On 1st May, Mr. X arrived in a new city and was looking for a place to stay. He met a landlord who offered to rent his apartment at a reasonable price but wanted Mr. X to pay the rent on a daily basis. Mr. X had a silver bar of 31 inches and an inch of the silver bar was exactly equal to a day's rent. He agreed to pay an inch of the silver bar towards the daily rent. Mr. X wanted to make minimum number of pieces of silver bar but did not want to pay any advance rent. How many pieces did he make?
(A)5 (B)8 (C)16 (D)20 (E)31
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Last edited by mathbyvemuri on Mon May 21, 2012 6:06 pm, edited 1 time in total.
RaviSankar Vemuri
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- eagleeye
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Mr.X can do this with 5 pieces.
On May 1st, he gives the landlord 1 inch piece.
On the 2nd day, he gives the landlord a 2 inch piece, and takes the 1 inch back.
On the 3rd, gives the landlord the 2 inch piece. (now she has 2+1)= 3 inches for 3 days.
On the 4th day, he gives the landlord the 4 inch piece, but takes back the 2 and 1 inch pieces.
In the same way he can do it with 1+2+4+8+16 = 31 inches for 31 days, with 5 pieces.
On May 1st, he gives the landlord 1 inch piece.
On the 2nd day, he gives the landlord a 2 inch piece, and takes the 1 inch back.
On the 3rd, gives the landlord the 2 inch piece. (now she has 2+1)= 3 inches for 3 days.
On the 4th day, he gives the landlord the 4 inch piece, but takes back the 2 and 1 inch pieces.
In the same way he can do it with 1+2+4+8+16 = 31 inches for 31 days, with 5 pieces.
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Yes, as 'eagleeye' said, 5-pieces is the answer.
Beauty of math is illustrated in this problem.
If Mr X cuts the silver bar of 31 inches in to the pieces with following sizes (in inches), it will result in minimum number of pieces- solution:
1, 2, 4, 8, 16
On day-1 Mr X gives 1-inch piece to the landlord
On day-2 he gives 2-inch piece and takes back the 1-inch piece
On day-3 he gives 1-inch piece in addition to the already given 2-inch piece, thus making it 3-inches in total
On day-4 he gives 4-inch piece and takes back the 1-inch and 2-inch pieces
On day-5 he gives 1-inch piece in addition to the already given 4-inch piece, thus making it 5-inches in total
This will continue till day-31.
Hence a minimum of five pieces is enough
Math logic:
Any number up to 2n-1 can be represented by the combinations of the 'n' numbers: 2^0,2^1,2^2,...2n-1.
For example, if 7 is considered, all numbers from 1 to 7 can be represented as combinations of 1,2, and 4:
1 = 1; 2 = 2; 3 = 1+2; 4 = 4; 5 = 1+4; 6 = 2+4; 7 = 1+2+4
If 9 is considered, all numbers from 1 to 9 can be represented as combinations of 1,2,4 and 2:
1 = 1; 2 = 2; 3 = 1+2; 4 = 4; 5 = 1+4; 6 = 2+4; 7 = 1+2+4; 8 = 2+2+4; 9 = 1+2+2+4;
If 15 is considered, all numbers from 1 to 15 can be represented as combinations of 1,2,4 and 8:
1 = 1; 2 = 2; 3 = 1+2; 4 = 4; 5 = 1+4; 6 = 2+4; 7 = 1+2+4; 8 = 8; 9 = 1+8; 10 = 2+8;
11 = 1+2+8; 12 = 4+8; 13 = 1+4+8; 14 = 2+4+8; 15 = 1+2+4+8
Answer (A)
Beauty of math is illustrated in this problem.
If Mr X cuts the silver bar of 31 inches in to the pieces with following sizes (in inches), it will result in minimum number of pieces- solution:
1, 2, 4, 8, 16
On day-1 Mr X gives 1-inch piece to the landlord
On day-2 he gives 2-inch piece and takes back the 1-inch piece
On day-3 he gives 1-inch piece in addition to the already given 2-inch piece, thus making it 3-inches in total
On day-4 he gives 4-inch piece and takes back the 1-inch and 2-inch pieces
On day-5 he gives 1-inch piece in addition to the already given 4-inch piece, thus making it 5-inches in total
This will continue till day-31.
Hence a minimum of five pieces is enough
Math logic:
Any number up to 2n-1 can be represented by the combinations of the 'n' numbers: 2^0,2^1,2^2,...2n-1.
For example, if 7 is considered, all numbers from 1 to 7 can be represented as combinations of 1,2, and 4:
1 = 1; 2 = 2; 3 = 1+2; 4 = 4; 5 = 1+4; 6 = 2+4; 7 = 1+2+4
If 9 is considered, all numbers from 1 to 9 can be represented as combinations of 1,2,4 and 2:
1 = 1; 2 = 2; 3 = 1+2; 4 = 4; 5 = 1+4; 6 = 2+4; 7 = 1+2+4; 8 = 2+2+4; 9 = 1+2+2+4;
If 15 is considered, all numbers from 1 to 15 can be represented as combinations of 1,2,4 and 8:
1 = 1; 2 = 2; 3 = 1+2; 4 = 4; 5 = 1+4; 6 = 2+4; 7 = 1+2+4; 8 = 8; 9 = 1+8; 10 = 2+8;
11 = 1+2+8; 12 = 4+8; 13 = 1+4+8; 14 = 2+4+8; 15 = 1+2+4+8
Answer (A)
RaviSankar Vemuri
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https://mathbyvemuri.blogspot.in/2012/05 ... -data.html
https://mathbyvemuri.blogspot.in/2012/05 ... dates.html
https://mathbyvemuri.blogspot.in/2012/05 ... es-of.html