Let the number we need to find be \( x \).
According to the problem, when \( x \) is added to both the numerator and the denominator of the fraction \( \frac{3}{5} \), the result should be \( \frac{4}{5} \).
We can express this mathematically as:
\[ \frac{3 + x}{5 + x} = \frac{4}{5} \]
To eliminate the fraction, we can cross-multiply:
\[ 5(3 + x) = 4(5 + x) \]
Expanding both sides gives:
\[ 15 + 5x = 20 + 4x \]
Now, we'll isolate \( x \) by first moving \( 4x \) to the left side and \( 15 \) to the right side:
\[ 5x - 4x = 20 - 15 \]
This simplifies to:
\[ x = 5 \]
Thus, the number we need is \( \boxed{5} \).
To verify, let's substitute \( x = 5 \) back into the fraction:
\[ \frac{3 + 5}{5 + 5} = \frac{8}{10} = \frac{4}{5} \]
Since this is correct, the solution \( x = 5 \) is verified.
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