Question

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

Answer 1

Hint: Let the unknown rational number be $\dfrac{p}{q}$ where $q \ne 0$ and then proceed as asked in the question. Simply first we do the multiplication of 2 rational numbers and then addition.

Complete step-by-step answer:
Let the unknown rational number be $\dfrac{p}{q}$ where $q \ne 0$ .
Multiplying the rational number with $\dfrac{5}{2}$, we get
$ \Rightarrow $ $\dfrac{p}{q} \times \dfrac{5}{2} = \dfrac{{5p}}{{2q}}$
It is given in the question that the sum of $\dfrac{2}{3}$ and the product of rational number and $\dfrac{5}{2}$ is $\dfrac{{ - 7}}{{12}}$
$ \Rightarrow $$\dfrac{{5p}}{{2q}} + \dfrac{2}{3} = \dfrac{{ - 7}}{{12}}$
$ \Rightarrow $$\dfrac{{(5p \times 3) + (2 \times 2q)}}{{2q \times 3}} = \dfrac{{ - 7}}{{12}}$
$ \Rightarrow $$\dfrac{{15p + 4q}}{{6q}} = \dfrac{{ - 7}}{{12}}$
$ \Rightarrow $$\dfrac{{15p + 4q}}{q} = \dfrac{{ - 7}}{2}$
$ \Rightarrow $$2(15p + 4q) = - 7q$
$ \Rightarrow $$30p + 8q = - 7q$
$ \Rightarrow $$30p = - 7q - 8q$
$ \Rightarrow $$30p = - 15q$
$ \Rightarrow $$p = \dfrac{{ - 15q}}{{30}}$
$ \Rightarrow $$p = \dfrac{{ - q}}{2}$
$ \Rightarrow $$\dfrac{p}{q} = \dfrac{{ - 1}}{2}$
Hence the unknown rational number$\dfrac{p}{q} = \dfrac{{ - 1}}{2}$.

Note: A rational number is a number which can be represented in the form of $\dfrac{p}{q}$ where$q \ne 0$. Multiplication of 2 rational numbers is done as
$\dfrac{a}{b} \times \dfrac{p}{q} = \dfrac{{ap}}{{bq}}$
Addition of 2 rational numbers is done by taking LCM of denominators
$\dfrac{a}{b} + \dfrac{p}{q} = \dfrac{{aq + pb}}{{bq}}$