Question

State what x​ represents, write an​ equation, and answer the question.
In a certain​ fraction, the denominator is 2 less than the numerator. If 1 is added to both the numerator and​ denominator, the resulting fraction is equal to nine sevenths
. What was the original​ fraction?

Answer 1

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 \( \frac{4}{3} \).