Decimal expansion of rational numbers

Step 1: Multiply (20) by (5) to make (100). Step 2: (\frac{3}{20}=\frac{15}{100}=0.15). Step 3: Converting the denominator to (10), (100), or (1000) is a quick method.

Step 1: Multiply (25) by (4) to make (100). Step 2: (\frac{7}{25}=\frac{28}{100}=0.28). Step 3: When converting to decimal, multiply numerator and denominator by the same number.

Step 1: Dividing (1) by (3) gives the digit (3) repeatedly. Step 2: Hence, (\frac{1}{3}=0.\overline{3}). Step 3: The bar means that the digit repeats continuously.

Step 1: (\frac{1}{11}=0.\overline{09}). Step 2: Multiplying by (2) gives (\frac{2}{11}=0.\overline{18}). Step 3: Put the complete repeating block under the bar.

Step 1: A decimal with a fixed repeating block is called a recurring decimal. Step 2: Every recurring decimal can be written as a fraction, so it is rational. Step 3: A fixed repeat is a strong sign of rationality.

Step 1: (14=2\times7). Step 2: The factor (7) prevents termination, and because the number is rational, the decimal is recurring. Step 3: Even one extra prime factor stops termination.

Step 1: (250=2\times5^3). Step 2: The larger exponent is (3), so the decimal terminates within three places. Step 3: Comparing exponents saves time in exams.

Step 1: (8=2^3). Step 2: The denominator has only (2), so (\frac{45}{8}) has a terminating decimal. Step 3: Even if the numerator is larger, apply the rule to the denominator.

Step 1: The fraction is in lowest form and the denominator is (7). Step 2: Since (7) is neither (2) nor (5), the decimal is non-terminating recurring. Step 3: Examples like (\frac{1}{7}) make the rule easy to remember.

Step 1: (0.6=\frac{6}{10}). Step 2: Reducing (\frac{6}{10}) by (2) gives (\frac{3}{5}). Step 3: After converting a decimal to a fraction, always reduce it.

Step 1: (80=2^4\times5). Step 2: The denominator contains only (2) and (5), so the decimal terminates. Step 3: Do not worry about a large denominator; just factor it.

Step 1: (\frac{12}{30}=\frac{2}{5}). Step 2: The reduced denominator is (5), so the decimal terminates. Step 3: Do not call it recurring just because the original denominator (30) contains (3).

Step 1: (64=2^6). Step 2: The denominator has only (2), so (\frac{17}{64}) gives a terminating decimal. Step 3: Choose the denominator that contains only (2) and (5).

Step 1: (12=2^2\times3). Step 2: (\frac{5}{12}) is in lowest form and the denominator contains (3), so its decimal is recurring. Step 3: Check both reduction and extra prime factors.

Step 1: (0.125=\frac{125}{1000}). Step 2: Reducing gives (\frac{125}{1000}=\frac{1}{8}). Step 3: For three decimal places, start with denominator (1000).

Step 1: (0.04=\frac{4}{100}). Step 2: Dividing by (4) gives (\frac{1}{25}). Step 3: In decimals with zeros, counting decimal places is very important.

Step 1: (\frac{7}{20}=\frac{35}{100}). Step 2: Its decimal is (0.35), which has exactly two decimal places. Step 3: When exact places are asked, verify by writing the decimal.

Step 1: (3125=5^5). Step 2: The denominator can be made (10^5) by multiplying by (2^5). Step 3: Therefore, the decimal terminates after five places.

Step 1: For a terminating decimal, the denominator should have only (2) and (5). Step 2: If (3) remains, the decimal will not terminate and will be recurring because the number is rational. Step 3: The remaining factors in lowest form decide the result.

Step 1: (500=2^2\times5^3). Step 2: It has only (2) and (5), so the decimal terminates, and the larger exponent is (3). Step 3: When both type and places are asked, check both.