The ratio of present age of Ramesh and his daughter is 8 : 5, after 8 years ratio of their ages will be 3 : 2, then the present age of the daughter is :

Question

Accepted Answer

To determine the present age of Ramesh's daughter, we start by using the information given in the problem:

1. **Present Age Ratio**: The ratio of the present ages of Ramesh and his daughter is 8:5. Let's assume the present age of Ramesh is 8x years and the present age of his daughter is 5x years.

2. **Future Age Ratio**: After 8 years, the ratio of their ages will be 3:2. Therefore, in 8 years, Ramesh's age will be (8x + 8) years and his daughter's age will be (5x + 8) years.

According to the problem, the ratio of their ages after 8 years is 3:2. Therefore, we can set up the equation:
   $$
   \frac{8x + 8}{5x + 8} = \frac{3}{2}
   $$

3. **Solving the Equation**: Cross-multiply to solve for x:
   $$
   2(8x + 8) = 3(5x + 8)
   $$
   $$
   16x + 16 = 15x + 24
   $$
   $$
   16x - 15x = 24 - 16
   $$
   $$
   x = 8
   $$

4. **Calculating Present Age of Daughter**: Now that we have x = 8, we can find the present age of Ramesh's daughter:
   $$
   \text{Present age of daughter} = 5x = 5 \times 8 = 40 \text{ years}
   $$

Thus, the present age of Ramesh's daughter is 40 years, which corresponds to Option 1.

**Explanation for Incorrect Options**:

- **Option 2 (64 Years)**: If the daughter's age were 64 years, the ratio of their ages would not satisfy the given conditions after 8 years. The calculations would not hold true for the future age ratio of 3:2.

- **Option 3 (56 Years)**: Similarly, if the daughter's age were 56 years, the future age ratio would not be 3:2 as required by the problem statement.

- **Option 4 (Insufficient data, cannot be determined)**: The data provided is sufficient to determine the present age of the daughter. The problem gives clear ratios and a future condition that allows us to solve for the exact ages.

Therefore, the correct answer is indeed Option 1: 40 Years.