Electric circuits

Why doesn't my method give the right answer?

Your method doesn't work because you have misinterpreted the information given to you.

The question really is: if my LDR has a resistance of 750ohms, does that mean that the p.d. across the resistor is such that it is above the 13V threshold?

Instead, what you have assumed is: if my LDR has a resistance of 750ohms, that means the p.d. across the resistor must be the 13V threshold. But then the rest of your work doesn't really make sense as you recalculate the resistor p.d. (and as I've alluded to above, the question doesn't say what the p.d. across the resistor is because that is what it wants you to calculate).

Either way: the correct approach then is just to consider voltage divider rules as all you are actually interested in is the p.d. across one component (the Resistor). So, you know all required resistances in the circuit to determine the total, and you know the supply voltage, so by voltage divider rules you can determine the p.d. across either component at this particular point.

Let me know if this makes sense.