12 Days of Christmas GMAT Competition 2025 - Day 3: On January 1 of

On January 1 of 2025, Website Nexus had 1,000 registered users, and Website Echo had 1,600 registered users. Each site gained a constant number of new users every month, and by May 1, 2025, Website Nexus had 250% more registered users than Website Echo.

In the table, select for Nexus the number of registered users it gained per month and select for Echo the number of registered users it gained per month that would be jointly consistent with the given information. Make only two selections, one in each column.

Using arithmetic growth formula here. We know that Nexus will be 3.5 times Echo (250% more).

Let $k_1$ equal growth factor for Nexus and $k_2$ equal growth factor for Echo.

$4k_1+1,000=\frac{7}{2}(4k_2+1,600)$

Simplifying, we get:

$2k_1-7k_2=2,300$

We're looking for values for $k_1$ and $k_2$ here that equal 2,300. We can tell from the equation that $k_1$ needs to be a far bigger value than $k_2$ and needs to make a multiple of 7 when subtracted from 2,300.

Trying 3,600 first we get:

$2(3,600)-7k_2=2,300$

$-7k_2=-4,900$

$k_2=700$

Thus Nexus grows at 3,600 a month and Echo grows at 700 a month.

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17 Dec 2025, 13:12

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Bunuel

January 1’2025 , the number of registered users

Website Nexus = 1000

Website Echo = 1600

The users get added each month by a constant number for nexus (a) and Echo (b).

Jan — Feb —- Mar —- Apr —- May.

So we have 4 additions in each category.

On May 1’2025: Nexus = 1000+4a , while Echo = 1600+4b

Given that: On May 1’2025: the value of Nexus is 250% more than the value of Echo.

250% more = 1 + (250/100) = 7/2

1000+4a = (7/2)* (1600+4b)

2000 + 8a = 11200 + 28b

8a - 28b = 9200

2a - 7b = 2300

2a = 2300+7b

The possible answer solutions are : 200, 300, 700, 1500, 2100 and 3600.

If I take a =2100, then 4200 = 2300 + 7b

7b =1900. Not possible.

Let’s take a = 3600, then 2a = 2*3600 = 7200.

7200 = 2300 + 7b

7b = 4900

b = 700 .

This gives a possible answer.

Hence, Nexus = 3600, Echo = 700.

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17 Dec 2025, 13:34

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Bunuel

1000 + 4N = 3.5 (1600 + 4E)

1000 + 4N = 5600 + 14E

4N - 14E = 4600

2N - 7E = 2300

Testing out the options we get

N = 3600
E = 700

Nexus = 3600
Echo = 700

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17 Dec 2025, 14:44

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say monthly gain for nexus is a and for echo is b
so at the end of april total nexus = 1000+ 4a
total echo = 1600+ 4b
now nexus is 3.5 times echo
100+ 4a = 3.5 (1600+4b)
2a+7b= 2300
test value from table
a-3600
b - 700

Bunuel

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17 Dec 2025, 15:18

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Bunuel

let nexus growth per month be x, echo be y
equation will be below, since 250% more signifes 3.5 times value
(1000 + 4x) / (1600 + 4y) = 3.5
simpliefes to
1000 + 4x = 5600 + 14y
4x = 46000 + 14y

solves for value of
x = 3600
y = 700

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17 Dec 2025, 19:26

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ratio afterwards $= \frac{350}{100} = \frac{7}{ 2}$

interms of starting values = $\frac{1000 + N*4 }{ 1600 + E * 4}$
$\frac{250 + N }{ 400 + E} = \frac{7}{2} $

$500 + 2N =2800 + 7E$

$2N =2300 + 7E$

N will be a value >1150

Let N = 1500
3000 = 2300 + 7E
E = 100

Let N = 3600
7200 = 2300 + 7E
4900 = 7E
E = 700

Ans: 3600, 700 or F, C

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17 Dec 2025, 19:27

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Bunuel

Total reg users on May 1 for Nexus be 1000 + 4*a , where a be constant increase per month
Total reg users on May 1 for echo be 1600 + 4*b, where b is constant increase per month

Given
1000 + 4a = 1600 + 4b + 250/100(1600 + 4b)
1000 + 4a = 3.5(1600 + 4b)

500 + 2a = 3.5(800+2b)
500 + 2a = 2800 + 7b

2a - 7b = 2300
a = (2300 +7b)/2

Try values for starting with smallest 200 as b is definitely smaller than a.
b=200, a = (2300 + 1400)/2 , We don't have an answer choice ending with 350, No
b= 300, a = (2300 + 2100)/2 = 4400/2 = 2200, We don't have this choice, No
b=700, a = (2300+4900)/2 = 7200/2 = 3600. We have this choice.

