Question

Arrange the rational numbers $ - \dfrac{7}{{10}},\dfrac{5}{{ - 8}},\dfrac{2}{{ - 3}},\dfrac{{ - 1}}{4},\dfrac{{ - 3}}{5} $ in ascending order.

Answer 1

Hint: First as we know that a rational number can be defined as the number which can be expressed in the form of $ \dfrac{p}{q} $ , where the denominator i.e. $ q \ne 0 $ . So the above fractions fits under the category of rational numbers where the denominator and numerators are integers and the denominator is not equal to zero. In this question we will find the L.C.M of the denominators and then convert the denominators in the same and then solve it.

Complete step-by-step answer:
Here we have the numbers $ - \dfrac{7}{{10}},\dfrac{5}{{ - 8}},\dfrac{2}{{ - 3}},\dfrac{{ - 1}}{4},\dfrac{{ - 3}}{5} $ .
For rearranging the numbers in ascending order first we have to make the denominator the same of all given rational numbers. So the L.C.M of $ 10, - 8, - 3,4 $ and $ 5 $ is $ 120 $ .
Now we have to make the denominator of each rational number equal to $ 120 $ . So we will multiply the numerator and denominator with the same number of each to make the denominator the same.
We can write the first number as : $ \dfrac{{ - 7 \times 12}}{{10 \times 12}} $ , second number: $ \dfrac{{5 \times 15}}{{ - 8 \times 15}} $ , third number $ \dfrac{{2 \times 40}}{{ - 3 \times 40}} $ . The fourth number as $ \dfrac{{ - 1 \times 30}}{{4 \times 30}} $ and the fifth number $ \dfrac{{ - 3 \times 24}}{{5 \times 24}} $ .
On simplifying we have: \[\dfrac{{ - 84}}{{120}},\dfrac{{ - 75}}{{120}},\dfrac{{ - 80}}{{120}},\dfrac{{ - 30}}{{120}},\dfrac{{ - 72}}{{120}}\].
We can arrange them according to the value of the numerator on the basis of this concept that: If rational numbers have all the denominators equal, then the number with a bigger numerator will be bigger. Also we know that with negative signs the bigger the number is, the smaller it is.
So we can write $ $ \[\dfrac{{ - 84}}{{120}} < \dfrac{{ - 80}}{{120}} < \dfrac{{ - 75}}{{120}} < \dfrac{{ - 72}}{{120}} < \dfrac{{ - 30}}{{120}}\], therefore by comparing this from the above we can write them in their original forms, i.e. $ - \dfrac{7}{{10}} < \dfrac{2}{{ - 3}} < \dfrac{5}{{ - 8}} < \dfrac{{ - 3}}{5} < \dfrac{{ - 1}}{4} $ .
Hence the ascending order is $ - \dfrac{7}{{10}} < \dfrac{2}{{ - 3}} < \dfrac{5}{{ - 8}} < \dfrac{{ - 3}}{5} < \dfrac{{ - 1}}{4} $ .
So, the correct answer is “ $ - \dfrac{7}{{10}} < \dfrac{2}{{ - 3}} < \dfrac{5}{{ - 8}} < \dfrac{{ - 3}}{5} < \dfrac{{ - 1}}{4} $ .”.

Note: We should note that we should not order the rational numbers in ascending order without equating their denominators because without them we will get our answers wrong. In the above we can write denominators in their factors as $ 10 = 2 \times 5,3 = 1 \times 3,8 = 2 \times 2 \times 2,5 = 1 \times 5,4 = 2 \times 2 $ , So we the LCM = $ 2 \times 2 \times 2 \times 5 \times 3 = 120 $ .