<PRE><B>Question 1120860</B>
&lt;br&gt;
I will assume that &quot;Ann&quot; and &quot;Sue&quot; are supposed to be the same person....&lt;br&gt;
Joy starts swimming at Ann&#39;s 8th session.&lt;br&gt;
Ann starts doing 10 lengths and increases the number by 2 each session.  So in her 8th session, the number of lengths she swims is 10+7(2) = 24.&lt;br&gt;
In Joy&#39;s first session, she swims 15 lengths, and she increases that number by 5 each session.&lt;br&gt;
Let x be the number of Joy&#39;s sessions after her first (i.e., after Ann&#39;s 8th).&lt;br&gt;
The number of lengths Ann swims x sessions after Joy&#39;s first is 24+2x; the number Joy swims that same session is 15+5x.&lt;br&gt;
The number of lengths they swim will be the same when 24+2x = 15+5x.&lt;br&gt;
24+2x = 15+5x
3x = 9
x = 3&lt;br&gt;
It will be 3 sessions after her first (i.e., her 4th session) when Joy swims the same number of lengths as Ann.&lt;br&gt;&lt;pre&gt;
Ann&#39;s sessions:
     session number  1   2   3   4   5   6   7   8   9  10  &lt;b&gt;11&lt;/b&gt;  12...
          lengths   10  12  14  16  18  20  22  24  26  28  &lt;b&gt;30&lt;/b&gt;  32...
Joy&#39;s sessions:
     session number                              1   2   3   &lt;b&gt;4&lt;/b&gt;   5...
          lengths                               15  20  25  &lt;b&gt;30&lt;/b&gt;  35...
&lt;/pre&gt;&lt;br&gt;
Of course, you could solve the problem without formal algebra, simply by writing out a table like the one shown.
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