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Eugene and Frank had a total of 272 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

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If both have the same  number at the end  then  each  has   (272/2)   = 136

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

And   F  = 272 - E

Eugene gives away  (1/3) of his

And from Frank he gets   (1/5) ( F  + (1/3) E )  =   (1/5) ( 272  - E + (1/3)E )     (2)

So  at the end,   what  Eugene started with  - what he gave away + what he got from Frank  = 136

So

E  - (1/3)E  +  (1/5) [ (272 - E + (1/3)E  ]  = 136

(2/3)E +  272/5 + (1/5)(-2/3)E  = 136

(2/3)E - (2/15)E  =   136   - 272/5

[ 8/15] E  =  408 / 5

8E  =  1224

E = 153 = stamps that  Eugene started with

And Frank started with 119

Proof

Eugene gives  (1/3) to Frank = (153) / 3  = 51

Eugene now has   102

And now Frank has  119 + 51   = 170

He gives 1/5  of these to Eugene  = 34

Eugene now has    102  + 34  = 136

Frank now has  119 + 51  - 34  =  136

Apr 18, 2022