Correct answer b=700, a=3600

[color=#000000]Nexus[/color]	[color=#000000]Echo[/color]
		[color=#000000]200[/color]
		[color=#000000]300[/color]
		[color=#000000]700[/color]
		[color=#000000]1,500[/color]
		[color=#000000]2,100[/color]
		[color=#000000]3,600 [/color]

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17 Dec 2025, 19:39

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On 1 Jan 2025 , Nexus had 1000 Registered users. Assume each month it was getting increased by x amount. Then by 1 May, it would have total 1000+4x registered users. Similarly , Echo had 1600 registered users as on 1 Jan 2025. Assume each month it was getting increased by y amount , then by 1 May , it would have total 1600+4y registered users. On 1 May , it is found that Nexus had 250% more registered users than Website Echo. So ,

1000+4x = (1+2.5)(1600+4y)

on simplify , it will reduced to 2x-7y=2300. From options , x=3600 , y=700. So answer is Nexus = 3600 , Echo =700

Bunuel

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17 Dec 2025, 20:33

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N - 700; E - 200
Initially I solved this algebraically but then I think I found a faster way.

But with the algebra, let k be the new users per month for site N, and let m be the new users per month for site E.

"X is 250% more than Y" means x = y+2.5y, or 3y. If it said "X is 250% of Y" it would be x = 2.5y.

So 10+4k=3.5(16 +4m) (I find it easier to exclude the 0s in the hundreds and add them back on later).
Expanding:
10+4k = 56+14m
4k = 46 + 14m
k = 11.5 + 3.5m

Plug in m = 1, and get k = 15.

Therefore total number of users for E is 16 + 4(1) = 20 and 10+4(15) = 70. Then I noticed that 700 = 3.5(200).

If the total number of registered users is 250% more for E than N, then doesn't the amount by which they're increasing constantly need to be that difference, as well? The only one of the answer choices that is 250% more than another one is 200 and 700.

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17 Dec 2025, 20:35

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Solution:

We are given 1st January, 2025 Website nexus had 1000 registered users and Website Echo had 1600 registered users. By May 1, 2025 Website Nexus had 250% more registered users than Website Echo means Nexus = 3.5Echo. Let's look at the available options to match up both.

The only choice that fits into it is Nexus = 3.5(200)= 700.

So, the for Nexus the number of registered users it gained per month is 700 and for Echo the number of registered users it gained per month is 200.

Bunuel

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17 Dec 2025, 20:41

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Let the increase for Nexus be X and Echo be Y, So in May Nexus users 1000+4x, Echo users 1600+4y
Also, 1000+4x=(1600+4y)(1+250%)

Solving this we get, 2X-7Y=2300

Inserting the values from table only X=3600 and Y=700 satisfies

Bunuel

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17 Dec 2025, 22:22

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N on Jan1: 1000 -> post 4n increment months -> 1000 + 4n
E on Jan1: 1600 -> post 4e increment months -> 1600 + 4e

Notice: even though E had good start, N overtakes E within 4 months hence n > e (will help in zooming in answers)
Acc. to question, (1000 + 4n) = 3.5(1600 + 4e) => 2n = 2300 + 7e => n = 3600 and e = 700 satisfies.

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17 Dec 2025, 22:24

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nexus=(1600*(4*substitute the option value for echo)*2.5-1000)/4 which will give the per month increase for nexus. check if the substituted value for echo and the value obtained for nexus is present in option.

2.5 used in formula is nothing but the 250% increase
1000 is the value of per month nexus subscription on 1st Jan

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17 Dec 2025, 22:26

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Because the number of new users each site gained every month is a constant so from Jan 1 to May 1, the number of new users Nexus gained is 4a and number of new users Echo gained is 4b.
by May 1, Nexus had 250% more registered users than Echo so we have 1000+4a = 250% * (1600+4b) => 4a= 10b+3000=>2a=5b+1500 (1)
Using the answers option, we have a=1500, b=300 jointly consistent with (1)

Bunuel

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17 Dec 2025, 23:22

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Setup A.P equations with E and N for month 4,
Simplify and substitute values for E (Start with smallest values bc we know E<N from the question) and check if corresponding values of N are present in the table
Imo : E = 700 and N=3600

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18 Dec 2025, 01:12

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Number of users in Jan
Nexus - 1000
Echo - 1600
Let the constant number of new users added every month for Nexus and Echo be n and e respectively.
Number of users by May
Nexus - 1000 + 4n
Echo - 1600 + 4e
Given that Nexus had 250% more users than Echo in May
Nexus = Echo + 250%Echo
= 350%Echo
= 3.5 * Echo
This implies
1000 + 4n = 3.5 * (1600 +4e)
1000 + 4n = 5600 + 14e
4n = 4600 + 14e
2n = 2300 + 7e

Using trial and error,
let e = 200, then n = (2300+1400)/2 = 18500 - not available
let e = 300, then n = (2300+2100)/2 = 12000 - not available
let e = 700, then n = (2300+4900)/2 = 3600 - available
So e = 700, n = 3600

Bunuel

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18 Dec 2025, 01:19

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Let X be the rate at which Nexus is gaining users per month and Y for Echo

For Nexus:
1000 being the initial users as on Jan 1 2025 the users by May 1 2025 is equals to 1000+4x

For Echo:
1600 being the initial users as on Jan 1 2025 and the users by May 1 2025 becomes equal to 1600+4Y

Now it is given in the question that Nexus user counts has become 250% of Echo users that means

1000+4X = 1.35*( 1600+4Y)

Frm the options this equation can be satisfied by using
X= 3600 and Y = 700
Hence the answers are 3600,700

Bunuel