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.

Answer 3

Hey 😊

Totally valid confusion—and you're actually very close to getting it. Here’s the clear rule you can use to understand when to subtract from 1 vs when to take a direct product:

✅ Rule of thumb:

👉 If the question is asking for cost savings as a percentage of the original, then:

You always want to end up comparing the new cost to the original cost
So if the cost goes from X → Y, then cost savings = (X - Y) / X = 1 - (Y / X)

Let’s now look at your examples:

🚚 Trucking:

Trucking is 10% of Total Cost, and within that:

5% is Fixed → unaffected
95% is Variable → reduced by 50%

So within Trucking, your new cost is: 0.05 + 0.95 * 0.5 = 0.05 + 0.475 = 0.525

That means total Trucking cost is now 52.5% of what it was before, or a 47.5% reduction → and that’s what they wrote.

💡 They wrote 95% x 50% = 47.5% savings, which skips a step but is mathematically the same because only 95% of cost is affected.

🏭 Refining:

You start with cost = X

Step 1: You’re refining half the sugar beets → cost becomes 0.5X

Step 2: But now the per-unit cost increases by 25% → cost becomes: 0.5X * 1.25 = 0.625X

So now the cost is 62.5% of original → savings = 1 - 0.625 = 0.375 = 37.5%

💡 Here, the changes are sequential multipliers that act on the total cost, so you must calculate the end result, then subtract to get the savings.

🔁 So when to subtract from 1?

Whenever you're calculating "savings", or "reduction" as a percentage of the original—you’ll always want to do:

Savings % = 1 - (new cost / original cost)

The trucking shortcut only works cleanly if you’re applying a single % cut to a part of the cost.

Let me know if you'd like to go through more examples or build an easy cheat sheet for these types of questions 😊

Best,
Alessa 🌟

Answer 4

When do we do “1 – …” vs. just take the percentage?

In both situations, the underlying concept is:

Savings % = 1 - (Fraction Remaining)

If you have the fraction (or percent) that remains (like 0.625 → 62.5%), you do 1 - 0.625 = 0.375 (37.5% savings).
If, on the other hand, you already computed how much is saved (like 0.95 * 0.50 = 0.475 → 47.5% saved), then you’re already at the final “savings fraction” and don’t need to do “1 – …”.

So it’s not that “one scenario must do it, and the other must not.” It’s just a different presentation style:

Trucking: The write-up jumped straight to the fraction saved: 95% * 50% = 47.5%.
Refining: The write-up jumped straight to the fraction remaining (0.625) and then subtracted from 1 to get 37.5%.

Either approach works; they’re just different ways of doing the fraction arithmetic. Whenever you see “we have a new fraction of the original left,” you do 1 - (that fraction) to see how much was removed/saved. Whenever you see “we are cutting X% out of a portion,” that already is the portion saved.