Question

Select those which can be written as a rational number with denominator \[4\] in their lowest form:
\[\dfrac{7}{8}\], \[\dfrac{{64}}{{16}}\], \[\dfrac{{36}}{{ - 12}}\], \[\dfrac{{ - 16}}{{17}}\], \[\dfrac{5}{{ - 4}}\], \[\dfrac{{140}}{{28}}\]

Answer 1

Hint: First we know that the rational numbers are said to be in the lowest form (simplest form), if both the numerator and denominator have no other common factor other than \[1\]. i.e., a Rational Number \[\dfrac{a}{b}\] is said to be in its simplest form if the HCF (Highest common factor) of \[a,b\] is \[1\] which mains \[a,b\] are relatively prime. So, each rational number can be reduced by multiplying and dividing by HCF of its numerator and denominator.

Complete answer:
HCF of \[7,8\] is \[1\]
Then the lowest form of \[\dfrac{7}{8}\]\[ = \dfrac{7}{8}\].
HCF of \[64,16\] is \[16\]
Then the lowest form of \[\dfrac{{64}}{{16}} = \dfrac{{64}}{{16}} \times \dfrac{{16}}{{16}} = 4\].
HCF of \[36,12\] is \[12\]
Then the lowest form of \[\dfrac{{64}}{{16}} = \dfrac{{64}}{{16}} \times \dfrac{{16}}{{16}} = 4\].
HCF of \[16,17\] is \[1\]
Then the lowest form of \[\dfrac{{ - 16}}{{17}} = \dfrac{{ - 16}}{{17}}\].
HCF of \[5,4\] is \[1\]
Then the lowest form of \[\dfrac{5}{{ - 4}} = - \dfrac{5}{4}\].
HCF of \[140,28\] is \[28\]
Then the lowest form of \[\dfrac{{140}}{{28}} = 5\].
Hence the rational number which can be written as a rational number with denominator 4 in their lowest form is \[\dfrac{5}{{ - 4}}\].

Note: Note that a rational number is a number that is of the form\[\dfrac{a}{b}\] where \[a\]and \[b\]are integers and \[b \ne 0\]. Also note that a real number is said to be a rational number if it has either finite decimal expansion or infinite repeating decimal expansion.