The present ages of Ratnabali and Shaukat are in the ratio 8 : 5. After 22 years, the ratio of their ages will be 10 : 9. What is Ratnabali's present age?

Introduction / Context:
This problem involves present and future ratios of the ages of two people, Ratnabali and Shaukat. We are given the ratio of their present ages and the ratio of their ages after 22 years. Using these two pieces of information, we must determine Ratnabali's present age.

Given Data / Assumptions:

- Present age ratio of Ratnabali : Shaukat = 8 : 5.
- After 22 years, the ratio of their ages will be 10 : 9.
- All ages are in years and positive.

Concept / Approach:
When present ages are given in a ratio, we can represent them as multiples of a common variable. We then add the same number of years (22) to each age to reflect the future and set up a new ratio equation. Solving this equation for the variable gives us their actual present ages. This is a standard algebraic method for handling ratio-based age problems.

Step-by-Step Solution:
Step 1: Let the present ages of Ratnabali and Shaukat be 8x years and 5x years respectively. Step 2: After 22 years, Ratnabali's age will be 8x + 22 years and Shaukat's age will be 5x + 22 years. Step 3: After 22 years, the ratio of their ages is given as (8x + 22) : (5x + 22) = 10 : 9. Step 4: Form the equation (8x + 22) / (5x + 22) = 10 / 9. Step 5: Cross-multiply: 9(8x + 22) = 10(5x + 22). Step 6: Expand both sides: 72x + 198 = 50x + 220. Step 7: Rearrange: 72x − 50x = 220 − 198 ⇒ 22x = 22 ⇒ x = 1. Step 8: Ratnabali's present age = 8x = 8 × 1 = 8 years.

Verification / Alternative check:
With x = 1, Shaukat's present age is 5 × 1 = 5 years. After 22 years, Ratnabali will be 8 + 22 = 30 years old and Shaukat will be 5 + 22 = 27 years old. Their ages will then be in the ratio 30 : 27, which simplifies to 10 : 9 when both numbers are divided by 3. This matches the given future ratio, confirming that Ratnabali's present age is indeed 8 years.

Why Other Options Are Wrong:
Values such as 5 years, 11 years, 14 years or 81 years for Ratnabali do not yield a pair of present ages that satisfy both the initial ratio of 8 : 5 and the future ratio of 10 : 9 after 22 years. Only 8 years leads to consistent future ages of 30 and 27 with the required ratio.

Common Pitfalls:
Students often forget to add the same number of years to both ages when forming the future ratio, or they mistakenly set up the initial ratio using actual numbers instead of multiples of x. Another error is to make arithmetic mistakes when cross-multiplying. Carefully writing down each algebraic step helps avoid these issues.

Final Answer:
Ratnabali's present age is 8 years.