A string has been cut into 4 pieces, all of different lengths. The length of each pieceis 2 times the length of the next smaller piece. What fraction of the original stringis the longest piece?
Question
A string has been cut into 4 pieces, all of different lengths. The length of each pieceis 2 times the length of the next smaller piece. What fraction of the original stringis the longest piece?
Solution
To solve this problem, we need to understand that the lengths of the pieces form a geometric sequence, where each term is twice the previous one. Let's denote the length of the smallest piece as x. Then, the lengths of the other pieces are 2x, 4x, and 8x.
The total length of the string is the sum of the lengths of all pieces, which is x + 2x + 4x + 8x = 15x.
The longest piece is 8x long. Therefore, the fraction of the original string that is the longest piece is 8x / 15x = 8/15.
So, the longest piece is 8/15 of the original string.
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