A metalworker has a metal alloy that is 25% copper and another alloy that is 55% copper. How many kilograms of each alloy should the metalworker combine to create 90 kg of a 49% copper alloy?
amount of 25% copper needed ---- x kg
amount of 55% copper needed = 90-x kg
.25x + .55(90-x) = .49(90)
solve for x
To solve this problem, let's assume the metalworker combines x kilograms of the 25% copper alloy and y kilograms of the 55% copper alloy to get a total of 90 kg of a 49% copper alloy.
Given:
Amount of 25% copper alloy = x kg
Amount of 55% copper alloy = y kg
Total amount of the alloy = 90 kg
We can set up the following equation based on the copper content:
0.25x + 0.55y = 0.49(90)
Simplifying the equation:
0.25x + 0.55y = 44.1
To find the values of x and y, we need one more equation. Since the total amount of the alloy is 90 kg, we can set up another equation based on the total weight:
x + y = 90
Now we have a system of two equations:
0.25x + 0.55y = 44.1 ---(1)
x + y = 90 ---(2)
We can use substitution or elimination method to solve these equations. Let's use the substitution method:
Rearrange equation (2) to express x in terms of y:
x = 90 - y
Substitute the value of x in equation (1):
0.25(90 - y) + 0.55y = 44.1
Simplify and solve for y:
22.5 - 0.25y + 0.55y = 44.1
0.3y = 21.6
y = 72
Substitute the value of y in equation (2) to find x:
x + 72 = 90
x = 18
Therefore, the metalworker should combine 18 kg of the 25% copper alloy with 72 kg of the 55% copper alloy to create 90 kg of a 49% copper alloy.
To solve this problem, we can use the concept of a weighted average.
Let's assume the metalworker needs x kilograms of the 25% copper alloy and y kilograms of the 55% copper alloy to make 90 kg of a 49% copper alloy.
The copper content in the 25% alloy is 25/100 * x kg, and the copper content in the 55% alloy is 55/100 * y kg.
Since the total amount of copper in the final alloy is equal to 90 kg * 49/100, we can set up the equation:
25/100 * x + 55/100 * y = 90 * 49/100
Simplifying this equation:
0.25x + 0.55y = 44.1
Now, we have two variables and one equation. To solve this system of equations, we need another equation formed by the fact that the total weight of the alloys is 90 kg:
x + y = 90
Now, we have a system of equations:
0.25x + 0.55y = 44.1
x + y = 90
There are several methods to solve this system, such as substitution or elimination. Let's solve it using substitution.
Rearrange the equation x + y = 90 to obtain:
x = 90 - y
Substitute this value of x in the first equation:
0.25(90 - y) + 0.55y = 44.1
Simplify the equation:
22.5 - 0.25y + 0.55y = 44.1
Combine like terms:
0.3y = 21.6
Divide by 0.3 to solve for y:
y = 72
Now that we have found the value of y, we can substitute it back into x + y = 90 to solve for x:
x + 72 = 90
x = 18
Therefore, the metalworker should combine 18 kg of the 25% copper alloy and 72 kg of the 55% copper alloy to create 90 kg of a 49% copper alloy.