Word problem about percents

Question

18 karat gold is 18/24 or 75% gold.  12 karat is 50%, how much 12 karat should be mixed with pure gold to obtain 60 grams of 16 karat gold?

Answer 1

Let the amount of 12 karat gold (in grams) be X, and the amount of pure gold (in grams) be Y.

 

We end up with 60 grams of metal (gold and impurities) total, so

X + Y = 60.

 

Now, we can also look at things from the perspective of how much pure gold we have.

In X grams of 12 karat gold, there are 0.5X grams of pure gold.  And there are Y grams of pure gold in Y grams of pure gold (redundant, but can't leave a term out of our equation).  We also know that in that 60 grams of 16 karat gold, there are (16/24) * 60 grams, or 40 grams of pure gold. 

 

The amount of pure gold contributed from X and Y then have to add up to the amount of pure gold in the 60 grams of 16 karat gold.  This can be written as,

 

0.5X + Y = 40.

 

This is a system of linear equations:

 

X + Y = 60

0.5X + Y = 40

 

Subtracting the bottom equation from the top, we have

0.5X + 0 = 20, or X = 40 grams of 12 karat gold.