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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 09:00
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At the start of Potions class, Hermione and Ron are each given a magical cauldron. Hermione’s cauldron contains 27 magic beans, and Ron’s contains 729. At the end of every 6 hours, the number of beans in Hermione’s cauldron instantly increases by 200 percent, and at the end of every 18 hours, the number of beans in Ron’s cauldron instantly increases by 800 percent. If no beans are taken from either cauldron, after how many hours from the start of the class will the two cauldrons first contain the same number of beans?

A. 18
B. 27
C. 30
D. 36
E. 54

M46-21

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C
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 10:00
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Bunuel
At the start of Potions class, Hermione and Ron are each given a magical cauldron. Hermione’s cauldron contains 27 magic beans, and Ron’s contains 729. At the end of every 6 hours, the number of beans in Hermione’s cauldron instantly increases by 200 percent, and at the end of every 18 hours, the number of beans in Ron’s cauldron instantly increases by 800 percent. If no beans are taken from either cauldron, after how many hours from the start of the class will the two cauldrons first contain the same number of beans?

A. 18
B. 27
C. 30
D. 36
E. 54
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GMAT Club Official Explanation:



Hermione's cauldron starts with 27 beans, which is 3^3. Every 6 hours, Hermione's cauldron increases by 200%, meaning it multiplies by 3. So:
• After 6 hours the number of beans = 3^3 * 3 = 3^4
• After 12 hours the number of beans = 3^4 * 3 = 3^5
• After 18 hours the number of beans = 3^5 * 3 = 3^6
• After 24 hours the number of beans = 3^6 * 3 = 3^7
• After 30 hours the number of beans = 3^7 * 3 = 3^8
• ...

Ron’s cauldron starts with 729 beans, which is 3^6. Every 18 hours, Ron’s cauldron increases by 800%, meaning it multiplies by 9 or 3^2. So:
• After 18 hours the number of beans = 3^6 * 3^2 = 3^8
• After 36 hours the number of beans = 3^8 * 3^2 = 3^10
• ...

So, after 30 hours, the two cauldrons will, for the first time, contain the same number of beans: 3^8. Hermione’s beans reach 3^8 after 30 hours, while Ron’s beans remain at 3^8 after increasing at 18 hours.

Answer: C.
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 09:15
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At the start of Potions class, Hermione and Ron are each given a magical cauldron. Hermione’s cauldron contains 27 magic beans, and Ron’s contains 729. At the end of every 6 hours, the number of beans in Hermione’s cauldron instantly increases by 200 percent, and at the end of every 18 hours, the number of beans in Ron’s cauldron instantly increases by 800 percent. If no beans are taken from either cauldron, after how many hours from the start of the class will the two cauldrons first contain the same number of beans?

A. 18
B. 27
C. 30
D. 36
E. 54

start of class
H has 27 beans and R has 729
200% @6 hours and 800% 18 hours
200% is 3 times and 800 % is 9 times
noting down time wise increase
0 to 6 for H = 27 ; R 729
6 to 12 H = 27*3 :81 ; R = 729
12 to 18 ; H = 81*3 : 243 ; R = 729
18 to 24 ; H = 243*3 ;729 R = 729*9 ; 6561
24 to 30 H = 729*3 : 2187 R = 6561
30 to 36 H = 2187*3 6561 ; R = 6561
at 30 hours both R & H have same count of beans
OPTION C ; 30
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 09:24
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Hermione cauldron has 27 beans
Every 6 hours, her beans increase by 200% or by 3 times
After 6 hours, 27*3=81
After 12 hours, 81*3=243
After 18 hours, 243*3=729
After 24 hours, 729*3=2,187
After 30 hours, 2187*3=6,561 ..........
After 36 hours, 6561*3=19,683

Ron has 729 beans
Every 18 hours, his beans increase by 800% or by 9 times
After 6 & 12 hours, 729 (no change happens)
After 18 hours, 729*9=6,561
After 24 hours, 6561
After 30 hours, 6561 ..........
After 36 hours, 6561*9=59,049

At 30 hour mark, both have exactly same beans for the first time, 6561.

C
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 09:31
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Bunuel
At the start of Potions class, Hermione and Ron are each given a magical cauldron. Hermione’s cauldron contains 27 magic beans, and Ron’s contains 729. At the end of every 6 hours, the number of beans in Hermione’s cauldron instantly increases by 200 percent, and at the end of every 18 hours, the number of beans in Ron’s cauldron instantly increases by 800 percent. If no beans are taken from either cauldron, after how many hours from the start of the class will the two cauldrons first contain the same number of beans?

