The ages of A and B are in the ratio $6 : 5$ and the sum of their ages is 44 years. What will be the ratio of their ages after 8 years?

Concept & Logic When given a ratio and a total sum, you can immediately find the absolute value of one ratio "part" by dividing the total sum by the sum of the ratio parts. Once present ages are found, simply add the future years to establish the new ratio. $$ 1 \text{ Part} = \frac{\text{Total Sum}}{\text{Sum of Ratio Parts}} $$ Step-by-Step Solution * **Given:** The present age ratio of A and B is $6 : 5$. * **Given:** The sum of their present ages is 44 years. * **Calculation:** Let the present ages of A and B be $6x$ and $5x$ respectively. * Set up the sum equation: $6x + 5x = 44$. * Combine the terms: $11x = 44$. * Solve for $x$: $x = 4$. * Calculate A's present age: $6(4) = 24$ years. * Calculate B's present age: $5(4) = 20$ years. * Add 8 years to find their future ages: A will be $24 + 8 = 32$, and B will be $20 + 8 = 28$. * Find the new ratio: $32 : 28$. * Simplify the ratio by dividing both numbers by their greatest common divisor (4): $8 : 7$. Exam Strategy & Shortcut Use the **Ratio Parts Method** for speed. Total ratio parts = $6 + 5 = 11$ parts. Total years = 44. So, 11 parts = 44 years $\Rightarrow 1$ part = 4 years. Instead of finding absolute ages and then simplifying the final ratio, convert the 8 future years directly into "ratio parts". Since 4 years = 1 part, 8 years = 2 parts. Simply add 2 parts directly to the present ratio: $(6 + 2) : (5 + 2) = 8 : 7$. Common Pitfall Students often calculate the future ages (32 and 28) correctly but fail to reduce the final fraction to its simplest form, leading to confusion when their exact numbers do not appear in the options. Always simplify ratios. Final Answer **Therefore, the correct answer is 8 : 7.**