To solve this problem, we'll first define the variable and the components based on the description provided.
Let \( x \) represent the numerator of the fraction. Then, according to the problem, the denominator can be expressed as \( x - 2 \).
The original fraction can therefore be represented as:
\[ \frac{x}{x - 2} \]
According to the problem, if 1 is added to both the numerator and the denominator, the new fraction becomes:
\[ \frac{x + 1}{(x - 2) + 1} = \frac{x + 1}{x - 1} \]
It is given that this new fraction is equal to nine sevenths:
\[ \frac{x + 1}{x - 1} = \frac{9}{7} \]
Now we can cross-multiply to solve for \( x \):
\[ 7(x + 1) = 9(x - 1) \]
Expanding both sides:
\[ 7x + 7 = 9x - 9 \]
Now, we'll rearrange this equation to isolate \( x \):
\[ 7 + 9 = 9x - 7x \] \[ 16 = 2x \] \[ x = 8 \]
Now that we have found the value of \( x \) (the numerator), we can find the denominator:
\[ \text{Denominator} = x - 2 = 8 - 2 = 6 \]
Thus, the original fraction is:
\[ \frac{x}{x - 2} = \frac{8}{6} \]
We can simplify this fraction:
\[ \frac{8}{6} = \frac{4}{3} \]
Final Answer: The original fraction is \( \frac{4}{3} \).