To solve this problem, we need to balance the total lengths of the timber on both sides of the cutting line $L$, as stated in the problem. The total length of the timber pieces to the left and to the right of $L$ must be the same after the cut.

Total Lengths Before the Cut:

• Top piece of timber: 500 cm
• Second piece of timber: 300 cm
• Third piece of timber: 200 cm
• Fourth piece of timber: 150 cm

Distances of the pieces to the right of $L$ (i.e., their lengths to the right of $L$):

• Top piece to the right of $L$: 300 cm
• Second piece to the right of $L$: 300 cm
• Third piece to the right of $L$: 500 cm
• Fourth piece to the right of $L$: 400 cm

Step 1: Find the total lengths to the right of $L$

Add the lengths of all the portions to the right of $L$:

$300 + 300 + 500 + 400 = 1500 \, \text{cm}$

Step 2: Total Length of All Timbers

Now, the total length of all timbers combined (i.e., all four pieces before the cut):

$500 + 300 + 200 + 150 = 1150 \, \text{cm}$

To make the cut balanced, the length to the left of $L$ must be:

$\frac{1500}{2} = 750 \, \text{cm}$