Two numbers are in the ratio 3: 2. If 2 is added to the first and 6 is added to the second number, they are in the ratio 4: 5. Find the numbers.

Let the numbers be 3x and 2x.

According to the problem:

$\frac{3x+2}{2x+6}$ = $\frac{4}{5}$

⟹ 5(3x + 2) = 4

⟹ 15x + 10 = 8x + 24

⟹ 15x – 8x = 24 - 10

⟹ 7x = 14

⟹ x = $\frac{14}{7}$

⟹ x = 2

Therefore, the original numbers are: 3x = 3 × 2 = 6 and 2x = 2 × 2 = 4.

Thus, the numbers are 6 and 4.