A pawn shop is selling gold for $12.00 an ounce and platinum for $9.00 an ounce. How much of each should be used to make a gold-platinum mixture of x pounds selling for $y an ounce?
Please explain to do the problem with an answer. :)
To solve this problem, we need to use a system of equations. Let's break it down step by step:
Step 1: Define the variables
- Let's assume that we will use "g" ounces of gold and "p" ounces of platinum to make the mixture.
- We also know that the total weight of the mixture will be "x" pounds, which is equal to 16x ounces.
Step 2: Set up the equations
- The first equation will represent the weight of the gold in the mixture: g = 16x.
- The second equation will represent the weight of the platinum in the mixture: p = 16x.
- The third equation will represent the price of the mixture: 12g + 9p = y(16x).
Step 3: Solve the system of equations
We can substitute the value of g and p from the first two equations into the third equation to eliminate the variables:
12(16x) + 9(16x) = y(16x)
192x + 144x = 16xy
336x = 16xy
Now we can solve for y by dividing both sides of the equation by 16x:
336/16 = y
21 = y
So, the mixture should sell for $21 an ounce.
Step 4: Find the amount of gold and platinum needed
Since we know the value of y, we can substitute this back into the original equation to solve for the amount of gold and platinum needed:
12g + 9p = 21(16x)
12g + 9p = 336x
Since g = p (from step 2), we can substitute this into the equation:
12g + 9g = 336x
21g = 336x
g = 16x
Therefore, we need 16x ounces of gold and 16x ounces of platinum to make the mixture.
In summary, to make a gold-platinum mixture of x pounds selling for $y an ounce, you would need 16x ounces of gold and 16x ounces of platinum. The mixture should sell for $21 an ounce.