Question

RAMU had money. he gave one rupee to a beggar and he spent half of the
remaining mony to buy for himself. the next day alsohe gave 0ne rupee
to the beggar and spent half of the remaining to buy himself.this
pattern continued for 6 days. onthe sixthday. when he gave one rupee
to the beggar, he found that there was no money left with him. find
out how much money did ramu had inthe beginning

Answer 1

Let the amount of money Ramu had in beginning = x

After giving Re. 1 to the beggar, amount of money he had = x-1

Amount Ramu spent on himself = (x-1)÷2

Money remaining after first day = (x-1)-((x-1)÷2) = (x-1)÷2

After again giving Re. 1 to the beggar, money he had = ((x-1)÷2)-1 - (x-3)÷2

Amount Ramu spent on himself = (x-3)÷4

Money remaining after second day = ((x-3)÷2)-((x-3)÷4) = (x-3)÷4

Similarly,Money remaining after third day = (x-7)÷8

Money remaining after fourth day = (x-15)÷16

Money remaining after fifth day = (x-31)÷32

Given, on sixth day, after giving Re. 1 to the beggar, he had no money left.

Money left with him after fifth day = Re. 1

∴ (x-31)÷32 = 1
⇒x-31 = 32
⇒x = 63

Total money Ramu had in beginning = Rs. 63

Answer 2

He gives 1 rupee to beggar first and then spends half.
On 6th day, he had only 1 rupee.
Going from day 6 to day 1;

on 6th day, he had = Re 1
on 5th day, he had = 1×2 + 1 = 2+1 = Rs. 3
on 4th day, he had = 3×2 + 1 = 6+1 = Rs. 7
on 3rd day, he had = 7×2 + 1 = 14+1 = Rs. 15
on 2nd day, he had = 15×2 + 1 = 30+1 = Rs. 31
on 1st day, he had = 31×2 + 1 = 62+1 = Rs. 63

So he had Rs. 63 in the beginning.