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
[as x+y=14]
grams
and 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
grams
and grams.
Hence, 12.28 grams of 82% gold and 1.72 grams of 50% gold added to make 14 grams of 75% gold.