Question

The given rational numbers are$\dfrac{1}{2},\dfrac{4}{{ - 5}},\,\dfrac{{ - 7}}{8}$. If these numbers are arranged in the ascending order of descending order, then the middle number is
(A) $\dfrac{1}{2}$
(B) $\dfrac{{ - 7}}{8}$
(C) $\dfrac{4}{{ - 5}}$
(D) None of these

Answer 1

Hint:: To find ascending order or descending order of given rational numbers we will first find L.C.M. of denominators of given rational numbers and then using this L.C.M. change respective numerators of corresponding rational with the help of which we can arrange in either of order to get respective position of given rational numbers.

Complete step-by-step answer:
To find middle numbers from given rational numbers we have to arrange them either in ascending order or in descending order.
Given rational numbers are $\dfrac{1}{2},\dfrac{4}{{ - 5}},\,\dfrac{{ - 7}}{8}$ or we can write them as
$\dfrac{1}{2},\dfrac{{ - 4}}{5},\,\dfrac{{ - 7}}{8}$
To find ascending order or descending order of given rational numbers we take L.C.M. of their denominators.
$\begin{array}{*{20}{c}}
  2&\hline & 2&5&8 \\
\hline
  2&\hline & 1&5&4 \\
\hline
  2&\hline & 1&5&2 \\
\hline
  2&\hline & 1&5&1 \\
\hline
  5&\hline & 1&1&1
\end{array}$
L.C.M. of denominators $2,5\,and\,\,\,8$is given as $2 \times 2 \times 2 \times 2 \times 5$
L.C.M. = $40$
Now, using L.C.M. finding corresponding numerator of given rational numbers.
$\Rightarrow \dfrac{1}{2}\,\,becomes\,\,\dfrac{{20}}{{40}}$ , $\dfrac{{ - 4}}{5}\,\,becomes\,\dfrac{{ - 32}}{{40}}$and $\dfrac{{ - 7}}{8}\,\,becomes\,\dfrac{{ - 35}}{{40}}$
From above we see that given rational numbers $\dfrac{1}{2},\dfrac{{ - 4}}{5},\,\dfrac{{ - 7}}{8}$ becomes $\dfrac{{20}}{{40}},\,\dfrac{{ - 32}}{{40}}\,,\dfrac{{ - 35}}{{40}}$
Since, denominators of above rational numbers are the same. So, their ascending or descending order can be found by arranging their numerator or denominator in either ascending or descending order.
So, ascending order of above rational numbers are written as:
$\dfrac{{ - 35}}{{40}} < \dfrac{{ - 32}}{{40}} < \dfrac{{20}}{{40}}$
Or in ascending order in the form of given rational numbers can be written as:
\[\Rightarrow \dfrac{{ - 7}}{8} < \dfrac{{ - 4}}{5} < \dfrac{1}{2}\]
Therefore, from above we see that the middle rational number of given rational numbers is$\dfrac{{ - 4}}{5}$ or$\dfrac{4}{{ - 5}}$.
Hence, from above we see that from given four options the correct option is (C)
So, the correct answer is “Option C”.

Note: To find the position of any rational number with respect to given rational numbers we arrange them in either ascending order or in descending order but while doing so one should take care of signs very carefully.