Question 1184459
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UNFORTUNATELY,  your writing is  SO  INACCURATE,  that  MISREPRESENTS  the meaning of the problem.

THEREFORE, &nbsp;below &nbsp;I &nbsp;placed &nbsp;<U>my editing of your post</U>, &nbsp;with my corrections.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Gillian &nbsp;and &nbsp;Ben had a total of &nbsp;4850 &nbsp;toys. 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Ben &nbsp;{{{cross(have)}}} <U>GAVE</U> &nbsp;3/7 &nbsp;of his toys to &nbsp;Gillian. 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;She then gave &nbsp;1/3 &nbsp;of &nbsp;{{{cross(here)}}} &nbsp;<U>HER</U> &nbsp;toys to &nbsp;Ben. 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The ratio of the number of toys &nbsp;Gillian left to the number of toys &nbsp;Ben {{{cross(left)}}} &nbsp;<U>finally had</U> &nbsp;was &nbsp;2:3.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;a) &nbsp;&nbsp;How many toys did &nbsp;Gillian have at the end?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;b) &nbsp;&nbsp;How many toys did &nbsp;Ben have at first?
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I will solve the problem using backward solution method.


To make the solution totally clear, I will present / (will re-tell) the problem's description in steps.


<pre>
    1.  At the start, the total number of toys was 4850.

    2.  Ben gave 3/7 of his toys to Gilian.

    3.  Gilian then gave 1/3 of her toys to Ben.

    4.  After that, the ratio of Gilian's toys to Ben toys was 2:3.
</pre>


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;At this point, I start my solution.



<pre>
Notice that, althought Gilian and Ben exchanged toys, the total number of toys remained 4850, with no change.

So, at the final step 4, there were totaly 4850 toys in the ratio Gilian to Ben as 2:3.


It means, there were 5 equal parts of toys, each part counted  4850/5 = 970 toys,

and Gilian had 2 such parts, or 2*970 = 1940 toys, while Ben had 3 such parts, or 3*970 = 2910 toys.


In particular, we just know the <U>ANSWER</U> to question (a):  Gilian had 1940 toys at the end.


Next, the last action that Gilian did was giving 1/3 of her toys to Ben.

    HENCE, her final number of toys, 1940, is 2/3 of what she had before this giving.

    So, we conclude that before giving to Ben, Gilian had  (3/2)*1940 = 2910 toys.

    Again, immediately before step 3, Gilian had 2910 toys.

    Hence, immediately before step 3, Ben had the rest  4850 - 2910 = 1940 toys.


Thus we analysed the final stage and then restored from the condition, that immediately before step 3,
Gilian had 2910 toys, while Ben had 1940 toys.


Next, moving back, we know that after step 2, Ben had 1940 toys; Gilian had 2910 toys.

    In other words, after giving Gilian 3/7 of his toys at the step 2, Ben left with 1940 toys; it is 4/7 what he had before his giving.

    HENCE, before this giving, Ben had 7/4*1940 = 3395 toys and Gilian had the rest 4850-3395 = 1455 toys.


    Again, immediately before step 2, Ben had 3395 toys anad Gilian had the rest 1455 toys.


It gives the <U>ANSWER</U> to question (a): at first, Ben had 3395 toys.
</pre>

Completed and solved.


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Yes, &nbsp;I solved the problem using the backward solution method,

and it required very accurate logic and accurate presentation.