Question 1170861
<font color=black size=3>
kt = karat
24 kt = 100% pure gold
18 kt = (18/24)*100% = (0.75)*100% = 75%
This indicates the 18 kt stock is 75% pure gold


Similarly,
10 kt = (10/24)*100% = (0.41667)*100% = 41.667%
The 10 kt stock is roughly 41.667% pure gold.


And,
16 kt = (16/24)*100% = 66.667%
The 16 kt stock is roughly 66.667% pure gold


I'll use the fractional form of each decimal value to avoid rounding errors.


In other words,
18 kt = (18/24) of pure gold = (3/4) of pure gold
10 kt = (10/24) of pure gold = (5/12) of pure gold
16 kt = (16/24) of pure gold = (2/3) of pure gold


Fractions can be a pain to work with, but they provide exact values.


--------------------------------------------------------------------------------


Let 
x = amount of the 18 kt gold stock
y = amount of the 10 kt gold stock
amounts are in grams


If steve has x grams of 18 kt gold, then he has (3/4)x grams of pure gold. Recall that in the previous section 3/4 = 75% was the amount of pure gold from this stock. Effectively we have (3/4)x grams of pure gold (24 kt gold) mixed in with some other metal, likely copper or iron perhaps. This means the other 1/4 is the other metal.


If steve has y grams of 10 kt gold, then he has (10/24)y grams of pure gold. The fraction 10/24 reduces to 5/12. So (10/24)y = (5/12)y


The two amounts of pure gold add to 
(3/4)x+(5/12)y


Steve wants to have 200 grams of 16 kt gold, so he wants 200*(2/3) = 400/3 grams of pure gold.
The 2/3 is from reducing 16/24.


Set the result 400/3 equal to the previous sum we got, and we end up with this equation
(3/4)x+(5/12)y = 400/3
This equation represents summing the pure gold amounts of each stock, getting a final amount of pure gold for the mixture. 


To make life easier, multiply both sides by the LCD 12 to clear out the fractions
(3/4)x+(5/12)y = 400/3
12 * [(3/4)x+(5/12)y] = 12*400/3
9x+5y = 1600
Now everything is a whole number which is fairly nice.


--------------------------------------------------------------------------------


Since he wants the final amount to be 200 grams, this must mean the subtotals of x and y have to add to 200


We can then solve for y getting
x+y = 200
x+y-x = 200-x
y = 200-x


Plug this into the equation we found at the conclusion of the last section. Then solve for x.
9x+5y = 1600
9x+5(y) = 1600
9x+5(200-x) = 1600
9x+1000-5x = 1600
4x+1000 = 1600
4x+1000-1000 = 1600-1000
4x = 600
4x/4 = 600/4
x = 150


He needs 150 grams of the 18 kt gold stock.


Use this to find y
y = 200-x
y = 200-150
y = 50


He needs 50 grams of the 10 kt gold stock


--------------------------------------------------------------------------------
--------------------------------------------------------------------------------

Answers:
150 grams of the 18 kt gold stock.
50 grams of the 10 kt gold stock.

</font>