A goldsmith has two gold alloys. The first alloy is 25% gold and the second is 55% gold. How many grams of each should be mixed to produce 40 grams of an alloy that is 43%?
To solve this problem, you can use a technique called the "mixture" or "average" equation.
Let's assume that x grams of the first alloy (25% gold) are mixed with (40 - x) grams of the second alloy (55% gold) to obtain 40 grams of the final alloy, which is 43% gold.
To find the solution, follow these steps:
1. Calculate the amount of gold in the first alloy: 25% of x grams = 0.25x grams
2. Calculate the amount of gold in the second alloy: 55% of (40 - x) grams = 0.55(40 - x) grams
3. Add the amounts of gold from both alloys to find the total amount of gold in the final alloy: 0.25x grams + 0.55(40 - x) grams
4. Set up the equation using the average equation:
Total gold in final alloy = Average gold percentage × Total weight of the final alloy
(0.25x grams + 0.55(40 - x) grams) = 0.43 × 40 grams
Now, you can solve this equation to find the value of x.
0.25x + 0.55(40 - x) = 0.43 × 40
Let's distribute 0.55 into (40 - x):
0.25x + 22 - 0.55x = 17.2
Combine like terms:
-0.3x + 22 = 17.2
Subtract 22 from both sides:
-0.3x = -4.8
Divide both sides by -0.3:
x = -4.8 / -0.3
x = 16
So, you would need 16 grams of the first alloy (25% gold) and (40 - 16) = 24 grams of the second alloy (55% gold) to obtain 40 grams of the final alloy that is 43% gold.