A metal worker has a metal alloy that 30% copper and another one is 55% copper. how many kilograms of each alloy should the metalworker combine to make 50kg of a 50% copper alloy?
To solve this problem, we can use a method called the mixture equation. Let's assume the metalworker needs to combine x kilograms of the 30% copper alloy and y kilograms of the 55% copper alloy to obtain the desired 50% copper alloy.
Now, let's determine the amount of copper in each alloy:
- The 30% copper alloy contains 30% copper. So, the amount of copper in x kilograms of this alloy is 0.3x kilograms.
- The 55% copper alloy contains 55% copper. Therefore, the amount of copper in y kilograms of this alloy is 0.55y kilograms.
Since we want to make a 50% copper alloy, the total amount of copper in the resulting mixture should be 50% of the total weight, which is 50 kilograms. So we can write the equation:
0.3x + 0.55y = 0.5(50)
Simplifying this equation, we have:
0.3x + 0.55y = 25
Now, we need to solve this system of equations to find the values of x and y. There are several methods to solve systems of equations, such as substitution or elimination. In this case, we will use the elimination method.
First, let's multiply all terms of the equation by 100 to eliminate the decimal points:
30x + 55y = 2500
To eliminate the decimals, we can multiply the entire equation by 10:
3x + 5.5y = 250
Now, let's multiply the entire equation by 2 to eliminate the decimal:
6x + 11y = 500
Next, we can multiply the initial equation by -6 to eliminate the x term:
-6x - 11y = -750
Now, let's add these two equations:
(6x + 11y) + (-6x - 11y) = 500 + (-750)
The x terms cancel out:
0 = -250
This equation is not possible because it suggests that -250 equals 0. Therefore, there is no solution to the system of equations. This means it is not possible to create a 50% copper alloy by combining the 30% copper and 55% copper alloys provided.
If there are x kg of 30% alloy, then consider the total amount of copper in each part. It must add up to the total in the final alloy.
.30x + .55(50-x) = .50*50