To solve this problem, we can use the following equation of motion for the vertical velocity at any given time $$t$$:

$$v = u - gt$$

Where:

- $$v$$ is the final velocity at time $$t$$

- $$u$$ is the initial velocity (150 m/s)

- $$g$$ is the acceleration due to gravity (10 m/s²)

- $$t$$ is the time in seconds

First, we need to find the velocities at $$t = 3 \mathrm{~s}$$ and $$t = 5 \mathrm{~s}$$.

For $$t = 3 \mathrm{~s}$$:

$$v_3 = 150 - (10)(3) = 150 - 30 = 120 \mathrm{~m} / \mathrm{s}$$

For $$t = 5 \mathrm{~s}$$:

$$v_5 = 150 - (10)(5) = 150 - 50 = 100 \mathrm{~m} / \mathrm{s}$$

Now we need to find the ratio of these velocities:

$$\frac{v_3}{v_5} = \frac{120}{100} = \frac{6}{5} = \frac{x + 1}{x}$$

Next, we can set up an equation to find the value of $$x$$:

$$\frac{6}{5} = \frac{x + 1}{x}$$

Now, we can solve for $$x$$:

$$6x = 5(x + 1)$$

$$6x = 5x + 5$$

$$x = 5$$

The value of $$x$$ is 5.