A. 18
B. 27
C. 30
D. 36
E. 54

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let h be hour after beans are same.
solve for the eqn
27 * 3 ^ (h/6) = 729 * 9 ^ (h/18)
= 3 ^ (h/6) = 27 * 3 ^ (h/9)
h/6 = 3 + h/9
solving eqn we get h = 54
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 09:32
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R's beans = 729, becomes 9 times every 18 hours
H's beans = 27, become 3 times every 6 hours OR 3x3x3 = 27 times every 18 hours

In the first 18 hours,
H's beans = 27 x 27
R's beans = 9 x 729 OR 27 x 27 x 9

After 36 hours
H's beans = 27 x 27 x 27
R's beans = 27 x 27 x 9 x 9 OR 27 x 27 x 27 x 3

After 54 hours
H's beans = 27 x 27 x 27 x 27
R's beans = 27 x 27 x 27 x 3 x 9 which is equal to H's beans

Hence, Answer Option E
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 09:39
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Initial stage:
Hermione's cauldron contains number of magic beans = 27
Ron's cauldron contains number of magic beans = 729

Every 6 hours, the number of magic beans in Hermione's cauldron becomes = 3 times (100%+200% = 300%)
Every 12 hours, the number of magic beans in Hermione's cauldron becomes = 3*3 = 9 times
Every 18 hours, the number of magic beans in Hermione's cauldron become = 3*3*3 = 27 times

Every 18 hours, the number of magic beans in Ron's cauldron becomes = 9 times (100% + 800%)

After 18 hours : -
The number of magic beans in Hermione's cauldron = 27*27 = 3^6
The number of magic beans in Ron's cauldron = 729*9 = 3^8

After 18+6=24 hours: -
The number of magic beans in Hermione's cauldron = 3^6*3 = 3^7

After 24+6=18+6*2=30 hours:-
The number of magic beans in Hermione's cauldron = 3^7*3 = 3^8

After 30 hours, the number of magic beans in the two cauldrons contain the same number of beans.

IMO C
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 09:43
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HR
Initial27 = 3^3729 = 3^6
Increase Rate 39 = 3^2
Duration618

After 18hrs
H = 3^3 * 3^3 = 3^6
R = 3^6 * 9 = 3^8

After 27hrs
H = 27 * 3^4 = 3^7. (It has increased 4 times so far 27/6 < 5)
R = 3^6 * 3^2 = 3^8 (It has increased only once, 27/18 < 2)

After 30hrs
H = 3^3 * 3^5 = 3^8 (it has increased 5 times 30/5 = 5)
R = 3^6 * 3^2 = 3^8 (it has increased only once, 30/18 < 2)

Answer C
Bunuel
At the start of Potions class, Hermione and Ron are each given a magical cauldron. Hermione’s cauldron contains 27 magic beans, and Ron’s contains 729. At the end of every 6 hours, the number of beans in Hermione’s cauldron instantly increases by 200 percent, and at the end of every 18 hours, the number of beans in Ron’s cauldron instantly increases by 800 percent. If no beans are taken from either cauldron, after how many hours from the start of the class will the two cauldrons first contain the same number of beans?

A. 18
B. 27
C. 30
D. 36
E. 54

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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 09:50
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Hermione=H STARTS AT 27 INCREASE BY 200% PER 6 HRS
Ron=R STARTS AT 729 INCREASES 800% PER 18 HRS


hrs H R
6 81(27 INCREASED BY 200%) 729
12 243( INCREASED BY 200% OR MULTIPLIED BY 3) 729
18 729 6561
24 2187 6561
30 6561

30 ANSWER

Bunuel
At the start of Potions class, Hermione and Ron are each given a magical cauldron. Hermione’s cauldron contains 27 magic beans, and Ron’s contains 729. At the end of every 6 hours, the number of beans in Hermione’s cauldron instantly increases by 200 percent, and at the end of every 18 hours, the number of beans in Ron’s cauldron instantly increases by 800 percent. If no beans are taken from either cauldron, after how many hours from the start of the class will the two cauldrons first contain the same number of beans?

