npr:n+2pr+2:n+4pr+4=3:14:14.
Split this into fractions:
npr/(n+2pr+2)=3/14; (n+2pr+2)/(n+4pr+4)=14/14=1; npr/(n+4pr+4)=3/14.
We can form more equations:
14npr=3(n+2pr+2); n+2pr+2=n+4pr+4; 14npr=3(n+4pr+4).
So 2pr+2=4pr+4, 2pr+2=0, pr=-1.
Therefore: -14n=3n, so n=0. However, we would have 0:0:0=3:14:14 which makes no sense, so the question may be wrongly stated.
Let the ratios be A:B:C (undefined), then:
(1) npr/(n+2pr+2)=A/B; (2) (n+2pr+2)/(n+4pr+4)=B/C; (3) npr/(n+4pr+4)=A/C.
Then:
(1) Bnpr=A(n+2pr+2); (2) C(n+2pr+2)=B(n+4pr+4); (3) Cnpr=A(n+4pr+4).
(2) n(C-B)-2pr(2B-C)=4B-2C=2(2B-C), (4) n=2(pr+1)(2B-C)/(C-B). Note that B≠C (otherwise pr=-1 and n is indeterminate), and 2B≠C (otherwise n=0).
Substitute for n in (1):
2Bpr(pr+1)(2B-C)/(C-B)=A(2(pr+1)(2B-C)/(C-B)+2(pr+1)),
2Bpr(pr+1)(2B-C)=A(2(pr+1)(2B-C)+2(pr+1)(C-B); divide through by 2(pr+1):
Bpr(2B-C)=A(2B-C+C-B)=AB, pr=A/(2B-C); n=2(A/(2B-C)+1)(2B-C)/(C-B)=2(A+2B-C)/(C-B).
Clearly, B=C=14 wouldn't work.