Define the variables

Let the original numbers be $5x$ and $8x$ based on the given ratio of $5:8$.

$5x, 8x$

Formulate the equation

According to the problem, adding 13 to the first and subtracting 6 from the second results in the ratio $4:6$.

$\frac{5x + 13}{8x - 6} = \frac{4}{6}$

Solve for x

Simplify the ratio $4/6$ to $2/3$ and cross-multiply: $3(5x + 13) = 2(8x - 6) \Rightarrow 15x + 39 = 16x - 12$. Rearranging gives $x = 39 + 12 = 51$.

$x = 51$

Calculate the sum

The original numbers are $5x = 5 \times 51 = 255$ and $8x = 8 \times 51 = 408$. Their sum is $255 + 408 = 663$.

$5x + 8x = 13x = 13 \times 51 = 663$