Solve the problem. Anne and Nancy use a metal alloy that is 24.9% copper to make jewelry. How many ounces of a 20% alloy must be mixed with a 27% alloy to form 100 ounces of the desired alloy?
To solve this problem, we need to set up a system of equations.
Let's represent the amount of the 20% alloy as x (in ounces). Since we want a total of 100 ounces of the desired alloy, the amount of the 27% alloy would be 100 - x.
Now, we need to set up the equation based on the copper content.
The amount of copper in the 20% alloy is 0.20x ounces.
The amount of copper in the 27% alloy is 0.27(100 - x) ounces.
The total amount of copper in the final alloy is 24.9% of 100 ounces, which is 0.249(100) ounces.
Setting up the equation:
0.20x + 0.27(100 - x) = 0.249(100)
Now we can solve for x:
0.20x + 27 - 0.27x = 24.9
0.20x - 0.27x = 24.9 - 27
-0.07x = -2.1
x = -2.1 / (-0.07)
x = 30
Therefore, 30 ounces of the 20% alloy should be mixed with 70 ounces of the 27% alloy to form 100 ounces of the desired alloy.
To solve this problem, we can set up a system of equations.
Let's say the amount of the 20% alloy that needs to be mixed is x ounces, and the amount of the 27% alloy is y ounces.
Based on the problem, we have two equations:
Equation 1: The total amount of alloy is 100 ounces: x + y = 100
Equation 2: The resulting alloy is 24.9% copper, so the amount of copper from the 20% alloy plus the amount of copper from the 27% alloy equals 24.9% of 100 ounces: 0.20x + 0.27y = 0.249(100)
Now we can solve this system of equations to find the values of x and y.
First, let's rearrange Equation 1 to solve for x: x = 100 - y
Now we substitute this value of x into Equation 2:
0.20(100-y) + 0.27y = 0.249(100)
Now we simplify the equation:
20 - 0.20y + 0.27y = 24.9
Combine like terms:
0.07y = 4.9
Divide both sides by 0.07:
y = 70
Now substitute this value of y back into Equation 1 to find the value of x:
x = 100 - y = 100 - 70 = 30
Therefore, to form 100 ounces of the desired alloy, you need to mix 30 ounces of the 20% alloy with 70 ounces of the 27% alloy.