Solving Age Ratio Problems: A and B's Ages

This problem involves finding the current ages of two individuals, A and B, based on their current age ratio and their age ratio after a specific number of years. We need to calculate the sum of the ages of A and B.

Step-by-Step Solution

1. Representing Current Ages

The problem states that the current ages of A and B are in the ratio 4:7. We can represent their current ages using a common multiplier, let's say '$x$'.

• Age of A = $4x$ years
• Age of B = $7x$ years

2. Representing Future Ages

The problem also mentions that after 8 years, the ratio of their ages will be 2:3. Let's calculate their ages after 8 years:

• A's age after 8 years = (Current age of A) + 8 = $4x + 8$ years
• B's age after 8 years = (Current age of B) + 8 = $7x + 8$ years

3. Setting Up the Equation Based on Future Ratio

We are given that the ratio of their ages after 8 years is 2:3. We can set up an equation using the future ages:

$$ \frac{\text{Age of A after 8 years}}{\text{Age of B after 8 years}} = \frac{2}{3} $$

Substituting the expressions for their future ages:

$$ \frac{4x + 8}{7x + 8} = \frac{2}{3} $$

4. Solving for the Multiplier 'x'

To find the value of '$x$', we cross-multiply:

$$ 3 \times (4x + 8) = 2 \times (7x + 8) $$

Distribute the numbers:

$$ 12x + 24 = 14x + 16 $$

Now, rearrange the equation to solve for '$x$'. Move the '$x$' terms to one side and the constant terms to the other:

$$ 24 - 16 = 14x - 12x $$

$$ 8 = 2x $$

Divide by 2:

$$ x = \frac{8}{2} $$

$$ x = 4 $$

5. Calculating Current Ages

Now that we have the value of '$x$', we can find the current ages of A and B:

• Current Age of A = $4x = 4 \times 4 = 16$ years
• Current Age of B = $7x = 7 \times 4 = 28$ years

6. Calculating the Sum of Ages

The question asks for the sum of the ages of A and B. Add their current ages:

Sum of Ages = (Current Age of A) + (Current Age of B)

Sum of Ages = $16 + 28$

Sum of Ages = $44$ years

Final Answer

The sum of the current ages of A and B is 44 years.