12 Days of Christmas GMAT Competition 2025 - Day 3: On January 1 of : Two-part Analysis (TPA)

Deconstructing the Question
Time period: Jan 1 to May 1 = 4 months.
Website Nexus: Start: 1,000 users. Monthly gain: $N$.
Final count: $1000 + 4N$.
Website Echo: Start: 1,600 users. Monthly gain: $E$. Final count: $1600 + 4E$.
Relationship: Nexus final is "250% more than" Echo final. $\text{Nexus} = \text{Echo} + 2.5(\text{Echo}) = 3.5(\text{Echo})$.

Step 1: Set up the Equation
$1000 + 4N = 3.5(1600 + 4E)$ $1000 + 4N = 5600 + 14E$
Subtract 1000 from both sides:
$4N = 4600 + 14E$

Divide by 2 to simplify:
$2N = 2300 + 7E$
$N = 1150 + 3.5E$

Step 2: Test Values from the Table
We need to pick an $E$ from the list $\{200, 300, 700, 1500, 2100, 3600\}$ such that the resulting $N$ is also in the list.
Try $E = 200$:
$N = 1150 + 3.5(200) = 1150 + 700 = 1850$ (Not in list).
Try $E = 300$:
$N = 1150 + 3.5(300) = 1150 + 1050 = 2200$ (Not in list).
Try $E = 700$: $N = 1150 + 3.5(700) = 1150 + 2450 = 3600$.
Is 3600 in the list? YES.

Verification
Echo: $1600 + 4(700) = 4400$.
Nexus: $1000 + 4(3600) = 15400$.
Check ratio: $15400 / 4400 = 154 / 44 = 3.5$. Correct.
Answer Selection: Nexus: 3,600 Echo: 700

ghimires28

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17 Dec 2025, 09:57

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we are given that
3.5(1600+x)=1600+x
the number that satisfies this and are in the ration 7/2 is only 200 and 700.

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17 Dec 2025, 09:58

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Let monthly increase by x for N and y for E

1000 + 4x = 350% * (1600 + 4y)
1000 + 4x = 7/2 * 4 (400 + y)
1000 + 4x = 14y + 14*400
500 + 2x = 7y + 2800
2x = 7y + 2300
x = (7y/2) + 1150

If y = 700, x = 7*350 + 1150 = 2450 + 1150 = 3600

Answer 3600, 700

Bunuel

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17 Dec 2025, 10:04

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Quote:

Let a & b be the number of users gained per month for Nexus and Echo respectively. Clearly b < a

By May 1, Nexus had 250% = 2.5 more registered user => 1000 + a*4 = (1600 + b*4)*(1+2.5) = 3.5*(1600 + b*4)
=> 4a + 1000 = 5400 + 14b => 4a = 4400 + 14b => a = 1100 + 3.5b
Try b = 200 => a = 1850. Incorrect (option not on the list)
b = 300 => a = 2200. Incorrect (option not on the list)
b = 700 => a = 3600. Correct.
At this point, no need to test the case of b = 1500 or 2100 as it will give the value of a being higher than 3600, which is not on the list.
=> Echo = 700, Nexus = 3600

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17 Dec 2025, 10:08

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after 4 months
users Nexus = 1000 + 4n
users Echo = 1600 + 4e

250% more = (250+100)% = 350%

1000 + 4n = 3.5 (1600 + 4e)
-> n = 1150 + 3.5e
Only n=3600 and e=700 verify the condition

to make the choice easier exclude all options where e>n

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17 Dec 2025, 10:17

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By May 1, Nexus had 250% more users than Echo
=> (1,000+4N)/(1,600+4E)-1=2.5
=> E = ((1,000+4N)/3.5-1,600)/4
Use the calculator and scan the options (start with the largest N because N must be larger than E)

Answer: 3,600 and 700

Bunuel

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17 Dec 2025, 10:21

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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.

make table---N---E
Jan--------1000----1600
Feb ---3600---700
March--3600--700
Ap--3600--700
May--3600--700
total 15400---4400
total N is 3.5 times of E
plug in options at value 3600 users per month 3600*4 = 14,400 + 1000 ; 15400 users of Nexus
users of E become 15,400/3.5 = 44,00
Jan remove E subscribers = 4400-1600 ; 2800 ; 2800/4 ; 700 per month
OPTION NEXUS 3600, ECHO 700 is correct

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17 Dec 2025, 10:26

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We get an equation like this
1000 + 4n = (1 + 250%) (1600 + 4e), assuming n is the constant increase for Nexus and e is the constant increase for Echo.
Simplifying gives
2n - 7y = 2300

Testing options, n = 3600, gives e = 700.

