The ages of Hari and Shyam are in the ratio 5: 7. After four years the ratio of their ages will be 3: 4. Find the present age of Hari :

Question

Accepted Answer

To determine the present age of Hari, we need to analyze the given information and verify why Option 1 (20 Years) is the correct answer.

### Explanation:

1. **Understanding the Ratio:**
   - The ages of Hari and Shyam are initially in the ratio 5:7. Let's assume their ages are 5x and 7x, respectively.

2. **Future Age Calculation:**
   - After four years, the ages of Hari and Shyam will be (5x + 4) and (7x + 4), respectively.
   - The problem states that after four years, the ratio of their ages will be 3:4.

3. **Setting Up the Equation:**
   - According to the future ratio, we have:
     $$
     \frac{5x + 4}{7x + 4} = \frac{3}{4}
     $$

4. **Solving the Equation:**
   - Cross-multiply to solve for x:
     $$
     4(5x + 4) = 3(7x + 4)
     $$
     $$
     20x + 16 = 21x + 12
     $$
   - Simplify the equation:
     $$
     20x + 16 = 21x + 12
     $$
     $$
     16 - 12 = 21x - 20x
     $$
     $$
     4 = x
     $$

5. **Calculating Hari's Present Age:**
   - Substitute x = 4 back into the expression for Hari's age:
     $$
     5x = 5 \times 4 = 20
     $$
   - Therefore, Hari's present age is 20 years.

### Justification for Option 1:

- **Option 1 (20 Years):** As calculated, when x = 4, Hari's age is 20 years, which matches the given correct answer.

### Explanation for Incorrect Options:

- **Option 2 (28 Years):** If Hari's age were 28 years, then 5x = 28, which implies x = 5.6. This value of x does not satisfy the future age ratio condition when substituted back into the equation.

- **Option 3 (30 Years):** If Hari's age were 30 years, then 5x = 30, which implies x = 6. This value of x also does not satisfy the future age ratio condition when substituted back into the equation.

- **Option 4 (33 Years):** If Hari's age were 33 years, then 5x = 33, which implies x = 6.6. This value of x does not satisfy the future age ratio condition when substituted back into the equation.

In conclusion, the only value of x that satisfies both the initial and future age ratio conditions is x = 4, making Hari's present age 20 years. Therefore, Option 1 is the correct answer.