A. 18
B. 27
C. 30
D. 36
E. 54

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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 09:54
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Hermoine (Let's call this H) = 27 beans = 3^3 beans
Ron (Let's called this R) = 729 beans = 81 * 9 = 9^3 = 3^6 beans

H increases by 200% every 6 hours.
So, after the 1st 6 hours, H = 2*27 + 27 = (2+1)*27 = 3*27
after the 2nd 6 hours, H = 2*(3*27) + (3*27) = (2+1)*3*27 = 3*3*27
For n intervals of 6 hours, H = (3^n) * 27 = (3^n) * (3^3) = 3^(n+3)

Similarly, for m intervals of 18 hours ---
1st interval, R = 8*729 + 729 = (8+1)*729 = 9*729
m intervals, R = (9^m) * 729 = (3^(2m)) * 3^6 = 3^(2m+6)

For H and R to be equal, we need to equate the powers.
n+3 = 2m+6
or, n = 2m+3

For m = 1, n is = 5
Therefore, total number of hours = n intervals of 6 hours = 5 * 6 = 30.
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 09:56
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Bunuel
At the start of Potions class, Hermione and Ron are each given a magical cauldron. Hermione’s cauldron contains 27 magic beans, and Ron’s contains 729. At the end of every 6 hours, the number of beans in Hermione’s cauldron instantly increases by 200 percent, and at the end of every 18 hours, the number of beans in Ron’s cauldron instantly increases by 800 percent. If no beans are taken from either cauldron, after how many hours from the start of the class will the two cauldrons first contain the same number of beans?

A. 18
B. 27
C. 30
D. 36
E. 54

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Hermione: starts with 27 beans
"Number increases by 200% means it becomes 27 + 2.27 = 3.27. It triples every 6 hours
so after n periods of 6hrs (time t = 6n)
H(t) = 27.3^n

Ron: starts with 729 beans, increases by 800% every 18hrs
R(t) = 729 + 8.729 = 729.9^m

for them to have same number of beans, 6n = 18m => n = 3m
27.3^3m = 729.9^m
27.3^3m => 3^(3m+3)
729.9^m => 3^(2m+6)

3m+3 = 2m+6
m=3
t=18m = 18.3 = 54

Option E
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 10:02
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54 is the answer
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 10:16
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C - 30 hours.

This problem requires good translation skills in order to solve. First, recognize that "increases by 200%" is equivalent to "multiply by 3," and similarly, "increases by 800%" is equivalent to "multiply by 9."

Then we discover that 27 = 3^3 and 729 = 3^6.

I created a chart and listed out the increase in every six hour interval.
Since the bases are the same, we only need to compare the rate at which the exponents are changing to find the answer. Hermione's exponent increased by one every 6 hours, and Ron's increases by 2 every 18 hours.

So after 30 hours, Hermione's beans will have increased to 3^8, and since the change happens instantenously, Ron's beans will still be at 3^8.

Therefore, the answer is 30.
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 10:17
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R has 729 Beans and H has 27 Beans. Now Let us assume T hours is taken from the start by both of them. We Know, R takes 18 hrs to get 800% increase while H takes 6hrs to get 200% increase.
Hence, Time interval for R= T/18 and H=T/6.
Now, 800% increase = 9 times and 200% increase= 3 times.
So, 27*3_power_(T/6)=729*9_power_(T/18)
3_power_(T/6)=27*3_power_(2T/18)
3_power_(T/6)=3_power_(3+T/9)
T/6=3+T/9
hence, T=54 Hrs taken from the start.
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 10:28
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Please check solution as explained in image.
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 10:34
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hermoine cauldron increases by 3 times after 6 hrs whereas ron cauldron increases by 9 times after 18 hrs

so at the end of 30 hrs from start both will have 6561 cauldrons

answer=30 hrs
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 10:36
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For Hermione at t= 0 its 3^3 -- increases at the end of 6 hrs by 200%
and for Ron at t=0 its 3^6 --- increases at the end of 18 hrs by 800%


so after t = 18hrs
Hermione will be at 3^3(1+200/100)^3 = 3^6
and ron will have increase as 3^6(1+8)^1 = 3^8 and will stay till t=36

after t = 30hrs
Hermione will reach at 3^8 as it is increasing by 3 at every 6 hrs.

So answer will be 30hrs (C)
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 10:56
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at start 18 hrs 36 hrs 54 hrs
H3^33^63^93^12
R3^63^83^103^12
since given
H--27=3^3 which increases by 200% or becomes 3 times every 6 hours or 3^3 times every 18 hrs.
R--729=3^6 which increases by 800% or becomes 9 times every 18 hrs i.e. 3^2 times every 18 hrs.
hence E
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 11:08
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Ron after 18 hrs have 9*729
Hermoine after n times increase \(27*3^n\)

\(27*3^n = 9*729\)
n=5
total 6*5 = 30 hrs.

Ans 30
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12 Days of Christmas GMAT Competition 2025 - Day 7: At the start of 23 Dec 2025, 11:08
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Option C: 30
For H =27*3*3*3*3*3=6561 i.e. 30 hours

For R= 729*9=6561
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