SOLUTION: How many grams of pure silver must be mixed with 450grams of an alloy that is 10% copper in order to produce a sterling silver that is 7.5% copper?

How many grams of pure silver must be mixed with 450 grams of an alloy that
is 10% copper in order to produce a sterling silver that is 7.5% copper?

We have three things to consider: 1. the first stuff, 2. the second 
stuff, and 3. the final stuff that we get after we mix the first stuff
with the second stuff.

Each of those "stuffs" contains some pure copper, and/or some pure silver,
and/or some other stuff.    

So we mix: 

1.  x grams of the first stuff that is 100% silver and therefore 0% copper
and 0% other metal.

with

2.  450 grams of the second stuff that is 10% copper and therefore 90% other
metal and 0% silver

and end up with

3. x+450 grams of the final stuff that is 7.5% copper and therefore 92.5%
silver and other metal.

Note: it isn't necessary to deal with anything but the copper, but I
think it makes it easier to deal with all the ingredients.

Make this chart:

                grams of   grams of    grams of    grams of
                 stuff      silver      copper     other metal
first stuff                                               
second stuff                                                
--------------------------------------------------------------
final stuff                                                  
                                             
Fill in the number of grams of the first, second and final stuff.  That is,
x grams of the first stuff, 450 grams of the second stuff, and therefore
x+450 grams of the final stuff:
    
                grams of   grams of    grams of    grams of
                 stuff      silver      copper     other metal
first stuff         x                                                       
second stuff      450                                      
--------------------------------------------------------------
final stuff     x+450                                       


Next fill in the number of grams of silver only, by multiplying
the grams of the first and second stuff by the percentage of silver 
expressed as a decimal. The first stuff is 100% silver. The second
stuff is 0& silver. Then since the second stuff contains no silver,
the final stuff contains exactly the same amount of silver as the 
first stuff.

                grams of   grams of    grams of    grams of
                 stuff      silver      copper     other metal
first stuff         x        1.00x                             
second stuff      450        0.00(450)                     
----------------------------------------------------------
final stuff     x+450        1.00x                        
                   
Next fill in the number of grams of copper only, by multiplying
the grams of each stuff by their percentage of copper 
expressed as a decimal. 

                grams of   grams of    grams of    grams of
                 stuff      silver      copper     other metal
first stuff         x        1.00x       0.00x         
second stuff      450        0.00(450)   0.10(450)      
----------------------------------------------------------
final stuff     x+450        1.00x    0.075(x+450)      

Next fill in the number of grams of other metal only, by multiplying
the grams of the first and final stuff by their percentage of silver 
expressed as a decimal. Then since the first stuff contains no
"other metal", the final stuff contains exactly the same amount of other
metal as the second stuff.

                grams of   grams of    grams of    grams of
                 stuff      silver      copper     other metal
first stuff         x        1.00x       0.00x      0.00x
second stuff      450        0.00(450)   0.10(450)  0.90(450)
---------------------------------------------------------------
final stuff     x+450        1.00x    0.075(x+450)  0.90(450)


Now we can make the "stuff equation" out of the first column:

x + 450 = x + 450

But that equation doesn't do us any good.

We can make the "silver equation" out of the second column:

1.00x + 0.00(450) = 1.00x

But that equation becomes x = x and does us no good either.

We can make the "copper equation" out of the third column:

.00x + .10(450) = 0.075(x+450)

Now that equation shows promise!!

We can make the "other metal equation" out of the fourth column:

.00x + .90(450) = .90(450)

That equation becomes 405 = 405 and does us no good.  So we only
have one equation that does us any good, the copper equation:

.00x + .10(450) = 0.075(x+450) or

       .10(450) = 0.075(x+450)

Solve that for x and we get x = 150 grams of pure silver  

Edwin