Respuesta :
Given that the jeweler has alloys that are 25% gold, 50% gold, and 82% gold.
As he wants to make 14 grams of an alloy by adding two different alloys that is precisely 75% gold, so one alloy must have a percentage of gold more than 75%.
One alloy is 82% gold and, the second can be chosen between 25% gold, 50% gold, so there are two cases.
Case 1: 82% gold + 50% gold
Let x grams of 82% gold and y grams of 50% gold added to make x+y=14 grams of 75% gold, so
75% of 14 = 82% of x + 50% of y
[tex]\Rightarrow 75/100 \times 14 = 82/100 \times x + 50/100 \times y \\\\[/tex]
[tex]\Rightarrow 75/100 \times 14 = 82/100 \times x + 50/100 \times (14-x)[/tex] [as x+y=14]
[tex]\Rightarrow 75 \times 14 = 82 \times x + 50 \times (14-x) \\\\\Rightarrow 75 \times 14 = 82 \times x + 50 \times14-50\times x \\\\\Rightarrow 75 \times 14 = 32 \times x + 50 \times14 \\\\\Rightarrow 32 \times x =75 \times 14 - 50 \times14 \\\\[/tex]
[tex]\Rightarrow x =(25 \times 14)/32=10.9375[/tex] grams
and [tex]y = 14-x= 14-10.9375=3.0625[/tex] grams.
Hence, 10.9375 grams of 82% gold and 3.0625 grams of 50% gold added to make 14 grams of 75% gold.
Case 2: 82% gold + 25% gold
Let x grams of 82% gold and y grams of 25% gold added to make x+y=14 grams of 75% gold, so
75% of 14 = 82% of x + 25% of y
[tex]\Rightarrow 75/100 \times 14 = 82/100 \times x + 25/100 \times y \\\\\Rightarrow 75/100 \times 14 = 82/100 \times x + 25/100 \times (14-x) \\\\ \Rightarrow 75 \times 14 = 82 \times x + 25 \times (14-x) \\\\\Rightarrow 75 \times 14 = 82 \times x + 25 \times14-25\times x \\\\\Rightarrow 75 \times 14 = 57 \times x + 25 \times14 \\\\\Rightarrow 57 \times x =75 \times 14 - 25 \times14 \\\\[/tex]
[tex]\Rightarrow x =(50 \times 14)/57=12.28[/tex] grams
and [tex]y = 14-x= 14-12.28=1.72[/tex] grams.
Hence, 12.28 grams of 82% gold and 1.72 grams of 50% gold added to make 14 grams of 75% gold.
