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 10 grams of an alloy that is 40% gold?

amount of 30% gold ___ g
amount of 80% gold ___ g

Let's assume that x grams of the 30% gold alloy are required and y grams of the 80% gold alloy are required to produce 10 grams of an alloy that is 40% gold.

To find the amount of gold in each alloy, we can multiply the percentage of gold by the weight of the alloy.

For the 30% gold alloy, the amount of gold is 0.30x grams.

For the 80% gold alloy, the amount of gold is 0.80y grams.

Since we want to produce 10 grams of an alloy that is 40% gold, the amount of gold in the final alloy will be 0.40 * 10 = 4 grams.

Now we can set up a system of two equations to solve for x and y:

Equation 1: x + y = 10 (since the total weight of the alloys is 10 grams)

Equation 2: 0.30x + 0.80y = 4 (since the total amount of gold in the final alloy is 4 grams)

From Equation 1, we can solve for x: x = 10 - y.

Substituting this value of x into Equation 2, we get:

0.30(10 - y) + 0.80y = 4

3 - 0.30y + 0.80y = 4

0.50y = 1

y = 2

Now we can substitute this value of y back into Equation 1 to solve for x:

x + 2 = 10

x = 8

Therefore, the goldsmith should mix 8 grams of the 30% gold alloy and 2 grams of the 80% gold alloy to produce 10 grams of an alloy that is 40% gold.

To find the amount of each gold alloy needed to produce an alloy that is 40% gold, we can use a basic equation:

(amount of 30% gold) + (amount of 80% gold) = 10 grams

We need to find the values for (amount of 30% gold) and (amount of 80% gold).

Let's assume that the amount of 30% gold alloy is x grams, and the amount of 80% gold alloy is y grams.

Therefore, we can set up two equations based on the given information:

1. The amount of gold in the first alloy (30% gold) can be found by multiplying the percentage by the total amount of the alloy:

0.30x

2. The amount of gold in the second alloy (80% gold) can be found in the same way:

0.80y

Now, let's set up the equation for the total amount of gold in the alloy mixture:

0.30x + 0.80y = 0.40 * 10

Simplifying the equation:

0.30x + 0.80y = 4

Since we also know that the total amount of the alloy mixture is 10 grams, we can set up another equation:

x + y = 10

Now we have a system of equations:

0.30x + 0.80y = 4
x + y = 10

To solve this system of equations, we can use substitution or elimination methods.

Let's solve using the elimination method:

Multiply the second equation by 0.30 to make the coefficients of x in both equations equal:

0.30(x + y) = 0.30 * 10
0.30x + 0.30y = 3

Now we have the system of equations:

0.30x + 0.80y = 4
0.30x + 0.30y = 3

Subtract the second equation from the first equation to eliminate x:

(0.30x + 0.80y) - (0.30x + 0.30y) = 4 - 3

0.50y = 1

Divide both sides of the equation by 0.50:

y = 2

Now substitute the value of y back into the second equation to find x:

x + 2 = 10

x = 10 - 2

x = 8

Therefore, the amount of 30% gold alloy required is 8 grams, and the amount of 80% gold alloy required is 2 grams to produce 10 grams of an alloy that is 40% gold.

.3 a + .8 b = .4 * 10

a + b = 10

.3 a + .8 b = 4
.8 a + .8 b = 8
---------------- subtract
-.5 a = - 4

a = 8
so b = 2