The problem in the image describes a mixture problem with alloys containing copper. Let's break it down and solve it step by step.

Problem Details:

1. First alloy contains 5% copper.
2. Second alloy contains 14% copper, and its mass is 10 kg greater than the mass of the first alloy.
3. These two alloys are mixed to create a third alloy containing 12% copper.
4. Find the mass of the third alloy, with the answer in kilograms.

Step 1: Define Variables

Let:

• $x$: Mass of the first alloy (in kilograms),
• $x + 10$: Mass of the second alloy (since it is 10 kg heavier than the first alloy),
• $M = x + (x + 10)$: Total mass of the third alloy (sum of the masses of the first and second alloys).

Step 2: Copper Content in Each Alloy

• Copper in the first alloy: $0.05x$,
• Copper in the second alloy: $0.14(x + 10)$,
• Copper in the third alloy: $0.12M$.

Step 3: Balance Equation for Copper

The total copper in the third alloy comes from the copper in the first and second alloys: $0.05x + 0.14(x + 10) = 0.12(x + (x + 10)).$

Step 4: Expand and Simplify

Expand the terms: $0.05x + 0.14x + 1.4 = 0.12(2x + 10).$ Simplify: $0.19x + 1.4 = 0.24x + 1.2.$

Step 5: Solve for $x$

Rearrange terms: $1.4 - 1.2 = 0.24x - 0.19x,$ $0.2 = 0.05x.$ Solve for $x$: $x = \frac{0.2}{0.05} = 4 \, \text{kg}.$

Step 6: Find the Total Mass of the Third Alloy

The first alloy weighs $x = 4 \, \text{kg}$, and the second alloy weighs $x + 10 = 14 \, \text{kg}$. Therefore, the mass of the third alloy is: $M = 4 + 14 = 18 \, \text{kg}.$

Final Answer:

The mass of the third alloy is 18 kilograms.

Would you like a detailed explanation of any step, or do you have any questions? 😊

Follow-Up Questions:

1. What is the proportion of copper in the first alloy compared to the third?
2. If the second alloy was only 5 kg heavier than the first, how would the solution change?
3. What would happen if the percentage of copper in the third alloy increased to 15%?
4. How can this problem be generalized for $n$ alloys?
5. Can you formulate the problem as a system of equations in terms of total masses?

Tip:

In mixture problems, always set up a clear equation for the conserved quantity—in this case, the copper content.