Four persons A, B, C and D have present ages whose sum is 176 years. Four years ago, the ratio of their ages was 11 : 9 : 6 : 14 respectively. Who is the eldest and who is the youngest among them at present?

Introduction / Context:
This question involves four people with a known sum of present ages and a given ratio of ages at a time in the past. The task is to determine which person is currently the eldest and which is the youngest. The problem tests the ability to move between past and present ages using ratios and simple algebra.

Given Data / Assumptions:

• The sum of the present ages of A, B, C and D is 176 years.
• Four years ago, the ratio of their ages was 11 : 9 : 6 : 14 for A : B : C : D.
• We need to identify who is eldest and who is youngest at present.
• All ages are in years and positive.

Concept / Approach:
We represent their ages four years ago as 11 * k, 9 * k, 6 * k and 14 * k. From those we obtain their present ages by adding 4 years to each. The total sum of present ages is given, allowing us to solve for k. Once we know k, we can compute each present age and easily compare them to determine the eldest and youngest.

Step-by-Step Solution:
Let the ages four years ago be:A = 11 * k, B = 9 * k, C = 6 * k, D = 14 * k.Present ages will then be:A now = 11 * k + 4.B now = 9 * k + 4.C now = 6 * k + 4.D now = 14 * k + 4.Given that the sum of present ages is 176 years, we write:(11 * k + 4) + (9 * k + 4) + (6 * k + 4) + (14 * k + 4) = 176.Combine like terms: (11 + 9 + 6 + 14) * k + 4 * 4 = 176.This is (40 * k) + 16 = 176.Subtract 16 from both sides: 40 * k = 160.So k = 160 / 40 = 4.Now compute present ages:A now = 11 * 4 + 4 = 44 + 4 = 48 years.B now = 9 * 4 + 4 = 36 + 4 = 40 years.C now = 6 * 4 + 4 = 24 + 4 = 28 years.D now = 14 * 4 + 4 = 56 + 4 = 60 years.Comparing these, D is the eldest at 60 years, and C is the youngest at 28 years.

Verification / Alternative check:
Check the sum: 48 + 40 + 28 + 60 = 176 years, matching the given total.Check the ages four years ago: 44, 36, 24 and 56.Their ratio four years ago is 44 : 36 : 24 : 56. Dividing each by 4 gives 11 : 9 : 6 : 14, confirming the given ratio.

Why Other Options Are Wrong:
A is eldest and C is youngest: A is not the eldest, as D is older.C is eldest and A is youngest: This is the opposite of the actual ordering and does not match the calculated ages.A is eldest and B is youngest: Neither A nor B holds those extreme positions according to the computed ages.

Common Pitfalls:
Some students mistakenly treat the given ratio as present ages rather than ages four years ago.Others forget to add 4 years to each age when converting from past to present.Algebraic errors when summing the ages and solving for k can lead to incorrect age values.

Final Answer:
The correct statement is that D is eldest and C is youngest.