The alloy of copper and zinc contains 20 kg of zinc. When 10 kg of zinc was added to the alloy, its percentage increased

The alloy of copper and zinc contains 20 kg of zinc. When 10 kg of zinc was added to the alloy, its percentage increased by 10%. Find the original weight of the alloy if it is less than 50 kg.

First of all, let us pay attention to the fact that the amount of copper before and after adding 10 kg of zinc to the alloy did not change.

Let’s designate the amount of copper in the alloy by x.

Then the initial mass of the alloy is x + 20.

The percentage of zinc in the alloy is equal to the ratio of the mass of zinc to the total mass of the alloy, therefore, in the original alloy, the percentage of zinc is 20 / (x + 20).

After adding 10 kg of zinc to the alloy, the mass of zinc in the alloy became
20 + 10 = 30 kg, the mass of the alloy also increased by 10 kg and amounted to x + 30. Consequently, the percentage of zinc became 30 / (x + 30).

According to the condition of the problem, after adding zinc to the alloy, the percentage of zinc increased by 10%:

30 / (x + 30) – 20 / (x + 20) = 0.1.

Solving the equation and applying the constraint condition
Multiply the left and right sides of the equation by the expression
(x + 20) (x + 30):

30 (x + 20) – 20 (x + 30) = 0.1 (x + 20) (x + 30);

30x + 600 – 20x – 600 = 0.1 (x ^ 2 + 30x + 20x + 600);

10x = 0.1 (x ^ 2 + 50x + 600);

x ^ 2 + 50x + 600 – 100x = 0;

x ^ 2 – 50x + 600 = 0;

D = 50 * 50 – 4 * 1 * 600 = 2500 – 2400 = 100;

x1 = (- (- 50) – √100) / 2 = (50 – 10) / 2 = 40/2 = 20;

x2 = (- (- 50) + √100) / 2 = (50 + 10) / 2 = 60/2 = 30.

So, the weight of copper in the alloy is either 20 kg or 30 kg. In the first case, the total weight of the alloy is 20 + 20 = 40 kg, in the second 20 + 30 = 50 kg. But according to the condition of the problem, the mass of the alloy is less than 50 kg. Only x1 satisfies this condition, so the second solution must be discarded.

Answer: the initial weight of the alloy is 40 kg.