The sum of two rational numbers is $\dfrac{-5}{8}$. If one of them is $\dfrac{7}{16}$, find the other.

Given:

Let p and q be two rational numbers.

One rational number = p = $\dfrac{7}{16}$

Other rational number = q = ?

Sum of two rational numbers = (p + q) = $\dfrac{-5}{8}$

q = $\dfrac{-5}{8}$ - p

Substituting the values in above, we get:

$q = \dfrac{-5}{8} - \dfrac{7}{16} \\[1em] = \dfrac{-5}{8} + \Big(\text{additive inverse of } \dfrac{7}{16}\Big) \\[1em] q = \dfrac{-5}{8} + \dfrac{-7}{16}$

L.C.M. of 8 and 16 is 16.

Now, expressing each fraction with denominator 16:

$\dfrac{-5 \times 2}{8 \times 2} + \dfrac{-7 \times 1}{16 \times 1} \\[1em] = \dfrac{-10}{16} + \dfrac{-7}{16} \\[1em] = \dfrac{-10 + (-7)}{16} \\[1em] = \dfrac{-17}{16}$

The other rational number q is $\dfrac{-17}{16}$.