A metalworker has a metal alloy that is 20% copper and another alloy that is 65% copper. How many kilograms of each alloy should the metalworker combine to create 120 kg of a 47% copper alloy?
Let x be the number of kilograms of the 20% copper alloy.
Then 120 - x is the number of kilograms of the 65% copper alloy.
The amount of copper from the 20% copper alloy is 0.2x.
The amount of copper from the 65% copper alloy is 0.65(120 - x).
The total amount of copper in the final alloy is 0.47(120) = 56.4 kg.
So we have the equation:
0.2x + 0.65(120 - x) = 56.4
Simplifying the equation, we get:
0.2x + 78 - 0.65x = 56.4
-0.45x = -21.6
Dividing by -0.45, we find:
x = 48
So 48 kg of the 20% copper alloy should be combined with 120 - 48 = 72 kg of the 65% copper alloy to create 120 kg of a 47% copper alloy.
Gabe Amodeo, a nuclear physicist, needs 60 liters of a 40% acid solution. He currently has a 20% solution and a 50% solution. How many liters of each does he need to make the needed 60 liters of 40% acid solution?
Let x be the number of liters of the 20% acid solution that Gabe needs.
Then 60 - x is the number of liters of the 50% acid solution that Gabe needs.
The amount of acid from the 20% acid solution is 0.2x.
The amount of acid from the 50% acid solution is 0.5(60 - x).
The total amount of acid in the final solution is 0.4(60) = 24 liters.
So we have the equation:
0.2x + 0.5(60 - x) = 24
Simplifying the equation, we get:
0.2x + 30 - 0.5x = 24
-0.3x = -6
Dividing by -0.3, we find:
x = 20
So Gabe needs 20 liters of the 20% acid solution and 60 - 20 = 40 liters of the 50% acid solution to make 60 liters of a 40% acid solution.