Question

Find five rational numbers between:
(i) \[\dfrac{2}{3}{\rm{ }}\,and\,{\rm{ }}\dfrac{4}{5}\]
(ii) \[\dfrac{{ - 3}}{2}{\rm{ }}\,and\,{\rm{ }}\dfrac{5}{3}\]
(iii) \[\dfrac{1}{4}{\rm{ }}\,and\,{\rm{ }}\dfrac{1}{2}\]

Answer 1

Hint: Firstly, find an appropriate number that we can use to make all the denominators the same and then multiply denominator and numeration with the same number to get five rational numbers.

Complete step-by-step answer:
 Let us do all the parts and calculate five rational numbers :
(i) \[\dfrac{2}{3}{\rm{ }}\,and\,{\rm{ }}\dfrac{4}{5}\]
 Here, we will multiply 15 to both denominator and numerator to make the denominator as 45 to \[\dfrac{2}{3}\] .
First of all multiply \[\dfrac{2}{3}\] with \[\dfrac{{15}}{{15}}\]
We get, \[\dfrac{{2 * 15}}{{3 * 15}}\] = \[\dfrac{{30}}{{45}}\]
Here, we will multiply 9 to both denominator and numerator to make the denominator as 45 to \[\dfrac{4}{5}\]. Now we will multiply \[\dfrac{4}{5}\] with \[\dfrac{9}{9}\] .
We get, \[\dfrac{{4 * 9}}{{5 * 9}}\] = \[\dfrac{{36}}{{45}}\]
Now we have to find five rational numbers between \[\dfrac{{30}}{{45}}\] and \[\dfrac{{36}}{{45}}\] .
So, five rational numbers are : \[\dfrac{{31}}{{45}}\] , \[\dfrac{{32}}{{45}}\] , \[\dfrac{{33}}{{45}}\] , \[\dfrac{{34}}{{45}}\] , \[\dfrac{{35}}{{45}}\] .
(ii) \[\dfrac{{ - 3}}{2}{\rm{ }}and{\rm{ }}\dfrac{5}{3}\]
 Here, we will multiply 3 to both denominator and numerator to make the denominator as 6 to \[\dfrac{{ - 3}}{2}\] .
First of all multiply \[\dfrac{{ - 3}}{2}\] with \[\dfrac{3}{3}\]
We get, \[\dfrac{{ - 3 * 3}}{{2 * 3}}\] = \[\dfrac{{ - 9}}{6}\]
Here, we will multiply 2 to both denominator and numerator to make the denominator as 6 to \[\dfrac{5}{3}\]. Now we will multiply \[\dfrac{5}{3}\] with \[\dfrac{2}{2}\] .
We get, \[\dfrac{{5 * 2}}{{3 * 2}}\] = \[\dfrac{{10}}{6}\]
Now we have to find five rational numbers between \[\dfrac{{ - 9}}{6}\] and \[\dfrac{{10}}{6}\] .
So, five rational numbers are : \[\dfrac{{ - 1}}{6}\] , \[\dfrac{1}{6}\] , \[\dfrac{2}{6}\] , \[\dfrac{3}{6}\] , \[\dfrac{5}{6}\] .
(iii) \[\dfrac{1}{4}{\rm{ }}and{\rm{ }}\dfrac{1}{2}\]
 Here, we will multiply 7 to both denominator and numerator to make denominator as 28 to \[\dfrac{1}{4}\] .
First of all multiply \[\dfrac{1}{4}\] with \[\dfrac{7}{7}\]
We get, \[\dfrac{{1 * 7}}{{4 * 7}}\] = \[\dfrac{7}{{28}}\]
Here, we will multiply 14 to both denominator and numerator to make the denominator as 28 to \[\dfrac{1}{2}\]. Now we will multiply \[\dfrac{1}{2}\] with \[\dfrac{{14}}{{14}}\] .
We get, \[\dfrac{{1 * 14}}{{2 * 14}}\] = \[\dfrac{{14}}{{28}}\]
Now we have to find five rational numbers between \[\dfrac{7}{{28}}\] and \[\dfrac{{14}}{{28}}\] .
So, five rational numbers are : \[\dfrac{8}{{28}}\] , \[\dfrac{9}{{28}}\] , \[\dfrac{{10}}{{28}}\] , \[\dfrac{{11}}{{28}}\] , \[\dfrac{{12}}{{28}}\] .

Note: As we see in this question you just have to make the denominator the same. Then use that to calculate the first five rational numbers between both the updated fraction numbers.