A rational number with a denominator of 9 is divided by (-2/3). The result is multiplied by 4/5 and then -5/6 is added. The final value is 1/10. What is the original rational?

Sep 23, 2017

$- \frac{7}{9}$

Explanation:

"Rational numbers" are fractional numbers of the form $\frac{x}{y}$ where both the numerator and denominator are integers, i.e. frac(x)(y); $x , y \in \mathbb{Z}$.

We know that some rational number with a denominator of $9$ is divided by $- \frac{2}{3}$.

Let's consider this rational to be $\frac{a}{9}$:

$\text{ " " " " " " " " " " " " " " " " " } \frac{a}{9} \div - \frac{2}{3}$

$\text{ " " " " " " " " " " " " " " " " " } \frac{a}{9} \times - \frac{3}{2}$

$\text{ " " " " " " " " " " " " " " " " " " } - \frac{3 a}{18}$

Now, this result is multiplied by $\frac{4}{5}$, and then $- \frac{5}{6}$ is added to it:

$\text{ " " " " " " " " " " " " } \left(- \frac{3 a}{18} \times \frac{4}{5}\right) + \left(- \frac{5}{6}\right)$

$\text{ " " " " " " " " " " " " " " " " } - \frac{12 a}{90} - \frac{5}{6}$

$\text{ " " " " " " " " " " " " " " " } - \left(\frac{12 a}{90} + \frac{5}{6}\right)$

$\text{ " " " " " " " " " " " " } - \left(\frac{6 \times 12 a + 90 \times 5}{90 \times 6}\right)$

$\text{ " " " " " " " " " " " " " " } - \left(\frac{72 a + 450}{540}\right)$

Lastly, we know that the final value is $\frac{1}{10}$:

$\text{ " " " " " " " " " " " " } - \left(\frac{72 a + 450}{540}\right) = \frac{1}{10}$

$\text{ " " " " " " " " " " " " } \frac{72 a + 450}{540} = - \frac{1}{10}$

$\text{ " " " " " " " " " " " " } 72 a + 450 = - \frac{540}{10}$

$\text{ " " " " " " " " " " " " } 72 a + 450 = - 54$

$\text{ " " " " " " " " " " " " " } 72 a = - 504$

$\text{ " " " " " " " " " " " " " " " } a = - 7$

Let's substitute $- 7$ in place of $a$ in our rational number:

$\text{ " " " " " " " " " " " " " " } \frac{a}{9} = - \frac{7}{9}$

Therefore, the original rational number is $- \frac{7}{9}$.