Ages ratio in the future: Two people are currently 36 years and 50 years old. After n years, their ages will be in the ratio 3:4. Find the value of n.

Introduction / Context:
Age-ratio problems are solved by forming an equation on future ages. Linear growth by n years preserves differences but changes ratios; we equate the target ratio and solve for n directly.

Given Data / Assumptions:

• Current ages: 36 and 50 years.
• After n years, ratio becomes 3:4.
• n is nonnegative and integral in typical age problems.

Concept / Approach:
Write (36 + n)/(50 + n) = 3/4 and cross-multiply to obtain a simple linear equation in n. Ensure the solution is feasible (age cannot be negative, but here it is clearly positive).

Step-by-Step Solution:

Verification / Alternative check:
After 6 years: ages are 42 and 56. Ratio 42:56 simplifies to 3:4, confirming the result.

Why Other Options Are Wrong:

• 3, 4, 7, 8: Substituting these values does not yield the target 3:4 ratio.

Common Pitfalls:
Accidentally using (50 − n) or mixing up who is older. Always add n to both ages for “after n years.”

Final Answer:
6