It takes exactly 1 minute to fill 2/7 of a vessel with a liquid at a constant rate. Assuming the filling rate does not change, how much time in minutes is required to completely fill the whole vessel?

Introduction / Context:
This is a simple ratio based time and work question involving the filling of a vessel. We know the fraction of the vessel filled in a given time at a constant rate. The aim is to use proportional reasoning to determine the total time required to fill the entire vessel completely at the same rate.

Given Data / Assumptions:

• In 1 minute, 2/7 of the vessel is filled.
• The rate of filling is constant over time.
• The vessel starts empty.
• We want the time to fill 1 whole vessel (fraction 1).

Concept / Approach:
When rate is constant, the amount of work done is directly proportional to time. If a fraction of work is completed in a known time, we can scale the time up or down according to how many times larger the target fraction is. Here, 2/7 of the vessel corresponds to 1 minute. To get from 2/7 to the full vessel 1, we multiply by a factor that converts 2/7 into 1, and we apply the same factor to the time.

Step-by-Step Solution:
Fraction filled in 1 minute = 2 / 7.We want the time to fill 1, so we need a factor k such that (2 / 7) * k = 1.Solve for k: k = 1 / (2 / 7) = 7 / 2.Therefore, the required time = 1 minute * 7 / 2 = 7 / 2 minutes.Convert 7 / 2 minutes if desired: 7 / 2 = 3.5 minutes.

Verification / Alternative check:
You can verify by reversing the reasoning. If the whole vessel takes 7 / 2 minutes to fill, then the rate per minute is 1 / (7 / 2) = 2 / 7 of the vessel per minute, which matches the given condition. This symmetry confirms that our proportional scaling is correct and that 7 / 2 minutes is the correct time for the full vessel.

Why Other Options Are Wrong:
One seventh of a minute and five sevenths of a minute are both smaller than one minute, and therefore would correspond to less than 2 / 7 of the vessel, not the full amount. Seven fifths of a minute (which is 1.4 minutes) would also fill only (2 / 7) * 1.4 = 0.4 of the vessel. None of these match the requirement that all of the vessel be filled at the constant rate.

Common Pitfalls:
Many learners mistakenly multiply rather than divide by the given fraction, or they attempt to add extra steps instead of directly using proportional reasoning. Another common error is to assume that 1 minute corresponds to 1 / 7 of the vessel instead of 2 / 7, which leads to incorrect time calculations. Always base your ratio directly on the fraction explicitly provided in the question.

Final Answer:
The vessel will be completely filled in 7/2 minutes (that is, 3.5 minutes).