Can Solve[...] be used for set math? If so, how? If not, how else?

Question

I'm a recreational user who loves using the product.

I'm looking for a strategy to solve the following question using Mathematica.

The question is from a sample actuarial exam. I can solve this with pencil and paper, but I'm fascinated (and stumped) about how to solve this in Mathematica. Any hints? Thanks!

THE QUESTION: An auto insurance company has 10,000 policyholders.

Each policyholder is classified as (i) young or old; (ii) male or female; and (iii) married or single.

Of these policyholders,

• 3000 are young,
• 4600 are male, and
• 7000 are married.

The policyholders can also be classified as

• 1320 young males,
• 3010 married males, and
• 1400 young married persons.

Finally, 600 of the policyholders are young married males.

Calculate the number of the company’s policyholders who are young, female, and single.

Answer 1

The tedious part of this is writing all the equations

ages = {young, old};
sexes = {female, male};
wed = {single, married};

equations = {Sum[n[young, j, k], {j, sexes}, {k, wed}] == 
    3000,
   Sum[n[i, male, k], {i, ages}, {k, wed}] == 4600,
   Sum[n[i, j, married], {i, ages}, {j, sexes}] == 7000,
   Sum[n[young, male, k], {k, wed}] == 1320,
   Sum[n[i, male, married], {i, ages}] == 3010,
   Sum[n[young, j, married], {j, sexes}] == 1400,
   n[young, male, married] == 600};

Solve[equations]
(* {{n[old, female, married] -> 3190, 
  n[old, male, married] -> 2410, n[old, male, single] -> 870, 
  n[young, female, married] -> 800, n[young, female, single] -> 880, 
  n[young, male, married] -> 600, n[young, male, single] -> 720}} *)

There might be a slightly more concise way of expressing this.