Decimal expansion of rational numbers

Step 1: (0.5) can be written as (\frac{5}{10}). Step 2: On simplifying, it becomes (\frac{1}{2}). Step 3: Exam tip: Convert a terminating decimal into a fraction using place value first.

Step 1: For a terminating decimal, the denominator must have only (2) and (5) as prime factors. Step 2: Since (8=2^3), (\frac{7}{8}) terminates. Step 3: Exam tip: Always prime-factorise the denominator first.

Step 1: The rule is applied only after reducing the fraction to lowest form. Step 2: If the denominator (q) is of the form (2^m5^n), the decimal terminates. Step 3: Exam tip: Do not decide before simplifying the fraction.

Step 1: (40=2^3\times5). Step 2: The larger power of (2) and (5) is (3), so the decimal ends after (3) places. Step 3: Exam tip: Use the larger exponent to count decimal places.

Step 1: (\frac{3}{6}) simplifies to (\frac{1}{2}). Step 2: The denominator is (2), so the decimal ends at (0.5). Step 3: Exam tip: Do not judge from the original denominator before reducing.

Step 1: (\frac{7}{28}=\frac{1}{4}). Step 2: Since (4=2^2), the decimal terminates. Step 3: Exam tip: The lowest form gives the correct decision.

Step 1: (\frac{5}{12}) is already in lowest form. Step 2: (12=2^2\times3), and the factor (3) prevents termination. Step 3: Exam tip: A factor other than (2) or (5) gives a recurring decimal.

Step 1: A rational number can be written as (\frac{p}{q}). Step 2: Its decimal either terminates or repeats a block of digits. Step 3: Exam tip: Non-terminating non-recurring decimals are linked with irrational numbers.

Step 1: (0.125=\frac{125}{1000}). Step 2: Simplifying (\frac{125}{1000}) gives (\frac{1}{8}). Step 3: Exam tip: For three decimal places, first use denominator (1000).

Step 1: Multiply denominator (25) by (4) to make (100). Step 2: (\frac{2}{25}=\frac{8}{100}=0.08). Step 3: Exam tip: Make the denominator (10), (100), or (1000) for quick conversion.

Step 1: (\frac{6}{15}) simplifies to (\frac{2}{5}). Step 2: (\frac{2}{5}=\frac{4}{10}=0.4). Step 3: Exam tip: Simplifying first makes decimal conversion faster.

Step 1: (20=2^2\times5). Step 2: The larger exponent is (2), so the decimal terminates after (2) places. Step 3: Exam tip: The actual decimal (0.45) confirms the same result.

Step 1: (125\times8=1000). Step 2: (\frac{7}{125}=\frac{56}{1000}=0.056). Step 3: Exam tip: When making the denominator (1000), keep three decimal places carefully.

Step 1: (16=2^4). Step 2: The denominator has only (2), with exponent (4), so the decimal ends after (4) places. Step 3: Exam tip: (2^4) usually points to checking up to (4) decimal places.

Step 1: For a terminating decimal, the denominator must have only (2) and (5). Step 2: (100=2^2\times5^2), so it is suitable. Step 3: Exam tip: Denominators like (10), (100), and (1000) give terminating decimals.

Step 1: (\frac{1}{6}) is in lowest form. Step 2: (6=2\times3), and factor (3) makes the decimal non-terminating recurring. Step 3: Exam tip: An even denominator does not always mean termination.

Step 1: (625\times16=10000). Step 2: (\frac{4}{625}=\frac{64}{10000}=0.0064). Step 3: Exam tip: Count zeros carefully while placing the decimal point.

Step 1: Multiply denominator (50) by (2) to make (100). Step 2: (\frac{37}{50}=\frac{74}{100}=0.74). Step 3: Exam tip: When changing the denominator, multiply the numerator by the same number.

Step 1: In (0.333\ldots), the digit (3) repeats. Step 2: A recurring decimal is rational, so it can be written as a fraction. Step 3: Exam tip: A repeating digit is a sign of a rational number.

Step 1: Dividing (\frac{21}{14}) by (7) gives (\frac{3}{2}). Step 2: (\frac{3}{2}=1.5). Step 3: Exam tip: Reduce larger fractions before converting to decimals.

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.