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02 Apr 2024, 08:07
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
75% (03:38) correct
25%
(03:15)
wrong
based on 8
sessions
History
Date
Time
Result
Not Attempted Yet
Hungry chipmunk notices that, along a straight path, there are walnuts placed at intervals of 1 meter. The chipmunk wants to gather all the walnuts in its burrow, which is exactly where the middle walnut is. The chipmunk starts gathering the walnuts from the leftmost walnut, beginning its task standing next to the leftmost walnut, and carries only one walnut at a time. If after completing the task, the chipmunk covered a distance of 300 meters and collected an odd number of walnuts, then how many walnuts did the chipmunk collect?
A. 11
B. 13
C. 15
D. 23
E. 25
A. 11
B. 13
C. 15
D. 23
E. 25
02 Apr 2024, 08:07
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Official Solution:
Hungry chipmunk notices that, along a straight path, there are walnuts placed at intervals of 1 meter. The chipmunk wants to gather all the walnuts in its burrow, which is exactly where the middle walnut is. The chipmunk starts gathering the walnuts from the leftmost walnut, beginning its task standing next to the leftmost walnut, and carries only one walnut at a time. If after completing the task, the chipmunk covered a distance of 300 meters and collected an odd number of walnuts, then how many walnuts did the chipmunk collect?
A. 11
B. 13
C. 15
D. 23
E. 25
Check the image below:
There are \(n\) walnuts to the left of the burrow, \(n\) walnuts to the right of the burrow, and 1 walnut exactly in the middle, at the burrow. Hence, the total number of walnuts is \(n + n + 1=2n + 1\).
The chipmunk starts where the leftmost walnut is, so the first leg of the travel covers \(n\) meters. After that, the chipmunk reaches the burrow. To collect the remaining \(n-1\) walnuts, placed to the left of the burrow, the chipmunk should travel \(n-1\) meters twice: to the walnut and back to the burrow; \(n-2\) meters twice, to the walnut and back to the burrow; and so on until the first walnut to the left, where it travels 1 meter to the walnut and 1 meter back to the burrow. Thus, the chipmunk covers a total of:
\(2*1 + 2*2 + 2*3 + ... + 2(n-2) + 2(n-1)=2(1+2+3+...+(n-2)+(n-1))\) meters.
The sum within the parentheses represents the sum of consecutive integers from 1 to \(n-1\), which equals to \(\frac{first + last}{2} * (number \ of \ terms)\). Hence, we get:
\(2(1+2+3+...+(n-2)+(n-1))= 2(\frac{1 + (n-1)}{2}*(n-1))=n(n-1)\)
Don't forget the \(n\) meters the chipmunk covered to collect the first walnut to get the total distance covered to collect all walnuts to the left:
\(n(n-1) + n = n^2\) meters.
Similarly, we can calculate the distance the chipmunk will cover to collect walnuts placed to the right of the burrow, with an expectation that the chipmunk has have to cover \(n\) meters twice, not once: to the rightmost walnut and back, making the distance covered to collect walnuts placed to the right equal to \(n^2 + n\) meters.
Thus, we have that \(n^2 + (n^2 + n) = 300\), which gives \(2n^2 +n- 300=0\). Solving gives \(n=12\). Therefore, the number of walnuts is \(2n+1=25\).
Answer: E
Hungry chipmunk notices that, along a straight path, there are walnuts placed at intervals of 1 meter. The chipmunk wants to gather all the walnuts in its burrow, which is exactly where the middle walnut is. The chipmunk starts gathering the walnuts from the leftmost walnut, beginning its task standing next to the leftmost walnut, and carries only one walnut at a time. If after completing the task, the chipmunk covered a distance of 300 meters and collected an odd number of walnuts, then how many walnuts did the chipmunk collect?
A. 11
B. 13
C. 15
D. 23
E. 25
Check the image below:
There are \(n\) walnuts to the left of the burrow, \(n\) walnuts to the right of the burrow, and 1 walnut exactly in the middle, at the burrow. Hence, the total number of walnuts is \(n + n + 1=2n + 1\).
The chipmunk starts where the leftmost walnut is, so the first leg of the travel covers \(n\) meters. After that, the chipmunk reaches the burrow. To collect the remaining \(n-1\) walnuts, placed to the left of the burrow, the chipmunk should travel \(n-1\) meters twice: to the walnut and back to the burrow; \(n-2\) meters twice, to the walnut and back to the burrow; and so on until the first walnut to the left, where it travels 1 meter to the walnut and 1 meter back to the burrow. Thus, the chipmunk covers a total of:
\(2*1 + 2*2 + 2*3 + ... + 2(n-2) + 2(n-1)=2(1+2+3+...+(n-2)+(n-1))\) meters.
The sum within the parentheses represents the sum of consecutive integers from 1 to \(n-1\), which equals to \(\frac{first + last}{2} * (number \ of \ terms)\). Hence, we get:
\(2(1+2+3+...+(n-2)+(n-1))= 2(\frac{1 + (n-1)}{2}*(n-1))=n(n-1)\)
Don't forget the \(n\) meters the chipmunk covered to collect the first walnut to get the total distance covered to collect all walnuts to the left:
\(n(n-1) + n = n^2\) meters.
Similarly, we can calculate the distance the chipmunk will cover to collect walnuts placed to the right of the burrow, with an expectation that the chipmunk has have to cover \(n\) meters twice, not once: to the rightmost walnut and back, making the distance covered to collect walnuts placed to the right equal to \(n^2 + n\) meters.
Thus, we have that \(n^2 + (n^2 + n) = 300\), which gives \(2n^2 +n- 300=0\). Solving gives \(n=12\). Therefore, the number of walnuts is \(2n+1=25\).
Answer: E
17 Apr 2024, 00:30
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Very tough question, honestly I dont think anyone can finish this in less than 4 mintues.
