Certainly! Let's solve the problem step-by-step:
Given:
- The denominator of a fraction is 3 more than the numerator.
- If 5 is added to both the numerator and the denominator, the new fraction becomes [tex]\(\frac{4}{5}\)[/tex].
Let's denote the numerator of the fraction by [tex]\(x\)[/tex]. Consequently, the denominator will be [tex]\(x + 3\)[/tex].
So, initially, the fraction can be expressed as:
[tex]\[
\frac{x}{x + 3}
\][/tex]
According to the given condition, if we add 5 to both the numerator and the denominator, the new fraction becomes [tex]\(\frac{4}{5}\)[/tex]. Thus, we have:
[tex]\[
\frac{x + 5}{x + 8} = \frac{4}{5}
\][/tex]
To find [tex]\(x\)[/tex], we can cross-multiply to solve this equation:
[tex]\[
5(x + 5) = 4(x + 8)
\][/tex]
Let's solve for [tex]\(x\)[/tex] step-by-step:
[tex]\[
5x + 25 = 4x + 32
\][/tex]
Subtract [tex]\(4x\)[/tex] from both sides:
[tex]\[
x + 25 = 32
\][/tex]
Next, subtract 25 from both sides:
[tex]\[
x = 7
\][/tex]
So, the numerator of the original fraction is [tex]\(x = 7\)[/tex].
Now, let's find the denominator:
[tex]\[
x + 3 = 7 + 3 = 10
\][/tex]
Therefore, the original fraction is:
[tex]\[
\frac{7}{10}
\][/tex]
So, the fraction is [tex]\(\frac{7}{10}\)[/tex].