Bunuel

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17 Dec 2025, 10:27

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	Registered users of Website Nexus	Registered users of Website Echo
January 1, 2025	1000	1600
May 1, 2025 (4 months later)	1000+4x	1600+4y

Where Website Nexus gained x users/month and Website Echo gained y users/month

(1000 + 4x) = 350% (1600 + 4y)
1000 + 4x = 3.5*1600 + 14y
4x = 4600 + 14y
x = 1150 + 3.5y

y = 200; x = 1850
y = 300; x = 2200
y = 700; x = 3600

Nexus	3600
Echo	700

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17 Dec 2025, 10:37

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Bunuel

1. Let's consider Nexus got x subscribers per month. By May 1 it will have 1000+4x subscribers.. similarly Echo will have 1600+4y subscribers

---> given 1000+4x=3.5(1600+4y) ---> 4x = 5600 -1000+ 14 y ---> 4x-14y=4600 ---> 2x-7y=2300

Trying 7y=2x-2300 -- > y=(2*x - 2300)/7 ---> let's try x = 1500, 2100, 3600 --- y = 100 (not give) , 1900/7 (not divisible ), 700 Answer (3600, 700)

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17 Dec 2025, 10:41

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Nexus has x user increase per month and Echo has y users increase per month.
On Jan 1, Nexus has 1000 users and Echo has 1600 users
From Jan 1 to May 1 is 4 months
On May 1 Nexus has 250% more user than Echo
Nexus = 3.5*Echo
1000+4x =3.5(1600+4y)
x=1150+3.5y
Check from options and try putting different values of y or echo users in above equation.
If y=700, then x=1150+3.5(700)=3,600
Both values are in options.

Nexus=3600
Echo=700

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17 Dec 2025, 10:48

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Bunuel

Let the monthly increase be: Nexus=x; Echo=y
Time period = 4 Months
Nexus = 1000+4x; Echo = 1600+4y
Nexus has 3.5 times (250%) more users than Echo
1000+4x=3.5(1600+4y)
1000+4x=5600+14y
2x=2300+7y
Hit & Trial (Using Options):
Trying Echo = 700
2x=2300+7(700)
x=3600
Both 700 and 3600 appear in the table.

Hence, Echo=700; Nexus=3600.

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17 Dec 2025, 10:53

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Let no. of new users per month for Nexus is x and for Echo is y.
On 1 May, 2025 -> total 4 months.
Total users for Nexus on 1 May will be 1000+4x and for Echo will be 1600+4y.
Given -> Nexus user on 1 May is 250% more than Echo. That also, means x>y because initial users for Nexus was lesser compared to Echo but on 1st May it is greater that Echo which is only possible when x>y.
So, 1000+4x = (1+250/100)*(1600+4Y)
=> 1000+4x = 3.5*(1600+4y)
=> 1000+4x = 5600+14y
=> 2x=7y+2300.
Now, check for each option for Echo (y) and calculate x.
For y = 700, x will be 3600.
Nexus (x) = 3600 & Echo (y) = 700

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17 Dec 2025, 11:01

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From the statement that Nexus had 250% more users than Echo we can say by May:
Nexus = Echo + 2.5 Echo.....1

Mentioned that the growth was constant no of user for each.
Growth rates of Nexus and Echo to be Gn and Ge respectively:
Growth of Nexus over a time of say 5 months T = Gn x T
Growth of Echo over a time of say 5 months T = Ge x T

From 1: Gn = 3.5 x (Ge x T)
Now use the values given in the options:
If : Ge = 200 : Gn = 700 (Bingo!!!)
Luckily found it in the first try so ans :
Nexus: 700 and Echo: 200.

Bunuel

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17 Dec 2025, 11:07

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3600 nexus and 700 echo

Bunuel