Got any grapes?

Algebra Level 3

I have n n grapes. I divide them into 3 equal piles, with 1 left over. I put the leftover grape into one pile, and eat all the grapes from one of the two smaller piles. I now have x x grapes.

Which of the following is an expression for x x in terms of n n ?

2 n 3 + 1 3 \frac { 2n }{ 3 } +\frac { 1 }{ 3 } 2 n 3 − 1 3 \frac { 2n }{ 3 } -\frac { 1 }{ 3 } 2 n 3 + 1 2 \frac { 2n }{ 3 } +\frac { 1 }{ 2 } 2 n 3 + 1 \frac { 2n }{ 3 } +1 2 n 3 \frac { 2n }{ 3 }

Michael Fuller by Michael Fuller , 22, United Kingdom

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1 solution

Michael Fuller
Mar 27, 2015

Let y y denote a small pile. Then:

n = 3 y + 1 y = n − 1 3 = n 3 − 1 3 n=3y+1\\ y=\frac { n-1 }{ 3 } =\frac { n }{ 3 } -\frac { 1 }{ 3 }

The sum of the small and large piles that are leftover is:

2 y + 1 = 2 ( n 3 − 1 3 ) + 1 = 2 n 3 + 1 3 2y+1\\ =2(\frac { n }{ 3 } -\frac { 1 }{ 3 }) +1\\ =\large\color{#20A900}{\boxed{\frac { 2n }{ 3 } +\frac { 1 }{ 3 }}}

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