[synergy]

You have a piece of gold jewelry weighing 9.35 g.  Its volume is 0.654 cm3.  Assume that the metal is an alloy (mixture) of gold and silver, which have densities of 19.3 g/cm3 and 10.5 g/cm3, respectively.  Also assume that there is no change in volume when the pure metals are mixed.  Calculate the percentage of gold by mass in the alloy.  The relative amount of gold in alloy is measured in karats.  Pure gold is 24 karats; an alloy of 50% gold is 12 karats.  State the proportion of gold in the jewelry as karats.

I can't figure out hot to get the answer?

I calculated the density of the alloy 14.3 g/cm3. The density of a 50/50 mix of gold and silver that had a volume of 0.654 g/cm3 would be 14.9 g/cm3.

So the alloy is slightly lighter than a 50/50 mix, so it obviously has a little bit more silver than gold. I just don't know how to find the exact values, or ow to express it as karats.

Any help is appreciated.

Thanks

[Yggdrasil]

Let G be the mass of gold in your sample and S be the mass of silver in your jewlry.

You know that G + S = 9.35g.  Since you have two variables, you need another equation to be able to solve for each variable.  Can you derive a formula that expresses the volume of the jewelry in terms of G and S?

[wereworm73]

Since the percentage of gold in the alloy is directly affected by the percentage of silver, you could say that

% of gold = x
% of silver = 1-x


Now it's just a matter of setting up the equation with the correct coefficents...

[synergy]

Well I think I figured out the answer but I don't know why or even if its correct, all I know is the way I solved it is not correct.

I think the answer is the alloy contains 56% silver and 44% gold.

I took 19.3x + 10.5x = 9.35

Solving for x I got, x=.56

.56(10.5) + .44(19.3) = 14.37

I'm totally lost. Thanks for the help so far, any other help would be appreciated.

Thanks

[wereworm73]

You're almost there.  Since you're using density as the unit for one side of the equation, it should also be the unit for the other side.  You already know that the sample is 9.35 g; now you just have to get its density to make the units match from both sides...

(x)(19.3 g/ cm³) + (1-x)(10.5 g/ cm³) = 9.35 g / ______ cm³

[synergy]

I think I figured it out!

19.3(x) + 10.5(1 + -1x) = 14.3
10.5 + 19.3x + -10.5x = 14.3
10.5 + 8.8x = 14.3
8.8x = 3.8
x = 0.4318181818

So 43% of the alloy is gold, and 57% is silver.

Now as for what karat the gold is tell me if this is correct.
Since 24 karat gold is 100% gold, for every 1% more gold added to an alloy the karat rating goes up .24 karats.

So since the alloy is 43% gold 7*.24 =1.68

A 50% gold alloy is 12 karats, so this alloy is 12 karats - 1.68 karats making the alloy 10.32 karat gold?

Does all of that look correct?

Thanks again for all the help, I tried a few more problem from the book and am understanding them all better now.

[wereworm73]

Yep, you got it. 

[synergy]

Thank you.

+1 scooby snack my friend