question_answer 14)

                A rational number is such that when you multiply it by \[\frac{5}{2}\] and add \[\frac{2}{3}\] to the product, you get \[-\frac{7}{12}\]. What is the number?

Answer:

                Let the number be \[x\]. According to the question, \[\frac{5}{2}x+\frac{2}{3}=-\frac{7}{12}\] \[\Rightarrow \]               \[\frac{5}{2}x=-\frac{2}{3}-\frac{7}{12}\]                               | Transposing \[\frac{2}{3}\] to RHS \[\Rightarrow \]               \[\frac{5}{2}x=\frac{-8-7}{12}\] \[\Rightarrow \]               \[\frac{5}{2}x=-\frac{15}{12}\] \[\Rightarrow \]               \[x=-\frac{15}{12}\times \frac{2}{5}\]                     | Multiplying both sides by \[\frac{2}{5}\] \[\Rightarrow \]               \[x=-\frac{1}{2}\] Hence, the desired rational number is \[-\frac{1}{2}\].                 Check: \[\left( -\frac{1}{2} \right)\times \frac{5}{2}+\frac{2}{3}=-\frac{5}{4}+\frac{2}{3}\] \[=\frac{-15+8}{12}\] \[=-\frac{7}{12}\].                            | as desired