m = a, b, c;
n = e, r, t, y, u, i, g, h, j;
k = Outer[Divide, m, m];
k/n

gives

1/e, a/(b r), a/(c t), b/(a y), 1/u, b/(c i), c/(a g), c/(b h), 
 1/j

I want to do this with very large matrices filled with numbers of arbitrary precision. Is there a faster way?

EDIT

The sizes I am looking at for my practical applications start at 20000 and 20000^2 for the vector and matrix, respectively (of course the examples don't have to be with that many).

I am also interested in any method that might parallelise well.

m = RandomReal[-1, 1, 2000];
n = RandomReal[-1, 1, 2000, 2000];
a = Outer[Divide, m, m]/n; // RepeatedTiming // First
b = Map[#/m &, MapThread[#1 #2 &, m, 1/n]]; // 
 RepeatedTiming // First
c = m /(ConstantArray[m, Length[m]] n); // RepeatedTiming // First
d = KroneckerProduct[m, 1./m]/n; // RepeatedTiming // First
a == b == c == d

0.958

0.128

0.0281

0.0236

True

Edit

A parallelized version

cf = Compile[x, _Real, y, _Real, 1, z, _Real, 1,
 x/(y z),
 CompilationTarget -> "C",
 RuntimeAttributes -> Listable,
 Parallelization -> True,
 RuntimeOptions -> "Speed"
 ];
e = cf[m, n, m]; // RepeatedTiming // First
a == e

0.0096

True

Timing has been measured on a Quad Core CPU which shows that this does not scale too well. Btw., the timing with CompilationTarget -> "C" is only 4% slower, so there is always no point to compile it into a library.

Pretty printing your result gives...

1/e, a/(b r), a/(c t), b/(a y), 1/u, b/(c i), c/(a g), c/(b h),1/j//MatrixForm

$left(
beginarrayccc
frac1e & fracab r & fracac t \
fracba y & frac1u & fracbc i \
fracca g & fraccb h & frac1j \
endarray
right)$

Try this, it avoids constructing the huge Outer[ ] matrix

Map[#/m &, MapThread[#1 #2 &, m, 1/n]] // MatrixForm

$left(
beginarrayccc
frac1e & fracab r & fracac t \
fracba y & frac1u & fracbc i \
fracca g & fraccb h & frac1j \
endarray
right)$