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12 Jan 2026, 10:58
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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
A. 18
B. 27
C. 30
D. 36
E. 54
12 Jan 2026, 10:58
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Official Solution:
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
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
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
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
09 Feb 2026, 01:59
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Bunuel
I think the wording of the question could be better.
The solution states that the answer is 30 hours, but this overlooks how the increases actually occur. Ron’s cauldron only increases once every 18 hours (at 18, 36, 54, etc.). At 30 hours, Ron’s number of beans has not changed. It is still the same amount he reached at 18 hours. So 30 hours is not a new moment when the two cauldrons first match. It is just a later time when Hermione happens to catch up to a number Ron already had.
To find when the cauldrons first contain the same number of beans, we must look at the times when the growth patterns truly line up.
27 × 3^3x = 729 × 9^x
This gives x = 3 and total time = 54 hours
Doing this correctly shows that the earliest time both cauldrons have the same number of beans is 54 hours, not 30.
I think 30 can also be a correct answer if the wording is tightened so there’s no ambiguity about “first". Maybe changing the question to - After how many hours from the start will Hermione’s cauldron first contain the same number of beans as Ron’s cauldron contains at that time?
