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