Decimal expansion of rational numbers

Step 1: (2^4=16). Step 2: The denominator has only (2), with exponent (4), so the decimal ends after (4) places. Step 3: Exam tip: Think of converting a (2^n) denominator into (10^n).

Step 1: (5^3=125). Step 2: To make (125) into (1000), multiply by (8), so there are (3) decimal places. Step 3: Exam tip: A (5^n) denominator usually gives (n) decimal places.

Step 1: (\frac{11}{30}) is in lowest form. Step 2: (30=2\times3\times5), and the denominator contains (3). Step 3: Exam tip: Even if (2) and (5) are present, an extra factor (3) prevents termination.

Step 1: (1250=2\times5^4). Step 2: The larger exponent is (4), so the decimal ends after (4) places. Step 3: Exam tip: Even for a large denominator, prime powers quickly give the decimal-place count.

Step 1: (14) and (35) have common factor (7). Step 2: (\frac{14}{35}=\frac{2}{5}), so the decimal is (0.4). Step 3: Exam tip: Writing the lowest form often earns the main mark.

Step 1: (\frac{35}{70}) simplifies to (\frac{1}{2}). Step 2: (\frac{1}{2}=0.5), so the decimal terminates. Step 3: Exam tip: Extra factors in the original denominator may disappear after simplification.

Step 1: (0.2=\frac{2}{10}). Step 2: Simplifying (\frac{2}{10}) gives (\frac{1}{5}). Step 3: Exam tip: For one decimal place, start with denominator (10).

Step 1: (0.05=\frac{5}{100}). Step 2: Dividing by (5), we get (\frac{1}{20}). Step 3: Exam tip: For hundredths, begin with denominator (100).

Step 1: (40\times25=1000). Step 2: (\frac{3}{40}=\frac{75}{1000}=0.075). Step 3: Exam tip: A small numerator may lead to zeros after the decimal point.

Step 1: A rational number has a decimal that either terminates or recurs. Step 2: Non-terminating non-recurring decimal expansion is not possible for a rational number. Step 3: Exam tip: This difference helps identify rational and irrational numbers.

Step 1: A decimal with a fixed repeated block is called a recurring decimal. Step 2: Every recurring decimal can be written as a fraction, so it is rational. Step 3: Exam tip: When you see repetition, think rational number.

Step 1: (200=2^3\times5^2). Step 2: The larger exponent is (3), so the decimal ends after (3) places. Step 3: Exam tip: (\frac{17}{200}=0.085) shows three decimal places.

Step 1: For a terminating decimal, the denominator may have only (2) and (5) as factors. Step 2: (64=2^6), so the rule is satisfied. Step 3: Exam tip: Having only (2) in the denominator is also enough.

Step 1: (\frac{8}{15}) is in lowest form. Step 2: (15=3\times5), and factor (3) prevents termination. Step 3: Exam tip: A denominator with (5) and also (3) gives a recurring decimal.

Step 1: (\frac{15}{75}) simplifies by (15) to (\frac{1}{5}). Step 2: (\frac{1}{5}=0.2). Step 3: Exam tip: Spotting a large common factor saves time.

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.