cancellation, the decimal will be recurring.

Step 1: The denominator has only factors (2) and (5). Step 2: The exponent of (2) is (5), and the exponent of (5) is (3), so the larger exponent is (5). Step 3: Exam tip: Think of making the denominator like (10^5).

Step 1: (80\times125=10000). Step 2: (\frac{7}{80}=\frac{875}{10000}=0.0875). Step 3: Exam tip: Missing the zero after the decimal point changes the answer.

Step 1: (\frac{27}{375}) simplifies by (3) to (\frac{9}{125}). Step 2: Since (125=5^3), the decimal terminates. Step 3: Exam tip: In assertion-reason questions, check whether the reason is true and also explains the assertion.

Step 1: (0.\overline{27}=\frac{27}{99}). Step 2: Simplifying (\frac{27}{99}) by (9) gives (\frac{3}{11}). Step 3: Exam tip: For a two-digit repeating part, denominator (99) quickly gives the answer.

Step 1: (\frac{16}{1250}) simplifies by (2) to (\frac{8}{625}). Step 2: Since (625=5^4), the decimal terminates after (4) places. Step 3: Exam tip: Simplification can change the number of decimal places.

Step 1: (\frac{77}{308}) simplifies by (77) to (\frac{1}{4}). Step 2: (\frac{1}{4}=0.25). Step 3: Exam tip: Spotting the common factor in large numbers saves the most time.

Step 1: The denominator has only (2) and (5), so the decimal terminates. Step 2: The powers are (2) for (2) and (3) for (5), so the larger exponent is (3). Step 3: Exam tip: Since numerator (23) is coprime with the denominator, count places directly from the denominator.

Step 1: Multiply denominator (20) by (5) to make (100). Step 2: (\frac{7}{20}=\frac{35}{100}=0.35). Step 3: Exam tip: With denominator (20), make (100) for a quick answer.

Step 1: (36=2^2\times3^2). Step 2: Since the fraction is in lowest form, (3^2) remains in the denominator. Step 3: Exam tip: If factor (3) remains in the reduced denominator, the decimal does not terminate.

Step 1: (0.0125=\frac{125}{10000}). Step 2: Simplifying by (125), we get (\frac{1}{80}). Step 3: Exam tip: Count zeros very carefully in small decimals.

Step 1: (125=5^3), so (5^3) cancels from (5^5) in the denominator. Step 2: The reduced denominator is (2^4\times5^2), whose larger exponent is (4). Step 3: Exam tip: Factors (2) or (5) in the numerator can reduce the denominator powers.

Step 1: (\frac{91}{182}=\frac{1}{2}), (\frac{39}{156}=\frac{1}{4}), and (\frac{68}{170}=\frac{2}{5}). Step 2: (\frac{51}{119}) does not reduce, and (119=7\times17). Step 3: Exam tip: In multi-option questions, reduce each fraction before choosing the non-terminating one.

Step 1: (6250=2\times5^5). Step 2: The fraction is in lowest form, and the larger exponent is (5). Step 3: Exam tip: For denominators like (6250), identifying the power of (5) quickly gives the answer.

Step 1: (14=2\times7) and (28=2^2\times7). Step 2: Factor (7) remains in both denominators, so both decimals do not terminate. Step 3: Exam tip: In comparison questions, prime-factorise both denominators separately.

Step 1: In (0.12\overline{3}=0.123333\ldots), (12) is the non-repeating part and (3) is the repeating part. Step 2: The fraction is (\frac{123-12}{900}=\frac{111}{900}=\frac{37}{300}). Step 3: Exam tip: Identify the non-repeating and repeating parts before placing (9) and (0) in the denominator.