The ratio of the ages of two boys is 3 : 4 at present. After 3 years, the ratio of their ages will become 4 : 5. What will be the ratio of their ages after 21 years from now?

Introduction / Context:
Age problems with ratios at different times are very common in competitive exams. Here, we know the current ratio of ages and the ratio after a certain number of years. We must find the ratio after an even later time. This tests setting up linear equations from age ratios and understanding how adding the same number of years affects each age but preserves a new ratio relationship.

Given Data / Assumptions:

• Current ratio of ages of two boys = 3 : 4.
• After 3 years, the ratio will be 4 : 5.
• We must find the ratio of their ages after 21 years from now.
• Ages are positive whole numbers.

Concept / Approach:
We start by expressing their present ages as 3k and 4k using the given current ratio. After 3 years, their ages become 3k + 3 and 4k + 3, and this ratio is given as 4 : 5. That gives an equation which we can solve for k. Once k is known, we can calculate their actual present ages. Finally, we add 21 years to each age and simplify the resulting ratio to answer the question.

Step-by-Step Solution:

Verification / Alternative check:
Check the intermediate condition. After 3 years, ages become 12 and 15. The ratio 12 : 15 simplifies to 4 : 5, which matches the given condition. This confirms that k = 3 is correct. Then, after 21 years, ages become 30 and 33, and their ratio is indeed 10 : 11.

Why Other Options Are Wrong:

• 14 : 17, 17 : 19, and 11 : 12 do not match the ages 30 and 33 when simplified.
• For example, 30 : 33 cannot simplify to 11 : 12 or 14 : 17, so those options are incorrect.

Common Pitfalls:
Students sometimes use 3 years directly in the final step instead of 21 years, or they incorrectly form the equation by mixing present and future ages. Another error is to forget to simplify the final ratio, even when the numbers have a common factor. Carefully separating the stages (present, after 3 years, after 21 years) and checking each ratio helps avoid confusion.

Final Answer:
The ratio of their ages after 21 years will be 10 : 11.