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# Help

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Eugene and Frank had a total of 772 stamps. Eugene gave 1/3 of his stamps to Frank. Frank then gave 1/5 of his stamps to Eugene. In the end, each of them had the same number of stamps. How many stamps did Eugene have at first?

Apr 18, 2022

#1
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If both have the same  number at the end  then  each  has   (772/2)   = 386

Let  E  be the number that Eugene started with and F what Frank started with

And   F  =  772  - E        (1)

Eugene gives away  (1/3) of his

And from Frank he gets   (1/5) ( F  + (1/3) E ]  =   (1/5) ( 772  -E + 1/3)E )     (2)

So  at the end,   what  Eugene started with  - what he gave away + what he got from Frank  = 336

So

E  - (1/3)E  +  (1/5) (772 - E + (1/3)E  ]  = 336

(2/3)E +  772/5 + (1/5)(-2/3)E  = 336

(2/3)E - (2/15)E  =   336   - 772/5

[ 8/15] E  =  908 / 5

8E  =  2724

E = 340.5   ?????

Are your numbers correct  ???  (it's also possible that I made an error !!! )   Apr 18, 2022
#2
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CPhill --

Number Eugene starts with:  E

Number Frank starts with:     F

Eugene gave away 1/3 of his stamps to Frank ...

Now Eugene has:      (2/3)E

and Frank has:          F + (1/3)E

Frank then gave 1/5 of his stamps to Eugene ...

Now Eugene has:      (2/3)E + (1/5)( F + (1/3)E )

and Frank has:           (4/5)( F + (1/3)E )

They now have the same number of stamps:

(2/3)E + (1/5)( F + (1/3)E )  =  (4/5)( F + (1/3)E )

(2/3)E + (1/5)F + (1/15)E  =  (4/5)F + (4/15)E

multiply by 15:     10E + 3F + E  =  12F + 4E

11E + 3F  =  12F + 4E

7E  =  9F

Since  E + F  =  772      --->     F  =  772 - E

7E  =  9(772 - E)

7E  =  6948 - 9E

16E  =  6948

E  =  434.25

Final result:  ?????

Apr 18, 2022