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A goldsmith has two gold alloys. The first alloy is 20% gold and the second alloy is 60% gold.
How many grams of each alloy should be mixed to produce 80 grams that is 52% gold?
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Take F grams of the 20% gold alloy and S grams of the 60% gold alloy.
Make sure that
F + S = 80, (1) and
0.2F + 0.6S = 0.52*80. (2)
Solve this system for F and S:
F + S = 80, (1) and
0.2F + 0.6S = 41.6. (2) ( 41.6 = 0.52*80 )
You can solve the system using the substitution method.
Express S = 80 - F from equation (1) and substitute it into equation (2). You will get
0.2F + 0.6*(80-F) = 41.6,
0.2F + 48 - 0.6F = 41.6,
-0.4F = 41.6 - 48,
-0.4F = -6.4,
F = = 16.
So, the goldsmith should take 16 grams of the 20% gold alloy and 80-16 = 64 grams of the 60% gold alloy.
Check. 0.2*16 + 0.6*64 = 41.6 = 0.52*80.