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Algebra Level pending

John picked a bad of oranges to give to his friends. To the first of his friends, he gave half of the oranges he had and another one beside. to his second friend he gave half of the remaining oranges and another one beside. By this time, John had one orange left. How many did he start with?


The answer is 10.

Sarang Kapre by Sarang Kapre , 25, India

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2 solutions

Let's tell the total number of oranges is n.

John gives to the first friend ( n 2 + 1 ) (\frac {n} {2} + 1) oranges

so the remaining oranges are ( n 2 − 1 ) (\frac {n} {2} - 1)

John gives the half of these plus one to the second friend:

( n 2 − 1 2 ) + 1 = n 4 + 1 2 (\frac {\frac {n} {2} - 1} {2}) + 1 = \frac {n} {4} + \frac {1} {2}

and remains with one orange. So we can say:

n − ( n 2 + 1 ) − ( n 4 + 1 2 ) = 1 ⇒ n − n 2 − n 4 = 5 2 ⇒ 4 n − 2 n − n 4 = 5 2 ⇒ n = 20 2 = 10 n - (\frac {n} {2} + 1) - (\frac {n} {4} + \frac {1} {2}) = 1 \Rightarrow n - \frac {n} {2} - \frac {n} {4} = \frac {5} {2} \Rightarrow \frac {4n-2n-n} {4} = \frac {5} {2} \Rightarrow n = \frac {20} {2} = 10

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Mahdi Al-kawaz
May 24, 2014

2 ⋅ ( 2 ⋅ ( 1 + 1 ) + 1 ) = 10 2 \cdot (2 \cdot (1+1)+1)=10

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