Question

Arrange the following rational numbers in ascending order:
$\dfrac{3}{5},\,\dfrac{{ - 17}}{{ - 30}},\,\dfrac{8}{{ - 15}},\,\dfrac{{ - 7}}{{10}}$

Answer 1

Hint: We are given rational numbers and we have to arrange them in ascending order. Arranging in ascending order means to arrange the numbers from the smallest to the largest. For this firstly we will express the given rational numbers with positive denominator. Then we will take the least common multiple (L.C.M) of these positive denominators and then we will express each rational number with this (L.C.M) as the common denominator. Then the number having the smaller numerator is smaller. In case we are having both positive and negative numbers then we start from the most negative number and go to the most positive number.

Complete step by step answer:
Step1: We are given the rational numbers:
$\dfrac{3}{5},\dfrac{{ - 17}}{{ - 30}},\dfrac{8}{{ - 15}},\dfrac{{ - 7}}{{10}}$
We first write the given rational numbers so that their denominators are positive for that we will multiply numerator and denominator by $( - 1)$
$\Rightarrow \dfrac{{ - 17 \times ( - 1)}}{{ - 30 \times ( - 1)}} = \dfrac{{17}}{{30}}$
$\Rightarrow \dfrac{{8 \times ( - 1)}}{{ - 15 \times ( - 1)}} = \dfrac{{ - 8}}{{15}}$
Thus the given rational numbers with positive denominator are:
$\Rightarrow \dfrac{3}{5}$,$\dfrac{{17}}{{30}}$,$\dfrac{{ - 8}}{{15}}$,$\dfrac{{ - 7}}{{10}}$
Step2: Now L.C.M of the denominator $5,30,15$and$10$is$5 \times 6 \times 3 \times 2 = 180$.
We now write the numerator so that they have a common denominator $180$as follows
In case of $\dfrac{3}{5}$we multiply numerator and denominator by $36$
$\Rightarrow \dfrac{{3 \times 36}}{{5 \times 36}} = \dfrac{{108}}{{180}}$;
In case of $\dfrac{{17}}{{30}}$ we multiply numerator and denominator by$6$
$\Rightarrow \dfrac{{17 \times 6}}{{30 \times 6}} = \dfrac{{102}}{{180}}$;
In case of $\dfrac{{ - 8}}{{15}}$ we multiply numerator and denominator by$12$
$\Rightarrow \dfrac{{ - 8 \times 12}}{{15 \times 12}} = \dfrac{{ - 96}}{{180}}$
In case of $\dfrac{{ - 7}}{{10}}$ we multiply numerator and denominator by$18$
$\Rightarrow \dfrac{{ - 7 \times 18}}{{10 \times 18}} = \dfrac{{126}}{{180}}$
Step3: Comparing the numerators of these numbers and also considering the signs as a start from a negative number we get (in case of negative numbers greater the number with negative sign smaller will be its value)
$ - 126 < - 96 < 102 < 108$
Therefore, the ascending order of whole fraction will be
$\dfrac{{ - 126}}{{180}} < \dfrac{{ - 96}}{{180}} < \dfrac{{102}}{{180}} < \dfrac{{108}}{{180}}$
Step5: hence the order of rational numbers is $\dfrac{{ - 7}}{{10}} < \dfrac{{ - 8}}{{15}} < \dfrac{{17}}{{30}} < \dfrac{3}{5}$

Hence ascending order is $\dfrac{{ - 7}}{{10}} < \dfrac{{ - 8}}{{15}} < \dfrac{{17}}{{30}} < \dfrac{3}{5}$.

Note:
In this type of questions to arrange rational numbers in ascending order students mainly do mistakes in finding L.C.M, they sometimes don’t take the denominator common and randomly arranging the numbers sometimes they do calculation mistakes in finding L.C.M or making L.C.M common of fractions they also get confused in arranging the numbers with negative signs because they ignore the signs and on the bases of values arrange them but the sign should be considered.