Solving the Family Professions and Income Puzzle

The question provides us with information about seven family members, their professions, gender distribution, and relative monthly incomes. We need to use the given clues to determine who earns the least among them.

Understanding the Members and Constraints

We have seven members: P, Q, R, S, T, V, and W. There are three female members in total. Each person has a unique profession from the list: Lawyer, Chartered Accountant (CA), Engineer, Teacher, Doctor, Architect, and Pharmacist.

Key constraints are:

• No Lady is either a Pharmacist or a Chartered Accountant.
• The Chartered Accountant earns the most.
• S, the Engineer, earns less than V, the Doctor ($\text{S} < \text{V}$).
• R, the Teacher, earns more than P and less than S ($\text{P} < \text{R} < \text{S}$).
• W’s wife earns the least.
• T is an unmarried lady Lawyer and earns less than P and more than only Q ($\text{Q} < \text{T} < \text{P}$).
• The Pharmacist’s income is not the lowest.

Deducing the Income Order

Let's combine the income clues:

• From R < S and P < R, we have $\text{P} < \text{R} < \text{S}$.
• From S < V, we extend this to $\text{P} < \text{R} < \text{S} < \text{V}$.
• From Q < T < P, we can combine the inequalities: $\text{Q} < \text{T} < \text{P} < \text{R} < \text{S} < \text{V}$.

This gives us the relative income order for six members. The seventh member is W, and we know the CA earns the most. Since V is the Doctor, V is not the CA. Thus, W must be the CA, earning the most. The complete income order is:

$\text{Q} < \text{T} < \text{P} < \text{R} < \text{S} < \text{V} < \text{W}$

Identifying Professions

We know the professions for several members:

• T is the Lawyer.
• S is the Engineer.
• R is the Teacher.
• V is the Doctor.
• W is the CA.

The remaining members are P and Q. The remaining professions are Architect and Pharmacist. From the income order, Q earns the least, and P earns more than Q. We are told the Pharmacist's income is not the lowest. Since Q's income is the lowest, Q cannot be the Pharmacist. Therefore, Q must be the Architect, and P must be the Pharmacist.

Determining Genders

We need to find the genders. We know there are three female members.

• T is a lady Lawyer (Female).
• No Lady is Pharmacist (P is the Pharmacist, so P is Male).
• No Lady is CA (W is the CA, so W is Male).
• W's wife earns the least. Q earns the least, so Q is W's wife (Female).

So far, we have identified three females (T, Q) and two males (P, W). This means we need one more female and one more male from the remaining members R, S, and V. However, knowing the genders of R, S, and V is not necessary to answer the specific question about who earns the least.

Conclusion: Who Earns the Least?

Based on the derived income order $\text{Q} < \text{T} < \text{P} < \text{R} < \text{S} < \text{V} < \text{W}$, it is clear that Q earns the least among all the family members.