# If 3 is added to the number and denominator of a fraction and the result subtracted 'by' the original fraction, the difference is 1/15. The numerator of the original fraction is 6 less than the denominator. What is the original fraction?

Oct 11, 2015

$\frac{9}{15}$

#### Explanation:

So if the original fraction's numerator is $6$ less than the denominator, we can write the original fraction as $\frac{x}{x + 6}$.

Then, the second fraction is $3$ added to the numerator and denominator. Written like how we wrote the first fraction, it is $\frac{x + 3}{x + 6 + 3}$ or $\frac{x + 3}{x + 9}$.

We are also told the the difference between the second fraction and the first fraction is $\frac{1}{15}$, so we can write the equation:

$\frac{x + 3}{x + 9} - \frac{x}{x + 6} = \frac{1}{15}$.

Now, we have to find the LCD to be able to work with fractions with different denominators. The LCD in this case would be $\left(x + 6\right) \left(x + 9\right)$. If we multiply both sides of the equation by that term, we get:

$\left(x + 6\right) \left(x + 9\right) \left(\frac{x + 3}{x + 9} - \frac{x}{x + 6}\right) = \frac{1}{15} \left(x + 6\right) \left(x + 9\right)$.

Let's distribute the $\left(x + 6\right)$ first on the left side, to get:

(x+9)(((x+3)(x+6))/(x+9)) -x)=1/15(x+6)(x+9)

Then distribute the $\left(x + 9\right)$ on the left side, to get:

$= \left(x + 3\right) \left(x + 6\right) - x \left(x + 9\right) = \frac{1}{15} \left(x + 6\right) \left(x + 9\right)$.

Now, we can expand both sides to get:

${x}^{2} + 9 x + 18 - {x}^{2} - 9 x = \frac{1}{15} \left({x}^{2} + 15 x + 54\right)$.

Now, if we move all the terms to one side and do some simplification, we get:

$0 = \frac{1}{15} \left({x}^{2} + 15 x + 54\right) - 18$.

If we multiply both sides by $15$ and combine like terms, we get:

$0 = {x}^{2} + 15 x - 216$.

Factoring, we get:

$0 = \left(x - 9\right) \left(x + 24\right)$

Thus, the solutions for x are $9$ and $- 24$. Using these values for $x$, we get:

$\frac{9}{15}$ and $\frac{- 24}{-} 18$. But, the negatives cancel in the second expression, which skews the entire setup (if would have been $\frac{- 24}{-} 18 - \frac{- 21}{-} 15$ but now its $\frac{27}{21}$ and the subtraction does not work.

Quickly checking the first expression, you get $\frac{12}{18} - \frac{9}{15}$, which does equate to $\frac{1}{15}$!