Question 113120
This year Helen is h years old, and Dan is d years old.  Helen's age next year then must be h + 1.  And Dan's age last year must have been d - 1.  The problem tells us:

Helen's age next year (h + 1) will be (=) 3 times Dan's age last year 3(d - 1).


So we can write:
Eq. 1) {{{h+1=3(d-1)}}}


We also know that their present ages, h and d, total 32, so we can write:
Eq. 2){{{h+d=32}}}


Start by solving Eq. 2) for one of the variables (doesn't matter which, I'll choose to solve for h)
{{{h=32-d}}}, Adding -d to both sides.


Now we have an expression that represents h in terms of d that can be substituted into Eq. 1), thus:
{{{(32-d)+1=3(d-1)}}}


Now, simplify and solve for d:
{{{33-d=3d-3}}}  Distributive and Associative Properties
{{{-d-3d=-3-33}}}  Adding -3d and -33 to both sides
{{{-4d=-36}}}   Collecting terms
{{{d=-36/-4=9}}} Divide both sides by -4, and now we know that this year, Dan is 9.  Since the sum of their ages is 32, Helen must be 32 - 9 = 23.


Let's check the answer.

Helen's age next year will be 24.  Dan's age last year was 8.  3 times 8 is 24.  Furthermore, 9 + 23 = 32.  So our numbers satisfy both of the conditions of the problem.  Answer checks.


Hope that helps,
John