A Problem of Gooseberries

           When I was a little girl, one day my mother had left a bowl of gooseberries to be shared between my two sisters Lalitha, Vasantha and myself. I went home first.

           I ate what I thought was my share of gooseberries and left. Then Lalitha arrived. She thought she was the first one to arrive and ate the number of gooseberries, she thought was her share and left. Lastly Vasantha arrived. She again though she was the first one to arrive and she took what she thought was her share and left 8 gooseberries in the bowl.

           When we three sisters met in the evening we realized what had happened and my mother distributed the remaining 8 gooseberries in a fair share.

            How did my mother do it?

Answer:     Lalitha got 3 and vasantha 5. I ate my share of the gooseberries which was 61/3. Therefore there were 2/3 f the gooseberries left in the bowl.

Lalitha took her 1/3 of these 1/3 of 2/3 = 2/9 of them.

            So when Vasantha arrived already 1/3 + 2/19 =5/9 of the original gooseberries had been eaten. Therefore only 4/9 of the original number of the gooseberries remained form which Vasantha proceeded to eat her share?

 Therefore Vasantha ate 1/3 of 4/9 and there remained 2/3 of 4/9 = 8/27.

But in the evening we saw that eight gooseberries remained in the bowl.

Therefore 8/27 of the original number = 8 

So there were 27 gooseberries in the bowl when I first took my share of 9.

I was the only one to have had my fair share of the gooseberries.

      Lalitha took what she thought was her share, from the remaining 18 gooseberries namely 6. And from the remaining 12 Vasanta had taken 4 gooseberries thinking that to be her share.

Now after Lalitha got her 3 and Vasantha her 5 gooseberries, we all had eaten an even share of 9 gooseberries each.