Answer
68 days
Work Step by Step
Noodles is $N$ and Freckles is $F$
$1/N + 1/F = 1/30$
$N = F - 14$
$1/N + 1/F = 1/30$
$1/(F-14) + 1/F = 1/30$
$F*(F-14)*30*(1/(F-14) + 1/F) = 1/30*F*(F-14)*30$
$F*(F-14)*30*(1/(F-14) + 1/F) = F*(F-14)$
$30*F + 30*(F-14) = F*(F-14)$
$30F +30F-420 = F^2-14F$
$60F-420 = F^2-14F$
$60F-420-60F+420 = F^2-14F-60F+420$
$0 = F^2-74F+420$
$x= (-b±\sqrt{b^2-4ac})/2a$
$a=1$, $b=-74$, $c=420$
$x= (-b±\sqrt{b^2-4ac})/2a$
$x=(-(-74)±\sqrt{(-74)^2-4*1*420})/2*1$
$x=(74±\sqrt{5476-1680})/2$
$x=(74±\sqrt{3796})/2$
$x=(74±61.1)/2$
$x=(74+61.1)/2$
$x=135.1/2$
$x=67.55$
$x=68$
$x=(74-61.1)/2$
$x=12.9/2$
$x=6.45$
This answer isn't valid since the other dog would eat the bag of food in $6.45-14$, or $-7.55$ days.
