It takes five minutes for one person to pass a rumour to two other persons. Each newly informed person then continues the process in the same way, so that every informed person passes the rumour on to two more people in the next five minutes. How many minutes will it take for the rumour to reach 729 persons in total?

Introduction / Context:
This arithmetic reasoning question models the spread of a rumour using a branching pattern. It is effectively a problem of geometric progression. The rumour starts with one person, and in fixed time intervals, each informed person passes the rumour to two new people. You must determine how many such five minute intervals are needed for the total number of people who know the rumour to reach 729.

Given Data / Assumptions:

• Initially, 1 person knows the rumour.
• Every five minutes, each informed person tells the rumour to 2 new people.
• Once a person knows the rumour, he or she continues to spread it to new people in the next step.
• The process repeats identically each five minute period.
• The target is a total of 729 informed persons.

Concept / Approach:
At each stage, the number of people who know the rumour multiplies by 3. This happens because each current knower remains informed and also informs two new people. Thus, if the total number of informed people after n steps is Tn, we get Tn = 3^n, assuming we start counting steps after the first five minute period. The problem becomes finding n such that 3^n = 729 and then multiplying n by 5 minutes to get the total time.

Step-by-Step Solution:
Step 1: After the first 5 minutes, the original person tells 2 people, so the total who know the rumour is 3 (1 original + 2 new). Step 2: After the second 5 minute period, these 3 people each tell 2 new people, adding 6 more, so the total becomes 9. This is 3^2. Step 3: Similarly, after the third 5 minute period, the total becomes 27 (3^3); after the fourth, 81 (3^4); after the fifth, 243 (3^5). Step 4: After the sixth 5 minute period, the total becomes 729 (3^6). Step 5: Therefore, it takes 6 such periods to reach 729 people. Each period is 5 minutes long. Step 6: Total time = 6 * 5 minutes = 30 minutes.

Verification / Alternative check:
Check the powers: 3^1 = 3, 3^2 = 9, 3^3 = 27, 3^4 = 81, 3^5 = 243, and 3^6 = 729. This confirms that after six rounds the total informed population is exactly 729. No smaller exponent of 3 gives 729, so the number of steps cannot be fewer than six.

Why Other Options Are Wrong:
Option B 25 minutes and option D 36 minutes correspond to non integer multiples of 5 or to wrong numbers of steps. Option A 45 minutes would correspond to nine steps (9 * 5), which would give 3^9 = 19,683 informed people, far more than required. Only 30 minutes matches the step count obtained from 3^6.

Common Pitfalls:
Some learners mistakenly assume that the number of informed people doubles instead of tripling, ignoring that each knower remains informed while also informing two more people. Others miscount the number of rounds needed to reach 729, often stopping at 243 or 81. Carefully tracking the total after each step and recognizing the geometric progression pattern 3^n helps avoid these mistakes.

Final Answer:
It takes 30 minutes for the rumour to reach 729 persons.