The present age of a man is three times the total of the present ages of his two daughters. Five years from now, his age will be twice the total of the ages of his daughters at that time. What is the current age of the father?

Introduction / Context:
This is a standard age puzzle involving a parent and two children. The father age is related to the combined ages of his two daughters at two different points in time: now and five years later. The problem tests understanding of linear equations and relationships involving sums of ages rather than individual ages.

Given Data / Assumptions:

• The present age of the father is three times the sum of the present ages of his two daughters.
• After 0.5 decades, that is after 5 years, the father age will be twice the sum of the daughters ages at that time.
• We need to find the present age of the father.
• All ages are in years and positive.

Concept / Approach:
Instead of tracking each daughter separately, we can work with the combined age of the daughters. This simplifies the algebra. We represent the current sum of daughters ages by a single variable and then express the father age in terms of this variable. Using the future condition, we set up a second equation and solve for the daughters total and then the father age.

Step-by-Step Solution:
Let the present total age of the two daughters be D years.Then the present age of the father is 3 * D years, since he is three times their combined age.After 5 years, each daughter will be 5 years older, so the total age of the daughters will be D + 10 years.After 5 years, the father age will be 3 * D + 5 years.The problem states that after 5 years, the father age will be twice the total age of the daughters at that time.So we write the equation: 3 * D + 5 = 2 * (D + 10).Expand the right side: 3 * D + 5 = 2 * D + 20.Rearrange: 3 * D - 2 * D = 20 - 5.So D = 15.Therefore the father present age is 3 * D = 3 * 15 = 45 years.

Verification / Alternative check:
Present ages: father = 45 years, total daughters ages = 15 years.After 5 years: father = 45 + 5 = 50 years, daughters total = 15 + 10 = 25 years.Check the future condition: 50 is indeed twice 25.This confirms that the father present age of 45 years is correct.

Why Other Options Are Wrong:
35 years: With this age, it is not possible to satisfy both the three times and twice conditions using positive daughter ages.40 years: Fails the future condition when we compute the daughters total age.47 years: This does not fit neatly into the required multiplicative relationships with an integer daughters sum.

Common Pitfalls:
Some students mistakenly treat three times the age of each daughter separately rather than the sum.Others forget that both daughters grow older, and therefore the daughters total increases by 10, not 5, after 5 years.Carelessness in forming the equation 3 * D + 5 = 2 * (D + 10) can lead to wrong algebraic rearrangements.

Final Answer:
The present age of the father is 45 years.