<PRE><B>Question 1201985</B>
&lt;br&gt;
I show two paths to the solution -- one working &quot;forward&quot; and one working &quot;backward&quot;.&lt;br&gt;
Working forward....&lt;br&gt;
K = # Kaden started with
D = # Dan started with&lt;br&gt;
Dan gave away 3/5 as many as Kaden.  We could use x as the number Kaden gave away, making (3/5)x the number Dan gave away.  But the numbers we have to work with will be easier if we let 3x and 5x be the numbers Dan and Kaden gave away, respectively.&lt;br&gt;
K-5x = # Kaden had after giving some away
D-3x = # Dan had after giving some away&lt;br&gt;
The number Dan had left was 4 less than the number he gave away:&lt;br&gt;
D-3x = 3x-4
D = 6x-4&lt;br&gt;
The number Dan had left was 2/5 the number Kaden had left:&lt;br&gt;
K-5x = (2/5)(D-3x) = (2/5)(6x-4-3x) = (2/5)(3x-4)
5(K-5x) = 2(3x-4)
5K-25x = 6x-8
5K = 31x-8
K = (31x-8)/5&lt;br&gt;
The difference between the numbers they had left at the end was 21:&lt;br&gt;
(D-3x)-(K-5x) = 21
D-3x-K+5x = 21
D-K = 21-2x&lt;br&gt;
Use that equation and the expressions above for D and K in terms of x to solve for x:&lt;br&gt;
(6x-4)-((31x-8)/5) = 21-2x
(30x-20)-(31x-8) = 105-10x
-x-12 = 105-10x
9x = 117
x = 13&lt;br&gt;
The number Dan started with was
D = 6x-4 = 6(13)-4 = 78-4 = 74&lt;br&gt;
The number Kaden started with was
K = (31x-8)/5 = (31(13)-8)/5 = (403-8)/5 = 395/5 = 79&lt;br&gt;
Dan started with 74 marbles; Kaden started with 79&lt;br&gt;
ANSWER: Kaden had 79 marbles at first&lt;br&gt;
That was a bit ugly... but it was a good algebra exercise.  Let&#39;s look at the other approach.&lt;br&gt;
Working backward....&lt;br&gt;
Kaden ended with 2/5 as many marbles as Dan; and the difference between the numbers of marbles they had was 21.&lt;br&gt;
Let 5x = # Dan ended with
Let 2x = # Kaden ended with&lt;br&gt;
5x-2x = 21
3x = 21
x = 7&lt;br&gt;
Dan ended with 5x = 35 marbles
Kaden ended with 2x =14 marbles&lt;br&gt;
The number Dan ended with was 4 less than the number he gave away, so the number he gave away was 35+4 = 39; that means the number he started with was 35+39 = 74.&lt;br&gt;
The number Dan gave away was 3/5 the number Kaden gave away.  Since Dan gave away 39 marbles, the number Kaden gave away was (5/3)(39) = 5*13 = 65.  So the number Kaden started with was 14+65 = 79.&lt;br&gt;
ANSWER: 79&lt;br&gt;
That solution method was a lot easier than the other one.  The numbers we had to work with were much &quot;nicer&quot;, and much of the analysis was so easy that we didn&#39;t need to use formal algebra.&lt;br&gt;
A lesson to be learned from this is that it is often advantageous to take a bit of time to analyze the problem to look for an easier path to the answer, instead of just plunging into the problem from the start.&lt;br&gt;
An inexperienced algebra student would probably work the problem &quot;forward&quot;, in the order the wording of the problem suggests.  But for more experienced students, in this problem the ending information that Kaden had 2/5 as many marbles as Dan and the difference between those numbers was 21 suggests an easier path to the answer.&lt;br&gt;
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