A metal worker has a metal alloy that 15% copper and another one is 70% copper. how many kilograms of each alloy should the metalworker combine to make 120kg of a 48% copper alloy?
15% copper alloy: 54 kg
70% copper alloy: 66 kg
To solve this problem, we can set up a system of equations based on the copper content in the alloys.
Let's assume the metal worker combines x kilograms of the 15% copper alloy and y kilograms of the 70% copper alloy to make 120kg of a 48% copper alloy.
The amount of copper in the 15% alloy can be calculated as 0.15x (15% is equal to 0.15 as a decimal).
The amount of copper in the 70% alloy can be calculated as 0.70y (70% is equal to 0.70 as a decimal).
We can then set up the following system of equations based on the copper content:
Equation 1: x + y = 120 (since the total weight of the alloys combined is 120kg)
Equation 2: 0.15x + 0.70y = 0.48 * 120 (since 48% of 120kg should be copper)
Now we can solve this system of equations to find the values of x and y.
First, let's simplify Equation 2 by multiplying both sides by 100 to get rid of the decimals:
15x + 70y = 48 * 120
We can further simplify this equation:
15x + 70y = 5760
Now we have a system of equations:
Equation 1: x + y = 120
Equation 2: 15x + 70y = 5760
We can solve this system of equations using various methods such as substitution or elimination. Let's use the substitution method:
From Equation 1, we can rewrite it as x = 120 - y.
Substituting this value for x in Equation 2, we get:
15(120 - y) + 70y = 5760
Simplifying this equation:
1800 - 15y + 70y = 5760
1800 + 55y = 5760
55y = 3960
y = 72
Now that we have the value of y, we can substitute it back into Equation 1 to find x:
x + 72 = 120
x = 48
Therefore, the metal worker should combine 48kg of the 15% copper alloy and 72kg of the 70% copper alloy to make 120kg of a 48% copper alloy.
To find out how many kilograms of each alloy the metalworker should combine, we can use a system of linear equations. Let's denote the amounts of the 15% copper alloy and the 70% copper alloy as x and y, respectively. Since we want to make 120 kg of a 48% copper alloy, we can write the following equations:
Equation 1: x + y = 120 (the total weight of the alloys is 120 kg)
Equation 2: (0.15x + 0.7y) / 120 = 0.48 (the ratio of copper in the final alloy is 48%)
Now, let's solve this system of equations step by step:
Step 1: Solve Equation 1 for x: x = 120 - y
Step 2: Substitute x in Equation 2 with the value from Step 1: (0.15(120 - y) + 0.7y) / 120 = 0.48
Step 3: Simplify Equation 2: (18 - 0.15y + 0.7y) / 120 = 0.48
Step 4: Combine like terms: (18 + 0.55y) / 120 = 0.48
Step 5: Multiply both sides of the equation by 120 to eliminate the denominator: 18 + 0.55y = 0.48 * 120
Step 6: Simplify the right side: 18 + 0.55y = 57.6
Step 7: Subtract 18 from both sides: 0.55y = 57.6 - 18
Step 8: Simplify: 0.55y = 39.6
Step 9: Divide both sides by 0.55: y = 39.6 / 0.55
Using a calculator, we find that y ≈ 72 kilograms.
Step 10: Substitute this value of y back into Equation 1 to find x: x = 120 - y = 120 - 72 = 48 kilograms.
Therefore, the metalworker should combine 48 kilograms of the 15% copper alloy and 72 kilograms of the 70% copper alloy to make 120 kilograms of a 48% copper alloy.