A metalworker has a metal alloy that is 25% copper and another alloy that is 65% copper. How many kilograms of each alloy should the metalworker combine to create 50 kg of a 57% copper alloy?
To solve this problem, we can use the method of mixtures.
Let's assume the metalworker combines x kilograms of the 25% copper alloy and (50 - x) kilograms of the 65% copper alloy.
Now, we can calculate the amount of copper in each alloy:
Amount of copper in the 25% copper alloy = 25% * x
Amount of copper in the 65% copper alloy = 65% * (50 - x)
Since we want to create a 57% copper alloy, the amount of copper in the final alloy will be:
Amount of copper in the final alloy = 57% * 50
Now, we can set up an equation to solve for x:
25% * x + 65% * (50 - x) = 57% * 50
0.25x + 0.65(50 - x) = 0.57 * 50
0.25x + 32.5 - 0.65x = 28.5
-0.40x = -4
x = -4 / -0.40
x = 10
Therefore, the metalworker should combine 10 kilograms of the 25% copper alloy and (50 - 10) = 40 kilograms of the 65% copper alloy to create 50 kilograms of a 57% copper alloy.
To solve this problem, we can use the concept of a mixture equation. Let's assume that the metalworker needs x kilograms of the 25% copper alloy and y kilograms of the 65% copper alloy.
To find the equation, we can start by figuring out the total amount of copper in each alloy:
For the 25% copper alloy:
Amount of copper = 25% x (x kg) = 0.25x kg
For the 65% copper alloy:
Amount of copper = 65% x (y kg) = 0.65y kg
Now we need to write the equation for the final alloy composition. The total amount of copper in the resulting 50 kg alloy should be 57% of the total weight:
Amount of copper = 57% x (50 kg) = 0.57 x 50 kg = 28.5 kg
Now we can set up the equation:
0.25x + 0.65y = 28.5
Since the total weight of the alloy is 50 kg, we also have:
x + y = 50
Now we have a system of two equations with two unknowns. We can solve this system to find the values of x and y.
One way to solve this system is by substitution. Let's solve the second equation for x:
x = 50 - y
Now we substitute this expression for x in the first equation:
0.25(50 - y) + 0.65y = 28.5
Simplifying this equation:
12.5 - 0.25y + 0.65y = 28.5
Combine like terms:
0.4y = 16
Divide by 0.4:
y = 40
Substituting this value for y in the second equation:
x + 40 = 50
x = 10
Therefore, the metalworker should combine 10 kg of the 25% copper alloy with 40 kg of the 65% copper alloy to create 50 kg of a 57% copper alloy.
.25 x + .65 y = .57 * 50
x+y = 50