Question 1184088
Juliet gave her 1/5 of her marbles to Ken. After that, Ken gave 2/3 of what
he had and an additional 4 more marbles to Juliet. Then, Juliet lost 29
marbles on the way home. In the end, Juliet had 97 marbles and Ken had 21
marbles left. How many marbles did each of them have at first?
<pre>
It's easier to work it backwards.

>>In the end, Juliet had 97 marbles and Ken had 21 marbles left. 

>>Juliet lost 29 marbles on the way home.

97+29=126.  So before she lost those, Juliet had 126 marbles and Ken had 21.

>>Ken gave 2/3 of what he had and an additional 4 more marbles to Juliet.

Suppose Ken had N marbles before that.
2/3 of N is 2/3N.  4 more is 2/3N+4.  So Ken had 21+2/3N+4 before he gave
that to Juliet.  But that must be equal to what Ken had before he gave that
to her, which was N. So we have the equation 

21+2/3N+4 =  N
  25+2/3N =  N
    75+2N = 3N
       75 =  N   

So Ken had 75 and then gave Juliet 2/3 of 75, which is 50, plus 4 more,
which makes it 54 that he gave to Juliet.  So before she got those from Ken,
Juliet had 54 fewer marbles, or 126-54=72.  And of course, Ken had 75.

Ken had 75 and Juliet had 72 back then.

>>>Juliet gave 1/5 of her marbles to Ken. 

Suppose Juliet had M before giving those to Ken.
1/5 of M is 1/5M.  So she had 1/5M more before she gave that to Ken. That
is, she had 72+1/5M. But that must be equal to what she had before she gave
that to Ken, which was M. So we have the equation
 
 72+1/5M = M 
   360+M = 5M
     360 = 4M  
      90 = M

1/5 of 90 is 18.  So before she gave that to Ken, he must have had 18 fewer
than 75.  75-18 = 57. 

So the answer is that at first, Juliet had 90 and Ken had 57.

Let's check:

Juliet began with 90 and Ken began with 57.

>>>Juliet gave her 1/5 of her marbles to Ken. 

1/5 of 90 is 18. So Juliet then had 90-18=72 and Ken had 57+18=75

>>>After that, Ken gave 2/3 of what he had and an additional 4 more marbles to Juliet. 

2/3 of 75 is 50.  An additional 4 makes it 54.  So Juliet then had 72+54=126
and Ken had 75-54=21.

>>>Then, Juliet lost 29 marbles on the way home. 

So she ended up with 126-29=97 and Ken still had his 21.

It checks!

Edwin</pre>