This problem involves determining the age ranking of seven family members (A, B, C, D, E, F, G) based on given clues. The members have different ages, arranged from highest to youngest, with a consistent age difference of 3 years between consecutive persons.

Determining the Age Sequence

We are told that the age difference between consecutively arranged persons is 3 years. There are 7 members in the family. The age of the youngest person is given as 46.

Since the ages are in an arithmetic progression with a common difference of 3, and the youngest is 46, we can list the ages from youngest to oldest:

• Youngest (1st): 46
• 2nd: $46 + 3 = 49$
• 3rd: $49 + 3 = 52$
• 4th: $52 + 3 = 55$
• 5th: $55 + 3 = 58$
• 6th: $58 + 3 = 61$
• Oldest (7th): $61 + 3 = 64$

Arranging these ages from highest to youngest (as per the problem statement):

Applying Clues to Identify Members' Ages

Now, let's use the specific clues provided to identify the members' positions:

• The age of B is 61. From our sequence, the 2nd oldest person is 61. Therefore, B is the 2nd oldest.
• B is older than A ($B > A$). Since B is 61, A must be younger than 61. This means A's possible ages are 58, 55, 52, 49, or 46.
• A's age is not 55. This eliminates 55 as a possibility for A. So A's possible ages are 58, 52, 49, or 46.
• A is older than E ($A > E$) and A is older than D ($A > D$). This means A must be older than at least two other people.
• E is younger than D ($D > E$).
• D's age is not 55.
• G's age is less than 52 ($G < 52$). This means G can be 49 or 46.

Deducing A's Position

Let's test the possible ages for A:

• If A is 58: This is the 3rd oldest position.
  • Check constraints: $B > A$ (61 > 58) is true. A is not 55 (A is 58) is true.
  • A (58) must be older than D and E. So D and E must be from the ages {55, 52, 49, 46}.
  • D's age is not 55. So D can be from {52, 49, 46}.
  • We also know $D > E$. If D = 52, then E can be 49 or 46. If D = 49, E must be 46.
  • Also, G < 52, so G could be 49 or 46.
  • Let's assign: A = 58. If D = 52, and E = 46, G = 49. This combination satisfies all conditions: $A>D$ (58>52), $A>E$ (58>46), $D>E$ (52>46), $D \neq 55$ (52 $\neq$ 55), $G < 52$ (49 < 52). This scenario works.
• If A is 52: This is the 5th oldest position.
  • Check constraints: $B > A$ (61 > 52) is true. A is not 55 (A is 52) is true.
  • A (52) must be older than D and E. So D and E must be from the ages {49, 46}.
  • Since $D > E$, D must be 49 and E must be 46.
  • This means the ages 49 and 46 are taken by D and E.
  • Now check the constraint $G < 52$. The only ages younger than 52 are 49 and 46. Since these are already assigned to D and E, G cannot be assigned an age, which contradicts the fact that there are 7 members with distinct ages. Thus, A cannot be 52.
• If A is 49 or 46: If A is 49, D and E must be younger than 49. The only age younger than 49 is 46. This would mean D=46 and E would have no age, or both D and E would have to be 46, which is not possible as ages are distinct. Similarly, if A is 46, D and E must be younger, which is impossible.

Based on the analysis, the only valid age for A is 58.

Identifying the Third Oldest Member

The ages arranged from highest to youngest are:

1. 64
2. B (61)
3. A (58)
4. (Position 4, Age 55)
5. (Position 5, Age 52)
6. (Position 6, Age 49)
7. (Position 7, Age 46)

Since A's age is 58, and this corresponds to the 3rd position in the highest-to-youngest age ranking, A is the third oldest member among them.