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How do you solve this question?

A company has 3 products. They contribute to 30%, 30% and 40% of sales respectively.

They have profit margins of 15%, 30%, and 50% respectively.

If the client raises prices by 10% for Product A (assuming costs remain constant), what is the change in the company's total gross profit?

Is sales the same as total revenue (TR) or is it the same as profit ($\pi = TR - TC$)?

If it's the same as total revenue, I know that product A contributes 30% to the total sales, i.e.

$$ 0.3 \cdot TR = TR_A $$

Profit margin is given as $\tfrac{TR - TC}{TR}$, and if the profit margin of profit A is 10% then $$ \frac{1.1 \cdot TR - TC}{1.1 \cdot TR} = 1 - \frac{TC}{1.1 \cdot TR} $$

If the client raises prices by 10% for product A, the total revenue (TR) is now $$ TR' = 1.1 \cdot P \cdot Q = 1.1 \cdot TR $$

But how do i summarize? How can I conclude what the change in company's total gross profit is?

Jamgreen
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1 Answers1

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You have confused yourself by notation. You say that if the profit margin of product A is 10% then $$\frac{1.1 \cdot TR - TC}{1.1 \cdot TR} = 1 - \frac{TC}{1.1 \cdot TR}$$.

  1. A profit margin of 10% is never mentioned in the question.
  2. Your formula is referring, I think, to the revenue and costs for product A: so it should be $$\frac{1.1 \cdot TR_A - TC_A}{1.1 \cdot TR_A} = 1 - \frac{TC_A}{1.1 \cdot TR_A}$$

But frankly, don't be so algebraic. It makes a mess.

Assume the company makes sales of £100. Then the sales of A are £30, the sales of B are £30, and the sales of C are £40.

The cost of A is £30/1.15, the cost of B is £30/1.30, and the cost of C is £40/1.50. Add those up, and you get the total costs. Subtract those costs from £100, and you get the total gross profit.

If the company raises prices by 10% for product A, and the number of units sold remains constant (which is not stated), then the sales of A are £33, the sales of B are £30, and the sales of C are £40. Total sales are £103. Subtract the same old costs from £103, and you get the new total gross profit.

Divide the new total gross profit by the original total gross profit, subtract 1 and multiply by 100, and you get the percentage increase.

(I'm not actually going to do the calculations, so that you can make a contribution to this answer).

Martin Kochanski
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  • The costs are $30/1.15 + 30/1.3 + 40/1.5 = 26.1 + 23.1 + 26.7 = 75.9$. The profit is $100 - 75.9 = \boxed{24.1}$. The new sales is $103$, so the new profit is $103 - 75.9 = \boxed{27.1}$. The change in profit is $\tfrac{27.1 - 24.1}{24.1} = \boxed{12%}$? – Jamgreen May 29 '16 at 07:38
  • I'm not sure if I understand why the costs are $\text{sales} / (1 + \text{profit margin})$? – Jamgreen May 29 '16 at 07:40