Question 1062300
<pre>  
You just had a period after the first phrase where you should 
have had a comma.

<font color="red"><b>
"When Bob was as old as Rob was when Bob was half as old
as Rob was when Bob was as old as Rob is, Rob was just a
third as old as Bob was when Rob was half as old as Bob
is. One is in their teens. What are their present ages?"
</font></b>
Suppose Bob is B years old and Rob is R years old.

Look at just the first phrase:
<font color="red"><b>
When Bob was as old as Rob was when Bob was half as old
as Rob was when Bob was as old as Rob is,
</font></b>
Let's translate these words on the end:
<font color="red"><b>
"when Bob was as old as Rob is,"
</font></b>
to a number of years ago.  Bob is older than Rob, so we
subtract their ages, B-R to find out how many years ago
that was.  So we rewrite the phrase as:
<font color="red"><b>
"When Bob was as old as Rob was when Bob was half as old
as Rob was B-R years ago."
</font></b>
Let's translate these words on the end:
<font color="red"><b>
"as old as Rob was B-R years ago."
</font></b>
to an age.  To find that, we subtract B-R years from
Rob's age now, which is R and get

R-(B-R)
R-B+R
2R-B

So now we have
<font color="red"><b>
"When Bob was as old as Rob was when Bob was half of 2R-B"
</font></b>
Let's translate the words on the end:
<font color="red"><b>
"half of 2R-B"
</font></b>
to an age as 

{{{expr(1/2)(2R-B)}}}
{{{R-expr(1/2)B}}} years old

So now we have
<font color="red"><b>
"When Bob was as old as Rob was when Bob was {{{R-expr(1/2)B}}}
years old."
</font></b>
Let's translate the words on the end:
<font color="red"><b>
"when Bob was {{{R-expr(1/2)B}}} years old"
</font></b>
to a number of years ago.  To find how many years ago
that was, we subtract {{{R-expr(1/2)B}}}" from Bob's age now, 
which is B:

{{{B-(R-expr(1/2)B)}}}
{{{B-R+expr(1/2)B}}}
{{{expr(3/2)B-R}}} years ago.

So now we have
<font color="red"><b>
"When Bob was as old as Rob was {{{expr(3/2)B-R)}}} years ago.
</font></b>
Let's translate the words on the end:
<font color="red"><b>
"as old as Rob was {{{expr(3/2)B-R)}}} years ago."
</font></b>
We subtract that from Rob's age now:

{{{R-(expr(3/2)B-R)}}}
{{{R-expr(3/2)B+R}}}
{{{2R-expr(3/2)B}}}

So now we have
<font color="red"><b>
"When Bob was {{{2R-expr(3/2)B}}} years old."
</font></b>
We translate that to a number of years ago by subtracting
it from B:

{{{B-(2R-expr(3/2)B)}}}
{{{B-2R+expr(3/2)B}}}
{{{expr(5/2)B-2R}}} years ago.

Now we look at the second phrase:
<font color="red"><b>
"Rob was just a third as old as Bob was when Rob was half
as old as Bob is"
</font></b>
Notice the first two words:
<font color="red"><b>
"Rob was".
</font></b>
Let's replace that by Rob's age {{{expr(5/2)B-2R}}} years ago.  We do that by subtracting it from Rob's age now,
which is R:

{{{R-(expr(5/2)B-2R)}}}
{{{R-expr(5/2)B+2R}}}
{{{3R-expr(5/2)B}}}

and put an = sign after it because that is the subject of
the second phrase:
<font color="red"><b>
"{{{3R-expr(5/2)B}}}{{{""=""}}} just a third as old as Bob was when 
Rob was half as old as Bob is"
</font></b>
Let's translate the words on the end:
<font color="red"><b>
"half as old as Bob is"
</font></b>
That's simply {{{expr(1/2)B}}} years old.

So now we have:
<font color="red"><b>
"{{{3R-expr(5/2)B}}}{{{""=""}}} just a third as old as Bob was when Rob 
was {{{expr(1/2)B}}} years old."
</font></b>
Let's translate the words on the end:
<font color="red"><b>
"when Rob was {{{expr(1/2)B}}} years old."
</font></b>
to a number of years ago.  We do that by subtracting {{{expr(1/2)B}}} 
years from Rob's age now, which is R:

{{{R-expr(1/2)B}}} years ago.

So now we have:
<font color="red"><b>
"{{{3R-expr(5/2)B}}}{{{""=""}}} just a third as old as Bob was 
{{{R-expr(1/2)B}}} years ago."
</font></b>
Let's translate the words on the end:
<font color="red"><b>
"as old as Bob was {{{R-expr(1/2)B}}} years ago"
</font></b>
to the age of Bob then.  We do that by subtracting {{{R-expr(1/2)B}}} 
from Bob's age now, which is B:

{{{B-(R-expr(1/2)B)}}}
{{{B-R+expr(1/2)B}}}
{{{expr(3/2)B - R}}}
 
So now we have:
<font color="red"><b>
{{{3R-expr(5/2)B}}}{{{""=""}}} just a third (of) {{{expr(3/2)B - R}}}
</font></b>
So we just put {{{1/3}}} in front of that:
<font color="red"><b>
{{{3R-expr(5/2)B}}}{{{""=""}}}{{{expr(1/3)(expr(3/2)B - R)}}}
</font></b>
Multiply both sides by 6

{{{18R-15B}}}{{{""=""}}}{{{2(expr(3/2)B - R)}}}

{{{18R-15B}}}{{{""=""}}}{{{3B - 2R)}}}

{{{20R}}}{{{""=""}}}{{{18B}}}

That simplifies to

{{{9B}}}{{{""=""}}}{{{10R}}}

{{{B}}}{{{""=""}}}{{{expr(10/9)R}}}
</b></font>
So R must be divisible by 9,

 So R =  9, 18, 27, 36, 45, 54, 63, 72, 81, 90,...

And B = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100,... 

Now we use this sentence:

"One is in their teens."

There is only one possibility,

Rob is 18 and Bob is 20.

Edwin</pre></b></font>