x=22%, y=30%, z=42%. You want 80 g of 34%. 30% is two times the amount of 22%.

Question

Stewart's metals has three silver alloys on hand. One is 22% silver, another is 30% silver, and the third is 42% silver. How many grams of each alloy is required to produce 80 grams of a new alloy that is 34% silver if the amount of 30% alloy used is twice the amount of 22% alloy used.

Answer 1

 

Hi Elizabeth,

 

If x, y, and z denote the amount of 22%, 30%, and 42% silver metal, respectively,

 

we get the following three equations.

 

     x + y + z = 80     (Eq. 1, from the total weight of the new alloy)

 

     0.22x + 0.30y + 0.42z = 0.34 · 80     (Eq. 2, from the amount of silver in the new alloy)

 

     2x = y     (Eq. 3, from the given condition)

 

By putting Eq. 3 into Eq. 1 and Eq. 2, we get

 

     3x + z = 80     (Eq. 4)

 

     0.82x + 0.42z = 27.2     (Eq. 5)

 

By solving the system of Eq. 4 and Eq. 5, we get

 

     x = 6.4/0.44 = 14.5455 and

 

     z = 16/0.44 = 36.3636

 

By putting x into Eq. 3, we get

 

     y = 12.8/0.44 = 29.0909

 

∴ x = 14.5455,   y = 29.0909,   z = 36.3636