Stumped by an IB technical I found while prepping for interviews

bump

Price $1.2A = $1B
Quantity (mm's) A = 70/1.2A // B = 30/$1B
Assuming the unit b is $1.00 that is 30 million "b" and 53.8m "a"
COGS is the same per unit so div that 70 by total units, like ~$0.80

So ~20% GM for "b" and ~35% GM for "a" using my phone calculator.

took me longer than it should, but i think (hope) i got it. There is probably a shorter way of doing it, would be cool if you could confirm should you have the right solution.

Let P be the Price of Segment B, so Price of Segment A is 1,2P, Let x1 be the quantity sold/produced by A and x2 be the quantity sold/produced by B.

We know that 70M+30M = 100M or in other words 1,2P*x1+P*x2 = 100. Furthermore, 1,2P*x1 = 70m so P*x1 equals 50m.

We need the cost of segment A to calculate the profit and then the margin. We know that the combined cost of segment A and B is 70m such that Combined Rev (100m) - Combined Costs (70m) / Rev (100m) = 30% Margin.

Let Pc be production costs which we assume to be equal for both A and B since they sell the same product. So the total costs for both segments is Pc*x1+Pc*x2 (assuming that we sell everything we produce) = 70M -> Can be rewritten as Pc*(x1+x2) = 70M.

We already know (see above) that P*x1 = 50M and P*x2 = 30M -> If we add both up we get P*(x1+x2) = 80M, so the sum of the produced/sold quantities of A and B  (x1+x2)  equals 80M / P, i.e 80M divided by the Price set by Segment B. 

We can now enter (x1+x2) = 80M/P for (x1+x2) in our equation for the combined costs. Thus, we get Pc*(80M/P)=70M which we can rewrite as Pc/P = 70M/80M = 0,875. So we know that the production costs make up 87,5% of the price of segment B.

So Pc = 0,875*P which we can enter back into our equation for the combined costs. We initialy formulated that equation as Pc*x1+Pc*x2 = 70m so know we get 0,875*P*x1+0,875*P*x2 = 70M -> We know that P*x1 =50M so the left term equals 0,875*50M = 43,75M which are our production costs for segment A. Thus, the production costs for segment B are 26,25M (70M-43,75M).

Since the combined costs (70M) equals the revenue of Segment A we know that the profit of segment A is 70M - 43,75M = 26,25M which equals the cost of segment B. 

Hence, the Margin of Segment A is (70M-43,75M)/70M = 26,25M/70M = 37,5%.

Profit of A = 26,25M and Margin of A = 37,5%

What a stupid interview question lol.

No clue if this is right or not:

30% mixed margin on 100 of revenue, so 30 of gross profit. If margins were the same for both (since they are identical products this means costs can be exactly the same as well) then 70% of this 30 gross profit would be contributed by A while the remaining would be contributed by B. So 21 gross profit from A and 9 from B.

However, we know A is priced 20% more then B. So that means A has higher margins then B by 20%, so under these assumptions A should contribute 70% x (1.2) of the total gross profit, 84%. The remainder is contributed by B.

84% of 30 is 25.2, the gross profit of A. You sold 70 of A, so gross margin is 25.2/70 = 36%

The remainder 16% of 30 is 4.8, which was contributed from sales of 30 of B, so gross margins of B are 4.8/30 = 16% 

Gross margin of A is higher then margins of B by 20% because you're selling the same stuff at a higher price. Had to make some assumptions about costs for this thought process to make sense though. 

Does this check out? 

This thread has revealed to me how bad I am at math, but I think I got it. 

- Segment A: 70 Revenue -> Qa = Quantity A -> 70 = 1.2P *Qa

- Segment B: 30 Revenue  -> Qb = Quantity B -> 30 = P * Qb

- Now let's solve for quantities in terms of P -> P = 70 / (1.2 * Qa) & P = 30 / Qb 

- Set them equal to each other -> 70/1.2 = 58.33 -> Qa = (58.33 / 30) * Qb -> Qa = 1.94433 Qb

- Now we know the total quantity is 1.94433 + 1 = 2.944333 -> Qa is (1.94433 / 2.94433) = .66036 or 66.04% 

- Total COGS is 70 (from gross margin) -> Cogs A = 66.04% * 70 = 46.228 & Cogs B = (1 - .6604) (70) = 23.772

- Total Revenue A - COGS A = Gross Profit A -> 70 - 46.228 = 23.772 & Gross Profit B -> 30 - 23.772 = 6.228 

- Gross Margin A = 23.772 / 70 = .3396 or 33.96% & Gross Margin B = 6.228 / 30 = .2075 or 20.75% 

- Sanity Check = (.3366) (70) + (.2075) (30) = 29.79 or ~30 -> some intermediate rounding error must have underweighted one

Let the equation 70x + 30y = 30 describe the problem, where there is 70M and 30M revenue for Segment A and B, respectively. The right side represents the margin. Then, x and y are the gross margins of Segments A and B. We know that Segment A has a 20% higher margin than B, and each segment produces the same goods with identical costs. Then, we know that the gross margin of Segment A is x = 0.2 + y to account for the higher fixed 20% margin in Segment A than in B.

The equation now reads: 70(.2+y) + 30y = 30, which simplifies to y = 16% and x = 36%. The gross margin of Segment A is 36%, and the gross profit is 70(.36) = $25.2M.

Not sure if this is the right method, but getting the same answer as @OOM.

I basically made x as the unit selling price for B. So since A is 20% higher, it sells for 1.2x.

Next, the number of units A sells would be 70/1.2x (Total Price / Price per unit = # of units). Similarily, for B it would be 30/x.

Now this is the part where I am unsure whether I got lucky. But, assume that the cost per unit is $1 for each of them (they're identical products). I assumed that it is WLOG, because the cost would just get reflected in the selling price. 

Then, the COGS for A would be 70/1.2x * 1 and B would be 30/x * 1. Adding this up gets 70M. 

70/1.2x + 30/x = 70

Solving for x, we get x = 1.261905. So the GP for A = 23.77358 M and the GP for B = 6.226415 M

The GM for A = 33.96226% and the GM for B = 20.7547%