a metalworker has a metal alloy thay is 25% copper and another alloy that is 60% copper. how many kilograms of each alloy should the metalworker combine to create 90kg of a 53% coppet alloy
To solve this problem, we can use the concept of the weighted average. The idea is to find the right quantities of the two alloys to achieve the desired copper concentration.
Let's denote the amount of the 25% copper alloy as x (in kg) and the amount of the 60% copper alloy as y (in kg). We know that x + y = 90kg, as we need a total of 90kg of the final alloy.
We can now set up equations based on the copper concentration:
0.25x + 0.60y = 0.53 * 90
Here, 0.25x represents the amount of copper in the 25% copper alloy, and 0.60y represents the amount of copper in the 60% copper alloy. The right-hand side of the equation represents the desired concentration of copper in the final alloy (53% of 90kg).
Now we can solve the equations to find the values of x and y.
To solve this problem, we can set up a system of equations based on the amount of copper in each alloy.
Let's assume the metalworker combines x kilograms of the 25% copper alloy and y kilograms of the 60% copper alloy.
The amount of copper in the 25% alloy is 0.25x kilograms, and the amount of copper in the 60% alloy is 0.60y kilograms.
Since we want to create 90 kilograms of a 53% copper alloy, the amount of copper in the final alloy is 0.53 * 90 = 47.7 kilograms.
Using the information above, we can set up the following equations:
Equation 1: 0.25x + 0.60y = 47.7
Equation 2: x + y = 90
Now, we can solve this system of equations to find the values of x and y.
Let's use substitution method.
From Equation 2, we can express x in terms of y: x = 90 - y
Substituting this value of x in Equation 1:
0.25(90 - y) + 0.60y = 47.7
22.5 - 0.25y + 0.60y = 47.7
0.35y = 47.7 - 22.5
0.35y = 25.2
y = 25.2 / 0.35
y ≈ 72
Now, substituting the value of y in Equation 2:
x + 72 = 90
x = 90 - 72
x = 18
Therefore, the metalworker should combine 18 kilograms of the 25% copper alloy and 72 kilograms of the 60% copper alloy to create 90 kilograms of a 53% copper alloy.