clearvars

global a1 a2 a3

a1=[2 3 4];

a2=[3 4 5];

a3=[4 5 6];

aa=[a1; a2; a3]

syms q r

eqs=[mix2(q,r)/mix1(q,r)==q, mix3(q,r) /mix1(q,r) ==r];

v=solve(eqs);

q=v.q(2);

r=v.r(2);

m=eval(a1+q*a2+r*a3)

m*aa/m(1)

minoff=999;

for i=1:10000

  val=i/m(1)*m;

  valb=val(2);valc=val(3);

  if abs(valb-round(valb))<.0000001

    if abs(valc-round(valc))<.0000001

      disp(val) 

    end

  end

  totoff=abs(valb-round(valb))+abs(valc-round(valc));

  if totoff<minoff

    minoff=totoff;

    mini=i;

  end

end

mini/m(1)*m

mround=round(mini/m(1)*m)

mround*aa/m(1)

function a=mix1(x,y)

global a1 a2 a3

  t=a1+x*a2+y*a3;

  a=t(1);

end

function a=mix2(x,y)

global a1 a2 a3

  t=a1+x*a2+y*a3;

  a=t(2);

end

function a=mix3(x,y)

global a1 a2 a3

  t=a1+x*a2+y*a3;

  a=t(3);

end

finds

The bests ratio of alloys to use is

 12.4807406984079 : 16.4300336161555 : 20.3793265339031

 

to get an element mixture of ratio

 12.4807406984079 : 16.4300336161555 : 20.3793265339031

 

The best rounting to an integer was

 2051:2699.99992476677:3348.99984953355

 

which, when rounded to the integers 

        2051  :  2700 : 3349

        

produced the elemental ratio 

 2051.0000663074 : 2700.00000915801 : 3348.99995200862

I was not able to get this method to work with the second part, so I went to the suggested working with trial integers:

a1=[2 3 4];

a2=[3 4 5];

a3=[4 5 6];

aa=[a1; a2; a3]

best=999;

for a=1:250

  for b=1:200

    for c=1:150

      if gcd(gcd(a,b),c)==1

        rslt=[a b c]*aa;           % matrix multiplication

        ratio=b/rslt(2);

        rsltN=ratio*rslt;          % normalize to the middle term being b

        tst=sum(abs(rsltN-[c b a]));  % element-by-element subtraction

        if tst<best

          best=tst;

          bestRats=[a b c];

          bestRslt=rsltN;

        end

      end

    end

  end

end

bestRats

bestRslt

finds that in its range of tests, the best integral ratios on the mixing would be

   111:88:65

for a resulting mixture element ratio of

   64.9980198019802 : 88 : 111.00198019802

   

   

   

   

The same method, used for the first part, yields

    79:104:129

 

 resulting in

 

    79.0015408320493 : 104 : 128.998459167951

    

    

The equation-solving method for the first part was for practical purposes instantaneous, but took a while to program, and I gave up trying to modify it for the second part. The second method, using trial integers, took about a minute or two to run but less than half hour to program, and modifying it for part two was trivial.