Decimal expansion of rational numbers

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.

Step 1: (\frac{84}{350}) simplifies by (14) to (\frac{6}{25}). Step 2: Since (25=5^2), the decimal terminates after (2) places. Step 3: Exam tip: Count decimal places only after simplifying the fraction.

Step 1: (0.0\overline{6}=0.0666\ldots). Step 2: It equals (\frac{1}{15}) because (\frac{1}{15}=0.0666\ldots). Step 3: Exam tip: Separate the non-repeating part and the repeating part after the decimal point.

Step 1: The denominator (q) has (3) along with (2) and (5). Step 2: Since (3) remains in lowest form, the decimal will not terminate. Step 3: Exam tip: If any prime factor other than (2) and (5) remains in the denominator, the decimal is recurring.

Step 1: (\frac{121}{440}) simplifies by (11) to (\frac{11}{40}). Step 2: Since (40=2^3\times5), the decimal terminates. Step 3: Exam tip: A factor like (11) in the original denominator may disappear after simplification.

Step 1: (\frac{18}{75}) simplifies by (3) to (\frac{6}{25}), which is terminating, so it must be checked again. Step 2: (\frac{35}{56}=\frac{5}{8}), (\frac{49}{98}=\frac{1}{2}), and (\frac{22}{125}) are also terminating. Step 3: Exam tip: Here no option is non-terminating recurring, so the given option set has no valid answer.

Step 1: (\frac{72}{540}) simplifies by (36) to (\frac{2}{15}). Step 2: Since (15=3\times5), factor (3) remains in the denominator, so the decimal will not terminate. Step 3: Exam tip: If a factor other than (2) and (5) remains in the reduced denominator, the decimal is recurring.

Step 1: (625\times16=10000). Step 2: (\frac{91}{625}=\frac{1456}{10000}=0.1456). Step 3: Exam tip: When the denominator is (625), multiply by (16) to make (10000).

Step 1: The denominator contains only (2) and (5), so the decimal terminates. Step 2: The exponents are (6) and (2), and the larger exponent is (6). Step 3: Exam tip: The maximum number of decimal places is decided by the larger exponent.

Step 1: For (\frac{7}{125}), (125=5^3). Step 2: It can be converted to denominator (1000), so its decimal ends after (3) places. Step 3: Exam tip: When exact places are asked, check both the lowest form and the larger exponent.

Step 1: The two-digit recurring decimal (0.\overline{18}) is first written as (\frac{18}{99}). Step 2: Simplifying (\frac{18}{99}) by (9) gives (\frac{2}{11}). Step 3: Exam tip: For two repeating digits, using denominator (99) is a quick method.

Step 1: (\frac{143}{550}) simplifies by (11) to (\frac{13}{50}). Step 2: Since (50=2\times5^2), the decimal terminates. Step 3: Exam tip: This calculation shows that the correct decision is terminating decimal.

Step 1: (\frac{26}{143}) simplifies by (13) to (\frac{2}{11}). Step 2: The denominator (11) does not contain only (2) and (5), so the decimal is non-terminating recurring. Step 3: Exam tip: After simplification, use the remaining denominator as the final basis.

Step 1: (128=2^7), and the fraction is already in lowest form. Step 2: The denominator has only (2), so the decimal terminates and can go up to (7) places. Step 3: Exam tip: For a denominator (2^n), the decimal usually goes up to (n) places.

Step 1: (18=2\times3^2), so (3^2) in the denominator is the obstacle. Step 2: Multiplying by (3) gives (\frac{15}{18}=\frac{5}{6}), which still does not terminate. Step 3: Exam tip: The correct least multiplier should be (9), so none of the given options is suitable.

Step 1: (18=2\times3^2), and (3^2) must be removed from the denominator for a terminating decimal. Step 2: (\frac{5}{18}\times9=\frac{45}{18}=\frac{5}{2}), whose denominator is (2). Step 3: Exam tip: Remove the full remaining power of the unwanted prime factor.