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Find 10 rational numbers between \[\dfrac{3}{5}\] and \[\dfrac{3}{4}\] .

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Hint: In this question, we have been given to find 10 rational numbers between two rational numbers. For doing this, we are going to find the LCM of the denominators of the two numbers. Here, the numerators of the two given numbers are equal. Then we are going to first make the denominators of both the numbers equal. Then, we are going to check if the difference between the numerators is at least 10. If not, then we are going to multiply the numbers again by \[n\] such that \[difference \times n > 10\] . By doing that, we are going to have two numbers with equal denominators but unequal numerators with more than 10 numbers between the easily perceivable two numbers. Then, we just have to write the 10 numbers between the given range of numbers.

Complete step-by-step answer:
In this question, we have to find 10 rational numbers between \[\dfrac{3}{5}\] and \[\dfrac{3}{4}\] .
First, we find the LCM of the denominators ( \[4,5\] ), which is \[20\] .
Then, we make the denominators of the two numbers equal.
 \[\dfrac{3}{5} = \dfrac{3}{5} \times \dfrac{4}{4} = \dfrac{{12}}{{20}}\]
And \[\dfrac{3}{4} = \dfrac{3}{4} \times \dfrac{5}{5} = \dfrac{{15}}{{20}}\]
Now, the difference between the two numerators is \[3\] which is less than \[10\] .
So, we multiply the difference \[\left( 3 \right)\] by the least number which makes the product greater than \[10\] ; \[12\] .
So, \[\dfrac{3}{5} = \dfrac{{12}}{{20}} \times \dfrac{4}{4} = \dfrac{{48}}{{80}}\]
And \[\dfrac{3}{4} = \dfrac{{15}}{{20}} \times \dfrac{4}{4} = \dfrac{{60}}{{80}}\]
Now, we just have to write 10 different numbers between \[\dfrac{{48}}{{60}}\] and \[\dfrac{{60}}{{80}}\] , which are:
\[\dfrac{{49}}{{80}},\dfrac{{50}}{{80}},\dfrac{{51}}{{80}},\dfrac{{52}}{{80}},\dfrac{{53}}{{80}},\dfrac{{54}}{{80}},\dfrac{{55}}{{80}},\dfrac{{56}}{{80}},\dfrac{{57}}{{80}},\dfrac{{58}}{{80}}\]
Hence, 10 rational numbers between \[\dfrac{3}{5}\] and \[\dfrac{3}{4}\] are \[\dfrac{{49}}{{80}},\dfrac{{50}}{{80}},\dfrac{{51}}{{80}},\dfrac{{52}}{{80}},\dfrac{{53}}{{80}},\dfrac{{54}}{{80}},\dfrac{{55}}{{80}},\dfrac{{56}}{{80}},\dfrac{{57}}{{80}},\dfrac{{58}}{{80}}\] .
So, the correct answer is “ \[\dfrac{{49}}{{80}},\dfrac{{50}}{{80}},\dfrac{{51}}{{80}},\dfrac{{52}}{{80}},\dfrac{{53}}{{80}},\dfrac{{54}}{{80}},\dfrac{{55}}{{80}},\dfrac{{56}}{{80}},\dfrac{{57}}{{80}},\dfrac{{58}}{{80}}\]
”.


Note: So, we saw that in solving these types of questions, we need to properly identify what is being asked from us. Here we had to find rational numbers, which are terminating or repeating non-terminating decimal numbers. If the question would have asked us to find irrational, then we would have included some square root in either part of the number, i.e., in either of the numerator or the denominator. If the question would have asked for natural numbers, whole numbers or integers, then the answer would have been that no number is there in such a setting of the two numbers because both the numbers are greater than 0 but less than 1 and there is no natural number, whole number or integer in such a range. So, we have to see what is possible in the given required range of numbers.