Step 1: Define the original numbers using a common variable

Since the ratio of the two numbers is given as $7: 5$, we can let the first number be $7x$ and the second number be $5x$.

$\text{First} = 7x, \quad \text{Second} = 5x$

Step 2: Apply the given changes to both numbers

The first number is increased by $14$ (becoming $7x + 14$) and the second number is decreased by $5$ (becoming $5x - 5$).

$\text{New}\ \text{First} = 7x + 14, \quad \text{New}\ \text{Second} = 5x - 5$

Step 3: Set up the proportion equation

According to the problem, the new ratio becomes $4: 2$, which simplifies to $2: 1$. We equate our expressions to this simplified ratio.

$\frac{7x + 14}{5x - 5} = \frac{4}{2} = \frac{2}{1}$

Step 4: Solve for $x$

Cross-multiplying gives $7x + 14 = 2(5x - 5)$. Expanding and simplifying: $7x + 14 = 10x - 10 \implies 3x = 24 \implies x = 8$.

$x = 8$

Step 5: Find the numbers and calculate their sum

Substitute $x = 8$ back into the original expressions: First number = $7(8) = 56$ and Second number = $5(8) = 40$. The sum is $56 + 40 = 96$.

$\text{Sum} = 56 + 40 = 96$