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Bunuel
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Town Z plans to install b identical benches on a straight path that 15 Jun 2026, 03:30
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95% (hard)

Question Stats:

43% (03:38) correct 57% (03:38) wrong based on 46 sessions

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Town Z plans to install \(b\) identical benches on a straight path that runs through a park from east to west. The distance between the first bench and the eastern end of the path must be equal to \(x\) yards, and the same must be true of the last bench and the western end of the path. Additionally, any two benches must be separated by a gap equal to \(x\) yards. The total length of the path is 296 yards and the length in yards of each bench is an integer. If \(x\) is equal to \(\frac{1}{3}\) the length of each bench, how many benches can the path accommodate?

(A) 3

(B) 8

(C) 9

(D) 24

(E) 37
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Town Z plans to install b identical benches on a straight path that 17 Jun 2026, 03:00
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Bunuel
Town Z plans to install \(b\) identical benches on a straight path that runs through a park from east to west. The distance between the first bench and the eastern end of the path must be equal to \(x\) yards, and the same must be true of the last bench and the western end of the path. Additionally, any two benches must be separated by a gap equal to \(x\) yards. The total length of the path is 296 yards and the length in yards of each bench is an integer. If \(x\) is equal to \(\frac{1}{3}\) the length of each bench, how many benches can the path accommodate?

(A) 3

(B) 8

(C) 9

(D) 24

(E) 37
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Town Z plans to install b identical benches on a straight path that 15 Jun 2026, 23:07
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Now, we've to look for a factor, which is of the form 4n+1, from the factors of 296
1 2 4 8 37 74 148 296
only 37 fits the description
Therefore, 4n+1 = 37, n = 9
Hence, C 9 benches
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Town Z plans to install b identical benches on a straight path that 02 Jul 2026, 03:09
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Can someone provide a better explanation to the part about the forming of equation- x+nx4x what was done for the two x yards at the end of each front
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Town Z plans to install b identical benches on a straight path that 02 Jul 2026, 03:39
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Town Z plans to install \(b\) identical benches on a straight path that runs through a park from east to west. The distance between the first bench and the eastern end of the path must be equal to \(x\) yards, and the same must be true of the last bench and the western end of the path. Additionally, any two benches must be separated by a gap equal to \(x\) yards. The total length of the path is 296 yards and the length in yards of each bench is an integer. If \(x\) is equal to \(\frac{1}{3}\) the length of each bench, how many benches can the path accommodate?

(A) 3

(B) 8

(C) 9

(D) 24

(E) 37

Let each bench have length L yards.

Since x is 1/3 of the bench length:

x = L/3

The full path consists of:

end gap + benches + gaps between benches + end gap

There are b benches and b + 1 gaps, so:

bL + (b + 1)(L/3) = 296

3bL + (b + 1)L = 888

L(4b + 1) = 888

Since L is an integer, 4b + 1 must divide 888.

Check the choices:

For b = 9:

4b + 1 = 4(9) + 1 = 37

888/37 = 24

So L = 24, an integer.

Therefore, the path can accommodate 9 benches.

Answer: C.
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