Question 754318: A goldsmith has two gold alloys. The first alloy is 20% gold; the second alloy is 60% gold. How many grams of each should be mixed to produce 40 grams of an alloy that is 30% gold?
Amount of 20% gold=_____g
Amount of 60% gold=_____g
Answer by anteater(7)   (Show Source):
You can put this solution on YOUR website! You can start by creating variables to stand for the amounts of 20% gold and 60% gold you have. For example, Let x be the amount of 20% gold and y be the amount of 60% gold.
You know that altogether you have 40 grams of gold. So, you can create this equation:
x + y = 40
The amount of gold in the 20% alloy is going to be 20%, or .20 times the amount of that alloy you have: .20 times x, or .20x
The amount of gold in the 60% alloy is going to be 60% or .60 times the amount of that alloy you have: .60 timex y, or .60y
Finally, you know that when you combine those alloys, that 30% of the resulting mix will be gold: .30 times 40, or 12 grams of gold
You can use that information to create this equation:
.20x + .60y = 12
Now you have two equations and two variables, which is sufficient information to figure out what quantities the variables stand for. You can use either substitution or elimination to solve this system of equations. In this case, I think elimination would be the simplest, since it will involve fewer steps.
We have these equations:
x + y = 40
.20x + .60y = 12
In order to make the second equation "nicer" to work with, I am going to multiply both sides of that equation by 10. Since we are doing this to both sides of the equation, the resulting statement will still be "true". However, we won't have decimals to worry about, then:
10(.20x + .60y) = 10(12)
2x + 6y = 120
So now, we can work with these two equations instead:
x + y = 40
2x + 6y = 120
In using elimination to find one of the variables, we choose a variable to "eliminate". For example, we could decide to eliminate x. A way to do that would be to multiply the first equation by -2, then "add" that equation to the second one, so that the x terms "cancel" each other out (add to 0):
-2(x + y) = -2(40)
2x + 6y = 120
-->
-2x + (-2y) = -80
2x + 6y = 120
______________
0 + 4y = 40
Solve for y:
4y/4 = 40/4 --> y = 10 grams
Now that we have y, we go back to one of our previous equations involving x and y, substitute 10 for y, and solve the resulting equation for x. The first equation, x + y = 40, should work:
x + y = 40
x + 10 = 40
x = 30
So, you have 30 grams of the 20% gold, and 10 grams of the 60% gold.
To check this, 30 + 10 = 40
.20(30) + .60(10) = 6 + 6 = 12 and .3(40) = 12
So (x,y) = (30, 10) is your solution. :)
I hope this was helpful!
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