Question 939595
In the Anderson family:
{{{"husband's"}}} {{{age=highlight(39)}}} ,
{{{"wife's"}}} {{{age=highlight(34)}}} ,
{{{"son's"}}} {{{age=highlight(14)}}} , and
{{{"daughter's"}}} {{{age=highlight(13)}}} .
 
In the Billings family:
{{{"husband's"}}} {{{age=highlight(42)}}} ,
{{{"wife's"}}} {{{age=highlight(40)}}} ,
{{{"son's"}}} {{{age=highlight(10)}}} , and
{{{"daughter's"}}} {{{age=highlight(8)}}} .
 
How did I find that?
I cheated and used a spreadsheet program (Microsoft Excel) to tabulate possibilities.
I started from
{{{h="husband's"}}} {{{age}}} ,
{{{w="wife's"}}} {{{age}}} ,
{{{s="son's"}}} {{{age}}} , and
{{{d=s-2="daughter's"}}} {{{age}}} for the Billings family,
while {{{d=s-1="daughter's"}}} {{{age}}} for the Anderson family.
I made a table for each family with {{{d}}} as one column,
the corresponding {{{s}}} as a second column,
and several columns with various values of {{{w}}} as a title at the top,
and the corresponding results for {{{h=sqrt(d^2+s^2+w^2)}}} below.
When I found an integer for {{{h}}},
I calculated {{{h+w+d+s}}} to see if it was {{{100}}} .
The possibilities can be narrowed, but I could not narrow them too much.
We also know that the husband must be the oldest one,
because {{{h^2=w^2+s^2+d^2>w^2}}}--->{{{h>w}}} .
Also, the children must be younger than the parents, so we expect them to be under 25.
I used {{{d}}} values from 1 to 25, and {{{w}}} values from 18 to 40 for my tables.
 
Maybe some horrible algebra would show an easier path to the solution.
There must be ways to narrow the possibilities further,
and maybe there is a table of Pythagorean quadruples that would help.
Maybe asking the question in some low level artofproblemsolving website forum would prompt a 13 year old to give a succinct brilliant solution that does not require tabulations.
All I could think of is that for the Billings family
we know that {{{h+w+s+s-2=100}}}<-->{{{h+w+2s=102}}}<-->{{{h+w=2(51-s)}}} ,
which tells you that the ages of husband and wife are either both even, or both odd.
On the other hand, using a similar reasoning, in the Anderson family,
if the husband's age is even, the wife's age is odd and viceversa.