One alloy is 2 parts iron to 3 parts silver and another alloy is 7 parts iron to 3 parts silver. How much of each should be combined to produce a 30 pound alloy that is 1 part iron to 1 part silver?Please show me how to do this
The 1st alloy (x) is 2/5 iron and the 2nd alloy (30-x) is 7/10 iron. So, you want to mix 'em up so that
(2/5)x + (7/10)(30-x) = (1/2)(30)
x = 20
Check:
20 oz of Alloy 1 contains 8 oz iron
10 oz of Alloy 2 contains 7 oz iron.
Total is 15 oz in 30 oz of the mix, or a ratio of 1:1.
To solve this problem, we can use a system of equations. Let's assume that x represents the amount of the first alloy (2 parts iron to 3 parts silver) and y represents the amount of the second alloy (7 parts iron to 3 parts silver) that should be combined.
First, we need to set up equations based on the given information:
1. The total weight of the alloys is 30 pounds:
x + y = 30
2. The ratio of iron to silver in the resulting alloy is 1:1:
(2x + 7y)/(3x + 3y) = 1/1
Now, let's solve the system of equations:
Rewriting equation 2, we have:
2x + 7y = 3x + 3y
Rearranging equation 1, we have:
x = 30 - y
Substituting x in the second equation, we get:
2(30 - y) + 7y = 3(30 - y) + 3y
Simplifying:
60 - 2y + 7y = 90 - 3y + 3y
60 + 5y = 90
5y = 90 - 60
5y = 30
y = 30/5
y = 6
Substituting y back into equation 1, we find:
x + 6 = 30
x = 30 - 6
x = 24
Therefore, to produce a 30-pound alloy that is 1 part iron to 1 part silver, you would need 24 pounds of the first alloy (2 parts iron to 3 parts silver) and 6 pounds of the second alloy (7 parts iron to 3 parts silver).