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705-805 (Hard)|   Algebra|   Word Problems|         
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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 12:11
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let the total number of people in the group at the starting of the year be "x"

Then, for the first 3 months, each month we have x people walking x km per month.

total distance = 3(x*x) = \(3x^2\)
for the next 6 months, number of people remaning = \frac{x}{2}
so total distance for next 6 months = 6*(x/2*x/2) = \((6x^2)/4\)
again, for the remaining 3 months, number of people left = x/4
so total distance covered in last 3 months = 3*(x/4*x/4) = \((3x^2)/16\)
total distance traveled by the group in the 12 months = \(3x^2 + (3x^2)/2 + (3x^2)/16\) = 10800
simlifying, we get x^2 = 36 * 64
or x = 6 * 8 = 48
Answer is E



Bunuel
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


 


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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 12:50
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Bunuel
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


 


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Correct Answer (E)

We understand that if we have N people then they will walk N km, recording N^2 km. We assume we have N participants.

First 3 months: 3N^2 (N participants * N km * 3 months)

Next 6 months: 6(N/2)^2 = 1.5 N^2 (N/2 participants * N/2 km * 6 months) (Half of N participants means N/2)

Next 3 months: 3(N/4)^2 = 3N^2/16 (N/4 Participants * N/4 Km * 3 months) (Half of N/2 participants is N/4)

Solving: 3N^2 + 1.5N^2 + 3N^2/16 = 10800

On solving we get N as 48.
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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 12:57
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Answer:E = 48

Step 1: Translate the stem to an equation

Each participants records the number of kilometer to each active participants (p) , each month.

2 P record 2 km each
3 p record 3 km each

So km per p recorded is P

Total Km recorded each month is equal to p * p = P^ 2

For 3 months = 3 * p^2

Next 6 months half of participants = 6 * (1/2 * P ) ^2

Remaining months half of remaining prior = 3 * (1/4 * p)^2

Step 2 : write the equation:

10800 = 3 P^2 + 6 * 1/4 * P^2 + 3 * 1/16 P^2

Step 3: Factor and simplify

10800 = P^2 (75/16)

Step 4: Solve

P^2 = 16 * 144

P = 4 *12 = 48
Bunuel
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


 


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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 13:15
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Bunuel
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


 


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let total = t. so based on the condition given in ques, the equation becomes:
3*t^2 + t^2*(6/4) + t^2(3/64) = 10800
3t^2(1+1/2+1/64) = 10800
193t^2/128 = 3600
t^2 = (128/193)^1/2 * 60
approx 48 OPTION E
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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 13:53
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Let's suppose the total number of participants in the group are x, the distance travelled by the entire group in each month will be \( x^2\)
So the question says after the 1st 3 months, half the participants leave. so it is x/2 participants covering \((x/2)^2\) distance. And then 6 months later half of the remaining leave, so you have x/4 participants left; distance covered by them is \((x/4)^2\)

So basically, if we assume the year long challenge was from Jan to Dec:
Jan, Feb, March - x members covering x^2 distance each month, so total is = \(3 * x^2\)
April to Sept - x/2 members covering (x/2)^2 distance each month, so total = \(6 * (x/2)^2\)
Oct, Nov Dec - x/4 members covering (x/4)^2 distance each month, so total = \(3 * (x/4)^2 \)

We know total distance is 10800km
So,
\(3x^2 + (6x^2)/4 + (3x^2)/16 = 10800\)
Taking 16 as the LCM,
\(48x^2 + 24x^2 + 3x^2 = 10800 * 16 \)
\(75x^2 = 10800 * 16 \)
\(25x^2 = 3600 * 16 \)
\(x^2 = 36 * 4 * 16 \)
\(x = 6 * 2 * 4 = 48 \)

Answer is E

PS: The reason we do \(x^2\) as the distance is coz assume that we have 4 participants: A, B, C, D
The question says - each participant covers a distance equal to the number of active participants.
So, A covers 4Km, B covers 4Km, C covers 4km, & D covers 4 Km
So the total distance covered by them is 4 * 4

The same will be the case if we have 5 participants - 5 * 5 and so on. So for x participants, together they cover \(x * x\)
Bunuel
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


 


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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 15:46
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Answer: D - 48.

Let x be the number of participants in the beginning of the challenge.

