A goldsmith has two gold alloys. The first alloy is 30% gold; the second alloy is 80% gold. How many grams of each should be mixed to produce 50 grams of an alloy that is 68% gold?
amount of 30% gold ______grams
amount of 80% gold ______grams
let x be the amount of 30% gold
then 50-x is the amount of 80% gold
solve for x:
.3x + .8(50-x) = .68(50)
a chemist has two alloys one of which is 5% gold and 15% lead and the other of which is 25% gold and 30% lead how many grams of each of the two alloys should be used to make an alloy that contains 40g of gold and 93g of lead ?
To determine the amounts of each alloy needed, we can set up a system of linear equations based on the given information.
Let's assign variables to represent the amounts of each alloy:
Let x represent the amount of the 30% gold alloy.
Let y represent the amount of the 80% gold alloy.
We know that the total alloy produced is 50 grams, so the sum of x and y should be equal to 50:
x + y = 50
We also know that the resulting alloy should be 68% gold. Since the 30% gold alloy and the 80% gold alloy are being combined, we can set up the following equation:
(0.3 * x + 0.8 * y) / 50 = 0.68
Now we can solve this system of equations to find the values of x and y.
First, let's rearrange the first equation for x:
x = 50 - y
Next, substitute this value of x in the second equation:
(0.3 * (50 - y) + 0.8 * y) / 50 = 0.68
Now, we can solve for y:
(15 - 0.3y + 0.8y) / 50 = 0.68
15 + 0.5y = 0.68 * 50
15 + 0.5y = 34
Subtracting 15 from both sides:
0.5y = 34 - 15
0.5y = 19
Dividing by 0.5:
y = 38
Now, substitute this value of y back into the first equation to find x:
x = 50 - y
x = 50 - 38
x = 12
So, the amount of the 30% gold alloy needed is 12 grams, and the amount of the 80% gold alloy needed is 38 grams.