Final answer:
4 ounces of 14-karat gold and 2 ounces of 8-karat gold should be added to make 6 ounces of 12-karat gold.
Step-by-step explanation:
Let's denote the amount of 14-karat gold to be added as ( x ) ounces and the amount of 8-karat gold as y ounces.
The total amount of gold in 14-karat gold is ( {14} / {24}x ) (since 14-karat gold is 14 parts out of 24 parts gold).
The total amount of gold in 8-karat gold is ( {8} / {24}y ) (since 8-karat gold is 8 parts out of 24 parts gold).
The total amount of gold in the resulting 12-karat gold is ( {12} / {24}(x + y) ).
Since we want to make 6 ounces of 12-karat gold, we can set up the equation:
{14} / {24}x + {8} / {24}y = {12} / {24}(x + y)
To solve for x , we can simplify the equation:
{14} / {24}x + {8} / {24}y = {12} / {24}x + {12} / {24}y
{2} / {24}x = {4} / {24}y
Now, multiply both sides by 12 to get rid of the fractions:
x = 2y
Now, we know the relationship between \( x \) and \( y \). We also know that the total amount of gold is 6 ounces:
x + y = 6
Substitute the relationship \( x = 2y \) into this equation:
2y + y = 6
3y = 6
y = 2
Now that we have the value for y , we can find x :
x = 2 x 2 = 4
So, 4 ounces of 14-karat gold and 2 ounces of 8-karat gold should be added to make 6 ounces of 12-karat gold.
Now, if you want to graph this, you can use a coordinate system where ( x ) represents the ounces of 14-karat gold, ( y ) represents the ounces of 8-karat gold, and plot the equation ( x + y = 6 ), along with the constraint ( x = 2y ). Unfortunately, I can't provide a visual graph in this text format, but you can use graphing software or online graphing tools to plot these equations.