Let's break down the problem step by step:

Step 1: Define Variables for Ages

Let the present age of A be $a$, the present age of B be $b$, and the present age of C be $c$.

Step 2: Use the Given Ratios

1. Ratio of ages of A and B: $\frac{a}{b} = \frac{1}{3} \quad \Rightarrow \quad b = 3a$

2. Ratio of ages of A and C: $\frac{a}{c} = \frac{2}{3} \quad \Rightarrow \quad c = \frac{3}{2}a$

Step 3: Use the Given Difference Between Ages

The problem states that the difference between the present age of A and C is 2 years: $c - a = 2$ Substitute $c = \frac{3}{2}a$ from the second equation: $\frac{3}{2}a - a = 2$ Simplifying: $\frac{1}{2}a = 2 \quad \Rightarrow \quad a = 4$

Step 4: Find the Present Ages of B and C

Now that we know $a = 4$:

• The present age of B is: $b = 3a = 3 \times 4 = 12$

• The present age of C is: $c = \frac{3}{2}a = \frac{3}{2} \times 4 = 6$

Step 5: Calculate the Sum of Ages 14 Years Hence

In 14 years, the ages of A, B, and C will be:

• A's age in 14 years: $a + 14 = 4 + 14 = 18$
• B's age in 14 years: $b + 14 = 12 + 14 = 26$
• C's age in 14 years: $c + 14 = 6 + 14 = 20$

Now, the sum of their ages 14 years hence is: $18 + 26 + 20 = 64$

Final Answer:

The sum of the ages of A, B, and C fourteen years hence is 64.

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Would you like further details or have any questions on this? Here are 5 related questions to explore further:

1. How does the ratio of ages change if the difference between A and C is 5 years instead of 2?
2. What would be the ages of A, B, and C 20 years ago, given the same ratios?
3. How can we find the current ages of three people given multiple ratios and age differences?
4. If the ratio of B's age to C's age were 4:5, how would that change the solution?
5. What is the general approach for solving problems involving ratios of ages?

Tip: When working with age ratio problems, always translate the ratios into algebraic expressions first, then use the given age differences to solve for the unknowns.