Basically we are dealing with (X*7) / (X*9). Let’s look at the various ones digits of X to see what the ones digits of 7X and 9X could be and if this makes sense.

Well, X can’t end in 0 or 5. If they did, N would equal S which usually isn’t true in these problems.  We also have a one to one pairing for N and S. Those are (N, S) = (7, 9) (4, 8) (1, 7) (8, 6) (2, 4) (9, 3) (6, 2) (3, 1)

It doesn’t seem like I reduced the problem much, but I think I did.

Next, I’d like to check something. Luckily, both numbers end in an N or an S, and also begin with an N or an S. Well, if I kind of approximate by looking at S/(10N), I should see something in the range of 0.7 – 0.9. Looking at all those (N, S) pairs, what I get for S/(10N) are 0.1286, 0.2, 0.7, 0.075, 0.2, 0.0333, 0.0333, 0.0333.

Well, this indicates to me that I should be focussing my attention on N = 1, S = 7.

Ok, so so far we have 7EVE1 / 1I1TH7 = 7/9

Well, to even get in the general range of these requirements, we might have some decent restrictions on X. Well, for SEVEN to start with 7, 10000 < X < 11428. For NINTHS to start with 1, 11112 < X. Putting this together, we have 11112 < X < 11428. Not only that, but X must end in 3 for SEVEN to end in a 1 and for NINTHS to end in a 7. So this only leaves 32 possibilities for X to be.

Factor in the fact that the thousands digit in NINTHS is also a 1, and that brings us down to 12 possibilities for what X could be : X = 11223 – 11333 only going up in 10s.

Looking at the 12 pairs of 7X and 9X, the next quality I will look for are the two Es in SEVEN. There are only two such cases in my narrowed search: 78981 (with 9x = 101547) and 79191 (with 9X = 101817). The second pair clearly has issues – both 7X and 9X have a 1 other than where there is an N in the problem. So that cannot be the solution.

But let’s look at 78981 and 101547 just to be sure. This seems to check out. There are no "double assignments" such as "well, we said both V and H equal the same thing," or "we have T equaling two different numbers" (not that it could).

So that’s my answer: 78981 / 101547 = 7/9