Notice that the total number of kilometers walked by the group for each month will be the number of participants, squared.

So we have:

Months 1 - 3: x^2 * 3
Months 4 - 9: (x/2)^2 * 6
Months 10 - 12: (x/4)^2 * 3

The sum of these three terms will equal the total number of kilometers walked by the group per year, which is given as 10,800.
Equation simplifies to:
6x^2 + (3x^2)/2 + (3x^2)/16 = 10,800

Multiply both sides by 4
3x^2 * 25/16 = 10,800

x^2 = 10,800*16/75

x^2 = 12^2 * 4^2
x = 12 * 4
x = 48

10,800/27 = 400

400 * 4 = 1600

Sqrt 1600 = 40.

Answer D.
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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 16:51
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let there be X participants at the beginning, we get
for first 3 months, kilometers recorded= 3x^2
for next 6 months, kilometers recorded = 6*0.25*x^2
for next 3 months, kilometers recorded= 3*x^2* 1/16
Adding all three = 10800
x= 48
Bunuel
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


 


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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 17:46
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At the start the number of people=n
After 3 months the remaining pop= n/2
After 6 more months the further remaining population=n/4
It is mentioned that people actually walk as much as the population at that time for that month
For the first 3 months the total distance= 3*n*n =3n^2
For the next 6 months the total distance= 6*(n/2)*(n/2)= 3/2 (n^2)
For the last 3 months of the year total distance= 3*(n/4)*(n/4)= 3/16(n^2)
3n^2+3/2(n^2)+3/16(n^2)=10800
Solving this results in n=48-- option E
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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 18:10
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IMO, Option E is correct

Let the initial number of participants be N

Each month, if there are K active participants, each person walks K km. So, the total distance walked that month is K^2.

  • Months 1-3: N participants
    Total distance = 3N^2

  • Months 4-9: Half remains, so N/2 participants
    Total distance = 6 * (N/2)^2 = 3N^2/2

  • Months 10-12: Half remains, so N/4 participants
    Total distance = 3 * (N/4)^2 = 3N^2/16

Adding everything -->
3N^2 + 3N^2/2 + 3N^2/16 = 10800
(75/16)N^2 = 10800
N^2 = 2304
N = 48

Answer: Option (E): 48
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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 18:27
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Let us denoted the initial number of participants by n.

First 3 months, each month n participants walked n km each, totaling n*n*3 = 3n^2 km
Next 6 months, each month n/2 participants walked n/2 km each, totaling (n/2)*(n/2)*6 = 6(n^2)/4 km
Last 3 months, each month n/4 participants walked n/4 km each, totaling (n/4)*(n/4)*3 = 3(n^2)/16 km

Total distance traveled by the group in a year = Sum of the above three distances = 75(n^2)/16 km
We are given that this distance is 10800 km.

Hence, 75(n^2)/16 = 10800
or, n^2 = 10800*16/75 = 144*16
or, n = 12*4 = 48
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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 20:16
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Say there were x participants at the beginning.

MonthPartipantsTotal walking in the period
1st-3rdx3*x*x=3x^2
4th-9thx/26*x/2*x/2=3x^2 /2
10th-12thx/43*x/4*x/4=3x^2 /16
Total km,
.: 3x^2+3x^2 /2 +3x^2 /16=10800
.: 75x^2/16=10800
.: x^2=2^8 * 3^2
.: x= 16*3=48

Ans is E.

Bunuel
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


 


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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 20:38
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--------For the first 3 months----------
Number of active participants in the group = 4x (Assumed value)
Distance walked by each active participant per month= Number of active participants in that month = 4x
So, total distance walked by the group per month = (4X)*(4X)= 16X^2.
Hence total distance walked by the group during the first 3 months = 3*(16X^2)= 48X^2
--------For the next 6 months----------
Number of active participants in the group= 2x
Distance walked by each participants per month= 2x
Total distance walked by the group that month = (2x)*(2x)= 4x^2
Total distance walked by the group during the next 6 months= 6*(4X^2)= 24X^2.
--------For last 3 months----------------
Number of active participants in the group= x
Distance walked by each participants per month=x
Total distance walked by the group per month= (x)*(x)=X^2
Total disTance walked by the group for the last 3 months = 3*(X^2).

