Let the initial mass be $m$ and initial speed be $40\,\text{m s}^{-1}$.

Since the block splits into two equal parts, each part has mass $\dfrac{m}{2}$.

Use conservation of momentum.

Initial momentum: $p_i = m \cdot 40 = 40m$

After splitting, one part moves with speed $60\,\text{m s}^{-1}$ in the same direction. Let the speed of the other part be $v$.

Final momentum: $p_f = \frac{m}{2}(60) + \frac{m}{2}v$

By conservation of momentum, $40m = \frac{m}{2}(60) + \frac{m}{2}v$

Divide by $m$: $40 = 30 + \frac{v}{2}$ $\frac{v}{2} = 10$ $v = 20\,\text{m s}^{-1}$

Calculate initial kinetic energy.

$K_i = \frac{1}{2}m(40)^2 = \frac{1}{2}m(1600) = 800m$

Calculate final kinetic energy.

$K_f = \frac{1}{2}\left(\frac{m}{2}\right)(60)^2 + \frac{1}{2}\left(\frac{m}{2}\right)(20)^2$

$K_f = \frac{m}{4}(3600) + \frac{m}{4}(400)$ $K_f = 900m + 100m = 1000m$

Find the change in kinetic energy.

$\Delta K = K_f - K_i = 1000m - 800m = 200m$

Find the fractional change in kinetic energy.

Fractional change means: $\frac{\Delta K}{K_i} = \frac{200m}{800m} = \frac{1}{4}$

Given this is $x:4$, we have $\frac{x}{4} = \frac{1}{4}$ so, $x = 1$

Final Answer

$\boxed{1}$

Comparison with stored answer

Stored correct answer = $1$.

Our derived answer also equals $1$, so they agree.