Decimal expansion of rational numbers

Step 1: (\frac{18}{45}) simplifies by (9) to (\frac{2}{5}). Step 2: The denominator is (5), so the decimal terminates. Step 3: Exam tip: Do not get misled by denominator (45); check the lowest form.

Step 1: (\frac{22}{55}) simplifies to (\frac{2}{5}). Step 2: (\frac{2}{5}=0.4), so the decimal terminates. Step 3: Exam tip: Reduce the fraction before applying the denominator rule.

Step 1: (80=2^4\times5). Step 2: The larger exponent is (4), so the decimal has (4) places. Step 3: Exam tip: You can also check by writing (\frac{3}{80}=0.0375).

Step 1: In (0.727272\ldots), the block (72) repeats. Step 2: A repeating decimal is rational. Step 3: Exam tip: Do not only see that a decimal is long; check whether a fixed pattern repeats.

Step 1: The denominator has only (2) and (5), so the decimal terminates. Step 2: The exponents are (3) and (2), and the larger one is (3). Step 3: Exam tip: Maximum decimal places equal the larger exponent.

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.