Dear Francesca,

To solve, the quantity of low-line Cognac that needs to be converted to medium-line Cognac to achieve an extra 10% of Profit Margin across the entire portfolio, you need to proceed linearly to:

A] Reference Quantity:
What is the baseline level of Profits from which we must proceed?

B] Algebraic Solution:
What is the quantitative logic to proceed from the baseline to calculate the quantity being sought.

But, first, let me state my assumption that units sold are per bottle.

A] Reference Quantity: what are Profits today?
This is the baseline level on which the 10% increase is sought.
We get there by proceeding linearly from the supplied data:

1) Sales Quantity by Product Line

1. 25M Low-Line bottles
2. 2M Medium-Line bottles
3. 0.5M High-Line bottles

2) Average Unit Sales Price by Product Line

1. $10 for each Low-Line bottle
2. $20 for each Medium-Line bottle
3. $50 for each High-Line bottle

And by extension, we now know:

3) Revenue by Product Line

This is given by multiplying Sales Quantity by Average Sales Price.
This we obtain by multiplying 1) & 2) at each Product Line:

1. Revenue for Low-Line bottles = 25M x $10 = $250M
2. Revenue for Medium-Line bottles = 2M x $20 = $40M
3. Revenue for High-Line bottles = 0.5M x $50 = $25M

...and by extension

4) Profits by Product Line.
This is the application of the per-Product Line Profit Margins of:

1. 10% for LL
2. 20% for ML
3. 30% for HL

This is given by multiplying Revenue by Product Line by Profit Margin per Product Line.
This we obtain by multiplying 3) & 4) at each Product Line:

1. Profit for LL = $250M × 10% = $25M
2. Profit for ML = $40M x 20% = $8M
3. Profit for HL = $25M x 30% = $7.5M

And by extension, we further know:

5) Total Profits across all 3 Product Lines.
This is the sum of Profits for LL + Ptofits for ML + Profits for HL = $25M + $8M + $7.5M = $40.5M.

Therefore, $40.5M of Profits today is the baseline number over which we seek a 10% increase in Profits by adjusting Sales of LL and ML.

Let us proceed to Part B.

B] What is the algebraic expression that quantitatively alters sales quantities of LL & ML to increase A] by 10%?
We know that the algebra must incorporate:

1. New Profit from LL from New Sales Quantity of LL as reduced by X bottles
2. New Profit from ML from New Sales Quantity of ML as increased by X bottles (see i) above)
3. Old Profit from HL
4. Total Profits from Part A (see above)
5. Total Profits Increase by 10%

Therefore:

New Profits from New Sales Quantity of LL + 
New Profits from New Sales Quantity of ML + Old Profits from HL

 = 

Old Total Profits across (LL + ML + HL) × (100% + 10%)

Well,

New Profits from New Sales Quantity of LL
=
New Sales Quantity × Old Average Unit Sales Price × Old Profit Margin
= [(25M - X) x $10 × 10%]

and

New Profits from New Sales Quantity of ML
=
New Sales Quantity × Old Average Unit Sales Price × Old Profit Margin
=
[(2M + X) x $20 × 20%]

Therefore:

New Profits from New Sales Quantity of LL + New Profits from New Sales Quantity of ML + Old Profits from HL

 = 

Old Total Profits across (LL + ML + HL) × (100% + 10%).

translates to:

[(25M - X) x $10 × 10%] + [(2M + X) x $20 × 20%] + $7.5M
= 
Old Total Profits across (LL + ML + HL) × (100% + 10%)

So,

[(25M - X) x $10 × 10%] + [(2M + X) x $20 × 20%] + $7.5M = $40.5M x 110%.

SOLVE FOR X.

[(25M - X) x $10 × 10%] + [(2M + X) x $20 × 20%] + $7.5M = $44.55M

Which is...

[(25M - X) x $1] + [(2M + X) x $4] + $7.5M = $44.55M

Which is...

($25M - $X) + ($8M + $4X) + $7.5M = $44.55M

Which is...

• $25M - $X + $8M + $4X + $7.5M = $44.55M; Or
• $3X + $40.5M = $44.55M; Or
• $3X = $4.05M

And, X = $4.05M/$3
(The dollar sign cancels out affirming X is in Units)

And X = 1.35 million units.

Therefore, we must convert 1.35 million units of LL Cognac into ML Cognac to raise Profits across the entire Product porfolio by 10%, all other conditions held the same.

But, do the numbers work?
Let's see.

If we are converting 1.35 million units of LL Cognac to ML Cognac, it means we are now selling:

1) (25M - 1.35M) bottles of LL or 23.65M bottles
and
2) (2M + 1.35M) bottles ML Cognac or 3.35M bottles.

Therefore new LL Profits are:
23.65M × $10 × 10% = $23.65M
and new ML Profits are:
3.35M × $20 × 20% = $13.4M

Do New LL Profits + New ML Profits + Old HL Profits = Old Profits across all 3 Lines + $4.05M?

Does $23.65M + $13.4M + $7.5M = $40.5 + $4.05M

Yes.

Curious why you posed the question, though: is it part of a larger Prep strategy, or...?

(edited)

 [ Profit (new) - profit (old) ] / profit (old) = 10%

Profit new = [10 * (25-x) * 0.1] + [20*(2+x) * 0.2] + [50*0.5*0.3] 

Profit (old) = 10*25*0.1 + 20*2*0.2 + 50*0.5*0.3

X=volume send from low to medium

Just solve for x and you have your answer

Hi Francesca,

Tyrion and Francesco gave great answers here!

Just remember this is a breakeven question which is probably the most type of question in cases. Breakeven comes in many forms.

Critically, when figuring out any math in a case, get organized. The 1st step is to set things up at the highest level, and work your way down.

Hi Francesca,

Looks like you're covered on the math here, but just a quick note to say - make sure you walk through the explanations and understand the calculations well! This type of problem comes up a lot in case interviews, so it's crucial to make sure you know how to solve it.