Question

Pure gold is often mixed with other metals to produce jewelry. Pure gold is 24 carat. 12 carat gold is 12/24 or 50% gold, 6 carat gold is 6/24 or 25% gold, etc. A jeweler has some 12 carat gold and some 21 carat gold and wants to produce 90 g of 75% gold.

a. What percentage of gold is 21 carat?

b. How many grams of 12 carat gold and 21 carat gold are needed to produce the mixture?

Answer 1

a. 87.5%

b. 30 g of 12 carat and 60 g of 21 carat.

Explanation:

a. The percentage of gold in 21 carat is given by:

b. Since the mass he wants to produce is 90 g, then the added mass of the 12 carat, "x", and the mass of the 21 carat "y", must be equal to that:

The sum of gold on each carat type must be equal to the amount of gold on the final mixture, since a 12 carat has 50% gold and a 21 carat has 87.5% gold, then we have:

We now have a system of equations which we can solve to find the necessary mass of each type.

He needs 30 g of 12 carat gold and 60 g of 21 carat gold.

Explanation:

Answer 2

a. 87.5%

b. 30 g of 12 carat and 60 g of 21 carat.

Explanation:

a. The percentage of gold in 21 carat is given by:

`gold_(21) = (21)/(24)*100 = 87.5 \%`

b. Since the mass he wants to produce is 90 g, then the added mass of the 12 carat, "x", and the mass of the 21 carat "y", must be equal to that:

`x + y = 90`

The sum of gold on each carat type must be equal to the amount of gold on the final mixture, since a 12 carat has 50% gold and a 21 carat has 87.5% gold, then we have:

`0.5*x + 0.875*y = 0.75*90\\\\0.5*x + 0.875*y = 67.5\\`

We now have a system of equations which we can solve to find the necessary mass of each type.

`x + y = 90 \text{ } *-0.5\\0.5*x + 0.875*y = 67.5\\\\-0.5*x - 0.5*y = -45\\0.5*x + 0.875*y = 67.5\\\\0.375*y = 22.5\\\\y = 60`

`x = 90 - y = 90 - 60\\\\x = 30`

He needs 30 g of 12 carat gold and 60 g of 21 carat gold.