SUPER confused on this one Math Question, help please !

Question

Question context (summarized for simplicity)

Trucking: 10% of Total Cost. 5% of trucking cost is FC and the other 95% is VC. VC is reducing by half

Refining: 30% of Total Cost. Refining cost is being reduced by 50%, but price will also increase by 25%

What is the cost savings from both initiatives?

Calculation from Casebook:

Variable trucking costs, which represent 95% of the total cost structure, will decrease by 50%, leading to an overall costs savings of 95% x 50% = 47.5%.

Refining costs drop by half because only half as many sugar beets are being refined. However, then there is a 25% increase in costs per beet. So the costs go from “X” to “.5X” to “.625X”, an overall savings of 37.5%.

My confusion

Why is it that for trucking costs, you are finding the cost savings directly after taking 95% x 50%. But for refining, you need to take 1-0.625 to get the answer?

I've been asking ChatGPT about this, and the points don't seem to make much sense (e.g. it's cause refining has sequential percentages multiplying).

Could someone please help me understand the underlying concept? I want to know when must I do '1-' vs just taking the percentage immediately, not only for this question but in general

Thank you !

Answer 1

For the trucking cost you only have a reduction in cost that equals to 50% of the 95% so you can calculate the savings simply by multiplying 95% by 50%

If you only had a cost reduction for the refining cost you would be able to calculate this the same way but since you also have an increase in price you will have to add that to the remaining cost, before you can calculate the savings.

1. Calculate the reduction in cost by 50% (1*0.5)

2. Add the price increase, which gives you the new cost (0.5*1.25 = 0.625)

3. Calculate the difference between old and new cost (1-0.625 = 0.375)

Hope this helps :)

Answer 2

Trucking cost savings:

Initial Cost: 95% + 5% = 100%

Final Cost: 95% * 50% + 5% = 52.5%

Savings = Initial Cost - Final Cost = 100% - 52.5% = 47.5%

Refining cost savings:

Initial Cost: 100%

Final Cost: 100% * (1 + 25%) * 50% = 62.5%

Savings = Initial Cost - Final Cost = 100% - 62.5% = 37.5%

So what's different vs. your casebook? In the truck savings it calculated the savings directly. In the refining savings it calculated the new cost (not the savings), so you need to do an additional calculation to get the savings.