A jeweler wants to make a silver alloy to be used to make necklaces. How many ounces of a silver alloy that costs $3.50 per ounce should be mixed with one that costs $7.00 per ounce to make a new 30-ounce alloy that costs $6.30 per ounce?
Hmmm. I may have been hasty. Anyway, if there are x oz. of $3.50 alloy, the rest (30-x) is the $7.00 alloy. So, check to be sure that the value of silver adds up:
3.50x + 7.00(30-x) = 6.30(30)
x = 6
we've done a lot of these mixture problems for you. Why not try some yourself, and we can check your work...
To solve this problem, we'll use a weighted average formula. Let's denote the number of ounces of the silver alloy that costs $3.50 per ounce as x.
1. Determine the amount of the silver alloy that costs $7.00 per ounce. Since the total weight of the mixture is 30 ounces, the amount of the $7.00 per ounce alloy will be 30 - x ounces.
2. Calculate the total cost of the alloy. The total cost is the sum of the cost of the $3.50 alloy and the cost of the $7.00 alloy.
Cost of $3.50 alloy = (cost per ounce) x (number of ounces) = $3.50x
Cost of $7.00 alloy = (cost per ounce) x (number of ounces) = $7.00(30 - x)
Total cost = $3.50x + $7.00(30 - x)
3. Set up the equation for the average cost. The average cost is given as $6.30 per ounce, so the equation becomes:
Total cost / Total ounces = average cost per ounce
($3.50x + $7.00(30 - x)) / 30 = $6.30
4. Solve the equation for x.
$3.50x + $7.00(30 - x) = $6.30 * 30
5. Simplify and solve for x.
$3.50x + $210 - $7.00x = $189
-$3.50x = -$21
x = 6
This means that 6 ounces of the silver alloy that costs $3.50 per ounce should be mixed with 30 - 6 = 24 ounces of the silver alloy that costs $7.00 per ounce to make a new 30-ounce alloy that costs $6.30 per ounce.