Combining all , total distance walked by the group throughout the year = 48X^2 + 24X^2 + 3X^2 = 10800
= 75X^2= 10800
= X^2=144
X=12.
Since, total number of participants at the beginning = 4X = 4*12 = 48 (Answer), Choice-E
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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 20:55
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Here is my approach
A. 12*12 = 144*3 = 432, cannot reach towards 10,800
B. 24*24 = 576*3 = 1,728, cannot reach towards 10,800 with half of the participants for the next 6 months
Similarly 30 and 40, the same as they are not square.
E = 10,800/48 = 225 (Perfect square, lets try)

48*48 = 2,304*3 months = 6,912
Then half of the participants continued for 6 months, 48/2 = 24*24 = 576*6 months = 3,456
The rest of the participants stay till the end. 12*12 = 144*3 months = 432

Total = 6,912 + 3,456 + 432 = 10,800

Answer: E
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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 21:39
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Initial number of participants = n

Group total after 3 month ( km) = 3*n^2

For the next 6 months,
Number of participants = n/2
Group total in next 6 months = 6*(n^2)/4
= 3*n^2/2

For the next 3 months,
Number of participants = n/2 - n/4 = n/4
Group total in next 3 months = 3*(n^2)/16


Total distance covered during the year,

3n^2 + 3n^2/2 + 3n^2/16 = 10800
48n^2 + 24n^2 + 3n^2 = 172800
75n^2 = 172800
n^2 = 2304
n = 48

Answer : E (48)

Bunuel
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


 


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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 23:01
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let n be the number of participants

as there are n number of people and each walks n number of km, we can say its n x n kilometers per month.

So after 3 months there would be 3n^2

next 6 months half remains, so there are n/2 participants

6x(n/2)^2 = 3n^2/2

Final 3 months
half of those remaining, so n/4 participants remain:
3x(n/4)^2 = 3n^2/16

total if 10,800km

3n^2 + 3n^2/2 + 3n^2/16 = 10800
rearrange n = 48
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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 23:01
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ans - option E
let the beginning no of participants be x
as mentioned in the ques the distance covered will be equal to participants in that month hence X participants * x distance
then distance covered by x people in first 3 months = (x)^2 *3
after 6 months 1/2 of participants leave so left are = x*1/2= x/2, distance covered by them for 6 months = x/2*x/2*6= ((x^2)/4)*6)
after 9 months so for 3 months since its for a year again 1/2 of remaining leave so x/2*1/2 = distance covered by them for 3 months = x/4*x/4*3= (x^2/16)*3
adding all of them for total distance = x^2*3 +(x^2/4)*6 + (x^2/16)*3 = 10800
75x^2=10800*16
x=48
Bunuel
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


 


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Anchal02
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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 23:27
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X no of participants
So x km for x participants

3months : 3 x^2
6 months : 6 (x^2)/4
3 months : 3(x^2)/16

Add all and solve x=48
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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 16 Jul 2026, 23:51
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(3+3/2+3/16)n^2 = 10800
n = 48.
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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 17 Jul 2026, 00:15
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say total num of people = x

now given that, every person walks and records num of kilometers = num of active people in challenge.


for first 3 month everyone walked in.
so total distance = x^2
but thats for each month. so for 3 months, its 3x^2

now after 3 months half people leaves. so left with only x/2.
total distance per month = (x/2)^2
so in 6 months = 6x^2 = 3x^2/ 2

now after 6 months another half leaves from x/2. so only left with x/4
total distance per months = (x/4)^2
so in last 3 months = 3x^2/16

now total distance throughout year = 10800

3x^2 + (3x^2/2) + (3x^2/16) = 10800


upon solving this we get x = 48

choice E

Bunuel
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


 


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GMAT Club World Cup 2026 (Day 9): At the beginning of a yearlong 17 Jul 2026, 00:32
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Let the initial no. of participants be x
For the first 3 months, x participants each walk x km every month, so the total is \(3x^2\)
Next 6 months, x/2 participants each walk x/2 km every month, so the total is \(6(x/2)^2\) = \(3x^2/2\)
Last 3 months, x/4 participants each walk x/4 km every month, so the total is \(3(x/4)^2\) = \(3x^2/16\)
So,
\(3x^2\) + \(3x^2/2\) + \(3x^2/16\) = 10,800
\(x^2\) = 2,304
\(x\) = 48

Ans : E
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