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07 Aug 2026, 05:23
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At the beginning of a yearlong walking challenge, a group of friends begins tracking the total distance walked by the group. In each month, every active participant walks and records a number of kilometers equal to the number of active participants in the challenge during that month. After the first 3 months, half of the participants leave the challenge, and after the next 6 months, half of the remaining participants leave. Any participant who leaves stops walking and recording distances for the challenge, but the distance that participant recorded earlier still counts toward the group’s total. If the group records a total of 10,800 kilometers during the year, how many participants were in the challenge at the beginning?
A. 12
B. 24
C. 30
D. 40
E. 48
A. 12
B. 24
C. 30
D. 40
E. 48
07 Aug 2026, 05:23
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Official Solution:
At the beginning of a yearlong walking challenge, a group of friends begins tracking the total distance walked by the group. In each month, every active participant walks and records a number of kilometers equal to the number of active participants in the challenge during that month. After the first 3 months, half of the participants leave the challenge, and after the next 6 months, half of the remaining participants leave. Any participant who leaves stops walking and recording distances for the challenge, but the distance that participant recorded earlier still counts toward the group’s total. If the group records a total of 10,800 kilometers during the year, how many participants were in the challenge at the beginning?
A. 12
B. 24
C. 30
D. 40
E. 48
Let \(n\) be the number of participants at the beginning.
For the first 3 months, there are \(n\) active participants, and each records \(n\) kilometers per month.
Total for the first 3 months:
\(3 * n * n = 3n^2\)
After the first 3 months, half of the participants leave, so \(\frac{n}{2}\) participants remain.
For the next 6 months, each of the \(\frac{n}{2}\) active participants records \(\frac{n}{2}\) kilometers per month.
Total for the next 6 months:
\(6 * \frac{n}{2} * \frac{n}{2} = \frac{3n^2}{2}\)
After the next 6 months, half of the remaining participants leave, so \(\frac{n}{4}\) participants remain.
For the final 3 months, each of the \(\frac{n}{4}\) active participants records \(\frac{n}{4}\) kilometers per month.
Total for the final 3 months:
\(3 * \frac{n}{4} * \frac{n}{4} = \frac{3n^2}{16}\)
Thus:
\(3n^2 + \frac{3n^2}{2} + \frac{3n^2}{16} = 10,800\)
\(\frac{48n^2}{16} + \frac{24n^2}{16} + \frac{3n^2}{16} = 10,800\)
\(\frac{75n^2}{16} = 10,800\)
\(n^2 = 10,800 * \frac{16}{75}\)
\(n^2 = 2,304\)
\(n = 48\)
Answer: E
At the beginning of a yearlong walking challenge, a group of friends begins tracking the total distance walked by the group. In each month, every active participant walks and records a number of kilometers equal to the number of active participants in the challenge during that month. After the first 3 months, half of the participants leave the challenge, and after the next 6 months, half of the remaining participants leave. Any participant who leaves stops walking and recording distances for the challenge, but the distance that participant recorded earlier still counts toward the group’s total. If the group records a total of 10,800 kilometers during the year, how many participants were in the challenge at the beginning?
A. 12
B. 24
C. 30
D. 40
E. 48
Let \(n\) be the number of participants at the beginning.
For the first 3 months, there are \(n\) active participants, and each records \(n\) kilometers per month.
Total for the first 3 months:
\(3 * n * n = 3n^2\)
After the first 3 months, half of the participants leave, so \(\frac{n}{2}\) participants remain.
For the next 6 months, each of the \(\frac{n}{2}\) active participants records \(\frac{n}{2}\) kilometers per month.
Total for the next 6 months:
\(6 * \frac{n}{2} * \frac{n}{2} = \frac{3n^2}{2}\)
After the next 6 months, half of the remaining participants leave, so \(\frac{n}{4}\) participants remain.
For the final 3 months, each of the \(\frac{n}{4}\) active participants records \(\frac{n}{4}\) kilometers per month.
Total for the final 3 months:
\(3 * \frac{n}{4} * \frac{n}{4} = \frac{3n^2}{16}\)
Thus:
\(3n^2 + \frac{3n^2}{2} + \frac{3n^2}{16} = 10,800\)
\(\frac{48n^2}{16} + \frac{24n^2}{16} + \frac{3n^2}{16} = 10,800\)
\(\frac{75n^2}{16} = 10,800\)
\(n^2 = 10,800 * \frac{16}{75}\)
\(n^2 = 2,304\)
\(n = 48\)
Answer: E
