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
grams 20 % = a
grams 60% = b
total gold = .2 a + .6 b
total mass = a + b = 40 so b = 40-a
so
.2 a + .6 b = .3 (40) = 12
.2 a + .6 (40-a) = 12
-.4 a + 24 = 12
.4 a = 12
a = 30
so
b = 10
It can be difficult to come up with formulas from word problems.
Your going to add some weight of the first alloy and some weight of the second allow to get 40 grams.
let x be the weight used from the first alloy and y the weight from the second.
so x+y=40
How much gold is in the 40 grams?
Well it's 30% gold, so
40*(.3)=12g of gold
likewise the amount of gold present from the first alloy would be 0.2x
the amount in the 60% sample is 0.6y
those two amounts add together to be the 12g of gold.
0.2x+0.6y=12
so two formulas, two variables.
if x+y=40 then
x=40-y
plug it in
0.2(40-y)+0.6(y)=12
8-0.2y+0.6y=12
0.4y=4
y=10
therefore x=30
check that it works to be safe
0.2(30)+0.6(10)=12
6+6=12
excellent.
hope this helps!
There is a rule for this class of problems of finding proportions of mixtures. Most of the time this can be done mentally, for example by chemists or nurses.
We have one ingredient A at 20%, and B at 60%. The required mixture is 30% (must be between those of A & B).
Take the difference between the ingredients, namely 60-20=40%.
The target percentage lies at 10% from A and 30% from B, or in the ratio 1:3
The proportions required for A and B will be 3:1, namely 30g of A and 10g of B.
Interesting approach Mathmate. Hadn't seen that before.
To solve this problem, we can use the concept of mixtures.
Let x represent the amount (in grams) of the 20% gold alloy.
Therefore, the amount of the 60% gold alloy will be (40 - x) grams since the total amount is 40 grams.
In the 20% gold alloy, 20% of x grams is gold, so we have 0.2x grams of gold.
In the 60% gold alloy, 60% of (40 - x) grams is gold, so we have 0.6(40 - x) grams of gold.
To find the total amount of gold in the mixture, we need to add these two amounts:
0.2x + 0.6(40 - x)
Since we want the final alloy to be 30% gold, the total amount of gold in the mixture should be 30% of the final 40 grams. So, we have:
0.2x + 0.6(40 - x) = 0.3(40)
Now we can solve this equation to find the values of x and (40 - x).
0.2x + 24 - 0.6x = 12
Combining like terms:
-0.4x + 24 = 12
Subtracting 24 from both sides:
-0.4x = -12
Dividing both sides by -0.4:
x = -12 / -0.4
x = 30
To find the amount of the 60% gold alloy, we subtract x from the total:
40 - x = 40 - 30 = 10
Therefore, you should mix 30 grams of the 20% gold alloy with 10 grams of the 60% gold alloy to produce 40 grams of an alloy that is 30% gold.