Question 919228
WHAT YOUR TEACHER/TEXTBOOK THINKS:
18K gold is a mixture of 18 parts by volume of gold and 6 parts by volume of something else (or 75 % gold by volume and 25% other stuff by volume). Volumes mixed add, and so do masses.
{{{a}}}= volume of gold in cubic centimeters
{{{c}}}= volume of copper in cubic centimeters
{{{19.3a}}}= mass of gold in grams
{{{9c}}}= mass of copper in grams
So {{{system(a+c=15,19.3a+9c=238)}}} is their system.
Their solution by substitution could be
solving {{{a+c=15}}} for {{{c}}} ,
{{{a+c=15}}}--->{{{c=15-a}}} ,
and substituting into {{{19.3a+9c=238}}} :
{{{19.3a+9(15-a)=238}}}
{{{19.3a+9(15-a)=238}}}
{{{19.3a+135-9a=238}}}
{{{19.3a-9a=238-135}}}
{{{10.3a=103}}}
{{{a=103/10.3}}}
{{{a=10}}}
Since 10 cubic centimeters is less that 75% of 15 cubic centimeters,
which is {{{0.75*15=11.25}}} cubic centimeters,
your teacher textbook concludes that it is nor 18k gold, but less than 18k.
The ratio {{{10/15=2/3=16/24}}} is the ratio that corresponds to 16k gold.
Unfortunately 18K gold is 75% by weight, not by volume, so the problem has uglier numbers and the conclusion is different.
 
A CHEMIST'S POINT OF VIEW
{{{x}}}= grams of gold in the bracelet
{{{y}}}= grams of copper in the bracelet
{{{x+y=238}}}
18K gold is supposed to be 75% gold (25% something else) by weight,
and in an alloy, the volumes of the metals mixed add up to the volume of the mix.
{{{x/19.3}}}= cubic centimeters of gold in the bracelet
{{{y/9}}}= cubic centimeters of copper in the bracelet
{{{x/19.3+y/9=15}}}-->{{{9x+19.3y=2605.5}}}
My system of equations (with the way I defined x and y) is
{{{system(x+y=238,x/19.3+y/9=15)}}}
To solve by substitution, I solve {{{x+y=238}}} for {{{x}}} ,
{{{x+y=238}}} ---> {{{x=238-y}}} ,
and then substitute that expression for {{{x}}} in {{{9x+19.3y=2605.5}}} ,
{{{9x+19.3y=2605.5}}}
{{{9(238-y)+19.3y=2605.5}}}
{{{2142-9y+19.3y=2605.5}}}
{{{10.3y=2605.5-2142}}}
{{{10.3y=463.5}}}
{{{y=463.5/10.3}}}
{{{y=45}}} (rounded)
so {{{x=238-45=193}}} .
The percent copper is {{{43/238*100=18.9}}}
CONCLUSION:
Since 18K gold is supposed to be 75% gold (25% something else) by weight, the bracelet has more gold than 18K gold.