Let's denote David's age \( D \) and his sister's age \( S \).
We're given:
1. Seven years ago, David's age was \( 25\% \) of his sister's age:
\[
D - 7 = 0.25(S - 7)
\]
2. Three years later, David's age will be \( 50\% \) of his sister's age:
\[
D + 3 = 0.5(S + 3)
\]
We need to find out when David's age will be \( 75\% \) of his sister's age:
\[
D + x = 0.75(S + x)
\]
First, solve the first equation for \( S \) in terms of \( D \), then substitute it into the second equation to solve for \( D \). Finally, substitute the found value of \( D \) into the third equation and solve for \( x \).
Let's calculate:
From the first equation:
\[ S = \frac{D - 7}{0.25} + 7 = 4(D - 7) + 7 = 4D - 21 \]
Now, substitute \( S \) into the second equation:
\[ D + 3 = 0.5(4D - 21 + 3) \]
\[ D + 3 = 0.5(4D - 18) \]
\[ D + 3 = 2D - 9 \]
\[ D = 12 \]
Now, substitute \( D = 12 \) into the third equation:
\[ 12 + x = 0.75(4(12) - 21 + x) \]
\[ 12 + x = 0.75(39 - 21 + x) \]
\[ 12 + x = 0.75(18 + x) \]
\[ 12 + x = 13.5 + 0.75x \]
\[ 0.25x = 1.5 \]
\[ x = 6 \]
So, it will be 6 years later when David's age is \( 75\% \) of his sister's age.
