Question

solve.
\\(\dfrac{\dfrac{5}{8} + \dfrac{3}{4}}{\dfrac{-2}{3} - \dfrac{5}{6}}\\)
\\(-2\dfrac{1}{16}\\) \\(-\dfrac{11}{12}\\) \\(2\dfrac{1}{16}\\) \\(\dfrac{11}{12}\\)

Explanation:

Step1: Simplify the numerator

To add \(\frac{5}{8}\) and \(\frac{3}{4}\), we first find a common denominator. The common denominator of 8 and 4 is 8. So we rewrite \(\frac{3}{4}\) as \(\frac{6}{8}\) (since \(4\times2 = 8\) and \(3\times2=6\)). Then we add: \(\frac{5}{8}+\frac{6}{8}=\frac{5 + 6}{8}=\frac{11}{8}\).

Step2: Simplify the denominator

To subtract \(\frac{5}{6}\) from \(\frac{-2}{3}\), we find a common denominator. The common denominator of 3 and 6 is 6. We rewrite \(\frac{-2}{3}\) as \(\frac{-4}{6}\) (since \(3\times2 = 6\) and \(- 2\times2=-4\)). Then we subtract: \(\frac{-4}{6}-\frac{5}{6}=\frac{-4-5}{6}=\frac{-9}{6}=\frac{-3}{2}\) (after simplifying by dividing numerator and denominator by 3).

Step3: Divide the simplified numerator by the simplified denominator

Now we have \(\frac{\frac{11}{8}}{\frac{-3}{2}}\). Dividing by a fraction is the same as multiplying by its reciprocal, so this becomes \(\frac{11}{8}\times\frac{2}{-3}\). We can simplify before multiplying: the 2 and 8 have a common factor of 2, so we divide 2 by 2 (get 1) and 8 by 2 (get 4). Now we multiply: \(\frac{11\times1}{4\times(-3)}=\frac{11}{-12}=-\frac{11}{12}\). Wait, no, wait. Wait, let's check again. Wait, the original denominator is \(\frac{-2}{3}-\frac{5}{6}\). Let's recalculate the denominator: \(\frac{-2}{3}-\frac{5}{6}=\frac{-4 - 5}{6}=\frac{-9}{6}=-\frac{3}{2}\). The numerator is \(\frac{5}{8}+\frac{3}{4}=\frac{5 + 6}{8}=\frac{11}{8}\). Then the fraction is \(\frac{\frac{11}{8}}{-\frac{3}{2}}=\frac{11}{8}\times(-\frac{2}{3})=\frac{11\times(-2)}{8\times3}=\frac{-22}{24}=-\frac{11}{12}\). Wait, but let's check the options. Wait, maybe I made a mistake in the sign? Wait, no. Wait, let's re-express the original problem: \(\frac{\frac{5}{8}+\frac{3}{4}}{\frac{-2}{3}-\frac{5}{6}}\). Let's compute numerator and denominator again.

Numerator: \(\frac{5}{8}+\frac{3}{4}=\frac{5}{8}+\frac{6}{8}=\frac{11}{8}\) (correct).

Denominator: \(\frac{-2}{3}-\frac{5}{6}\). Let's convert \(\frac{-2}{3}\) to sixths: \(\frac{-4}{6}\). Then \(\frac{-4}{6}-\frac{5}{6}=\frac{-4 - 5}{6}=\frac{-9}{6}=-\frac{3}{2}\) (correct).

Now, dividing \(\frac{11}{8}\) by \(-\frac{3}{2}\) is \(\frac{11}{8}\times(-\frac{2}{3})=\frac{11\times(-2)}{8\times3}=\frac{-22}{24}=-\frac{11}{12}\) (after simplifying by dividing numerator and denominator by 2). So the result is \(-\frac{11}{12}\).

Answer:

\(-\frac{11}{12}\) (corresponding to the option \(-\frac{11}{12}\))