Decimal expansion of rational numbers

Step 1: (\frac{1}{11}) is in lowest form. Step 2: The denominator (11) is not made of (2) or (5), so the decimal is non-terminating recurring. Step 3: Exam tip: Understand the difference between (0.09) and (0.\overline{09}).

Step 1: Look carefully at the decimal: (46), then (46), then (46) appears. Step 2: So the recurring block is (46). Step 3: Exam tip: Identifying the recurring block is the first step in converting it to a fraction.

Step 1: Let (x=0.\overline{7}). Step 2: Then (10x=7.\overline{7}), so (9x=7) and (x=\frac{7}{9}). Step 3: Exam tip: When one digit repeats, the denominator is often (9).

Step 1: Multiply denominator (5) by (2) to make (10). Step 2: (\frac{3}{5}=\frac{6}{10}=0.6). Step 3: Exam tip: Convert denominator (5) into (10) for quick answers.

Step 1: (0.75=\frac{75}{100}). Step 2: Simplifying by (25), we get (\frac{3}{4}). Step 3: Exam tip: It is useful to remember fraction forms of decimals like (0.25), (0.5), and (0.75).

Step 1: (250=2\times5^3). Step 2: The larger exponent is (3), so the decimal ends after (3) places. Step 3: Exam tip: Thinking of converting (250) to (1000) gives the same answer.

Step 1: Check the denominator of the fraction in lowest form. Step 2: (14=2\times7), and factor (7) prevents termination. Step 3: Exam tip: If (2) is joined by another prime like (7), the decimal will recur.

Step 1: (\frac{1}{125}) is in lowest form. Step 2: The denominator (125=5^3), so it has only factor (5), and the decimal terminates. Step 3: Exam tip: In assertion-reason questions, also check whether the reason truly explains the assertion.

Step 1: (\frac{45}{120}) simplifies by (15) to (\frac{3}{8}). Step 2: Since (8=2^3), the decimal expansion is terminating. Step 3: Exam tip: Do not decide quickly from the original denominator; reduce the fraction first.

Step 1: The denominator has only (2) and (5) as prime factors, so the decimal terminates. Step 2: The exponents are (2) and (4), and the larger exponent is (4). Step 3: Exam tip: For a terminating decimal, use the larger exponent of (2) and (5) to count decimal places.

Step 1: (\frac{27}{150}) simplifies by (3) to (\frac{9}{50}). Step 2: Since (50=2\times5^2), the denominator has only (2) and (5). Step 3: Exam tip: Always reduce the fraction to lowest form before deciding the decimal type.

Step 1: (\frac{56}{180}) simplifies by (4) to (\frac{14}{45}). Step 2: Since (45=3^2\times5), the denominator still contains (3). Step 3: Exam tip: Even if (5) is present, a remaining factor (3) makes the decimal recurring.

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

Step 1: (\frac{39}{520}) simplifies by (13) to (\frac{3}{40}). Step 2: Since (40=2^3\times5), the larger exponent is (3). Step 3: Exam tip: Simplifying first gives the correct number of decimal places.

Step 1: (24=2^3\times3), so factor (3) in the denominator is the obstacle. Step 2: (\frac{7}{24}\times3=\frac{21}{24}=\frac{7}{8}), whose denominator is (2^3). Step 3: Exam tip: In such questions, think of cancelling unwanted denominator factors after multiplication.

Step 1: The denominator has only (2) and (5), so the decimal terminates. Step 2: The power of (2) is (4), and the power of (5) is (2), so the larger power is (4). Step 3: Exam tip: Focus more on prime powers of the denominator than the numerator.

Step 1: (\frac{65}{312}) is in lowest form because there is no common factor. Step 2: (312=2^3\times3\times13), so the denominator contains (3) and (13). Step 3: Exam tip: A non-terminating decimal of a rational fraction is always recurring.

Step 1: (16\times625=10000). Step 2: (\frac{7}{16}=\frac{4375}{10000}=0.4375). Step 3: Exam tip: For denominators like (16), (32), and (64), making a power of (10) is useful.

Step 1: (0.3125=\frac{3125}{10000}). Step 2: Dividing numerator and denominator by (625), we get (\frac{5}{16}). Step 3: Exam tip: For four decimal places, start with denominator (10000) and then simplify.

Step 1: (40=2^3\times5). Step 2: The larger exponent is (3), so there can be at most (3) decimal places. Step 3: Exam tip: Whatever (a) is, the denominator in lowest form decides the decimal places.