Question

A rational number lie between \[\dfrac{1}{4}\] and \[\dfrac{1}{3}\] is--
A. \[\dfrac{7}{{24}}\]
B. \[0.29\]
C. \[\dfrac{{13}}{{48}}\]
D.All of these.

Answer 1

Hint: Firstly note the definition of rational number and irrational number. Rational number is a number that can be expressed in the form of \[\dfrac{p}{q}\] , where \[p\] and \[q\] are integers. More importantly \[q \ne 0\] . If \[q\] becomes zero we will get infinity. Irrational means no ratio. While in case of irrational numbers it cannot be expressed as a simple fraction. For example, \[\sqrt 2 \] and \[\sqrt 3 \] .

Complete step-by-step answer:
We know that if \[a\] and \[b\] are rational then \[\dfrac{{a + b}}{2}\] is also rational which lies between \[a\] and \[b\] .
Now, the rationales that lies between \[\dfrac{1}{4}\] and \[\dfrac{1}{3}\] is
 \[ = \dfrac{{\left( {\dfrac{1}{3} + \dfrac{1}{4}} \right)}}{2}\]
 L.C.M. of 3 and 4 is 12. On simplifying we get,
 \[ = \dfrac{{\left( {\dfrac{{4 + 3}}{{12}}} \right)}}{2}\]
 \[ = \dfrac{{\left( {\dfrac{7}{{12}}} \right)}}{2}\]
 \[ = \dfrac{7}{{24}}\]
Since \[\dfrac{7}{{24}} = 0.29166\] , \[\dfrac{1}{4} = 0.25\] and \[\dfrac{1}{3} = 0.33\] .
Since given two options are correct. Let's find out the other rational number lies between \[\dfrac{1}{4}\] and \[\dfrac{1}{3}\] .
Since \[\dfrac{7}{{24}}\] lies in between \[\dfrac{1}{4}\] and \[\dfrac{1}{3}\] . Then the rationales that lies between \[\dfrac{1}{4}\] and \[\dfrac{7}{{24}}\] is also lies between \[\dfrac{1}{4}\] and \[\dfrac{1}{3}\] .
Then, we haves
 \[ = \dfrac{{\left( {\dfrac{1}{4} + \dfrac{1}{{24}}} \right)}}{2}\]
L.C.M. of 4 and 24 is 24. On simplifying we get,
 \[ = \dfrac{{\left( {\dfrac{{6 + 7}}{{24}}} \right)}}{2}\]
 \[ = \dfrac{{\left( {\dfrac{{13}}{{24}}} \right)}}{2}\]
 \[ = \dfrac{{13}}{{48}}\] .
Thus, all the obtained rational between \[\dfrac{1}{4}\] and \[\dfrac{1}{3}\] are
 \[\dfrac{7}{{24}}\] , \[0.29\] and \[\dfrac{{13}}{{48}}\] .
So, the correct answer is “Option D”.

Note: We can also find the rational between \[\dfrac{1}{3}\] and \[\dfrac{7}{{24}}\] , which will also lies in \[\dfrac{1}{4}\] and \[\dfrac{1}{3}\] . But we had obtained the answer earlier so we stopped there only. Irrational numbers are not a finite number. Rational numbers are finite. We know that the sum of two rationals is rational. Follow the same procedure for finding rationales that lies between given two rationales.