The ratio of the present ages of Simi and Seema is 5 : 4. After 9 years, the ratio of their ages will be 8 : 7. What is Simi's present age (in years)?

Introduction / Context:
This is a typical age-ratio problem involving two people, Simi and Seema. We know their present age ratio and the ratio that their ages will have after 9 years. From these two statements, we must determine Simi's present age. The problem tests your skill in forming and solving linear equations from ratio information.

Given Data / Assumptions:

- Present age ratio of Simi : Seema = 5 : 4.
- After 9 years, the ratio of their ages will be 8 : 7.
- All ages are in positive whole years.

Concept / Approach:
We represent their present ages as multiples of a common variable using the current ratio. Then, we add 9 years to each to obtain their ages after 9 years and set up a new ratio equation based on the future ratio. Solving this algebraic equation gives the common factor and thus the actual ages. This approach is standard for problems involving time-shifted ratios.

Step-by-Step Solution:
Step 1: Let the present ages of Simi and Seema be 5x years and 4x years respectively. Step 2: After 9 years, Simi's age will be 5x + 9 years. Step 3: After 9 years, Seema's age will be 4x + 9 years. Step 4: The ratio of their ages after 9 years is given as (5x + 9) : (4x + 9) = 8 : 7. Step 5: Form the equation (5x + 9) / (4x + 9) = 8 / 7. Step 6: Cross-multiply: 7(5x + 9) = 8(4x + 9). Step 7: Expand both sides: 35x + 63 = 32x + 72. Step 8: Rearrange: 35x − 32x = 72 − 63 ⇒ 3x = 9 ⇒ x = 3. Step 9: Simi's present age = 5x = 5 × 3 = 15 years.

Verification / Alternative check:
With x = 3, Seema's present age is 4 × 3 = 12 years. After 9 years, Simi will be 15 + 9 = 24 years and Seema will be 12 + 9 = 21 years. The ratio 24 : 21 simplifies to 8 : 7 when both numbers are divided by 3, which matches the given future ratio. Therefore, the calculated present age of Simi, 15 years, is consistent with all data.

Why Other Options Are Wrong:
Values such as 12, 18, 21 or 24 years for Simi would produce different present ages for Seema and would not yield the correct ratio of 8 : 7 after 9 years. Only 15 years, along with Seema's 12 years, satisfies both the present and future ratio conditions.

Common Pitfalls:
Students may forget to add the full 9 years to both ages when forming the future ratio, or they might incorrectly assign the ratio 5 : 4 to Seema : Simi instead of Simi : Seema. Another common error is mishandling the cross-multiplication. Writing each step clearly and keeping track of who is older helps avoid these mistakes.

Final Answer:
Simi's present age is 15 years.