Dimensional analysis is based on the principle that physical quantities that are equated must be represented in measures of the same basic quantities (e.g mass, length, or time) in order to draw conclusions about their relationships.
If Gold cost $8.00 per ounce
Then $15.00 will cost:

x = 1.875 ounces
Assuming x to be the amount of ounce, then Gold of $15.00 will cost 1.875 ounces.
Now, given that:
1000g = 1kg
1 kg = 2.2 lbs
1 lb = 16 oz
Then, to determine the amount of gram that is equivalent to 1.875 ounces, we have:
Since 1 lb = 16 oz
2.2 lbs = (2.2*16) oz
= 35.2 oz
Since 1 kg = 2.2lbs = 35.02 oz
∴
x(g) = 1 kg
1000 g = 35.02 oz
1 (g) = 35.02/1000 oz
1 (g) = 0.03502 oz
Thus, If 1 (g) = 0.03502 oz
x(g) = 1.875 oz

x = 53.54 g
Thus, we can conclude that amount of the grams you can purchase for $15.00 is 53.54 grams
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