Chapter A.8, Problem 43AYU

To determine

To find: How much 12-karat gold should be mixed with pure gold to obtain 60 grams of 16-karat gold?

Answer to Problem 43AYU

Mix 20 g of pure gold with 40 g .

Explanation of Solution

Given:

The purity of gold is measured in karats, with pure gold being 24 karats. Other purities of gold are expressed as proportional parts of pure gold. Thus, 18-karat gold is $\frac{18}{24}$, or $75\%$ pure gold; 12-karat gold is $\frac{12}{24}$, or $50\%$ pure gold; and so on.

Calculation:

Gold $=60.\left(\frac{16}{24}\right)=40 g$

Now set up an equation to solve for pure gold $+$ 12k gold.

We know we have 60 grams total weight and 40 g of gold.

Let X be the amount of pure gold in the mixture, then the amount of 12 k gold is $60-x$.

Gold content $=\left(pure Gold.X\right)+12k Gold.\left(60-x\right)$

$40=\left(\frac{24}{24}\right).X+\left(\frac{12}{24}\right).\left(60-X\right)$

$40.24=24X+720-12X$

$960-720=12X$

$X=20$

So we need to mix 20 g of pure gold with 40 g of 12 k gold to get 60 g of 1.