Question

The given rational numbers are $\frac{1}{2},\frac{4}{-5},\frac{-7}{8}.$ If these numbers are arranged in the ascending order or descending order, then the middle number is:

• A. $\frac{1}{2}$
• B. $\frac{-7}{8}$
• C. $\frac{4}{-5}$
• D. None of these

Answer 1

Answer: (C)

$\frac{4}{-5}$

The rational number $\frac{4}{-5}$ is same as $\frac{-4}{5}$.

Now consider the given rational numbers $\frac{1}{2},\frac{-4}{5}$ and $\frac{-7}{8}$ and make their denominator same by taking the LCM of the denominators as follows:

LCM$(2,5,8)=40$

The given fractions now with denominator $40$ can be written as:

$\frac{1\times 20}{2\times 20}=\frac{20}{40}\frac{-4\times 8}{5\times 8}=\frac{-32}{40}\frac{-7\times 5}{8\times 5}=\frac{-35}{40}$

The ascending order of the rational numbers is:

$\frac{-35}{40},\frac{-32}{40},\frac{20}{40}\Rightarrow \frac{-7}{8},\frac{4}{-5},\frac{1}{2}.......\left(1\right)$

The descending order of the rational numbers is:

$\frac{20}{40},\frac{-32}{40},\frac{-35}{40}\Rightarrow \frac{1}{2},\frac{4}{-5},\frac{-7}{8}.......\left(2\right)$

From equations 1 and 2, we conclude that the middle rational number in both ascending and descending order is same that is $\frac{4}{-5}$.

Hence, the middle number is $\frac{4}{-5}$.