A goldsmith has two alloys that are different purities of gold. The first is 3/4 pure and the second is 5/12 pure. How many ounces of each should be melted and mixed in order to obtain a 6 oz mixture that is 2/3 pure gold?
solve
(3/4)x + (5/12)(6-x) = (2/3)(6) ,each term times 12
9x + 5(6-x) = 48
etc
Since there are different quantities of each purity wouldn't there be an x and a y in the equation instead of both being x?
Ok,
x + y = 6 ----> y = 6-x
(3/4)x + (5/12)y = (2/3)(6)
sub in the value of y
mmmhhh!
oooh. okay! thanks!
To solve this problem, we can use a system of equations. Let's denote the number of ounces of the first alloy as x, and the number of ounces of the second alloy as y.
Step 1: Set up the equations
We have two equations based on the information given in the problem:
Equation 1: (3/4)x + (5/12)y = (2/3)(6) -- for the purity of gold
Equation 2: x + y = 6 -- for the total weight of the mixture
Step 2: Solve the system of equations
We can start by simplifying equation 1:
(3/4)x + (5/12)y = 4
Multiplying both sides of the equation by 12 to eliminate the denominators, we get:
9x + 5y = 32
Now, we can solve the system of equations using either substitution or elimination method. Let's use the substitution method here.
Rearrange equation 2 to express x in terms of y:
x = 6 - y
Substitute x in equation 1 with the value obtained:
9(6 - y) + 5y = 32
Simplify the equation:
54 - 9y + 5y = 32
-4y = 32 - 54
-4y = -22
Divide both sides of the equation by -4:
y = -22 / -4
y = 5.5
Substitute the value of y in equation 2 to find x:
x + 5.5 = 6
x = 6 - 5.5
x = 0.5
Step 3: Answer the question
According to the solution, we need 0.5 ounces of the first alloy and 5.5 ounces of the second alloy to obtain a 6 oz mixture that is 2/3 pure gold.