David and his wife each receives an $8$ percent annual rise. If David receives a raise of ₹ $800$ and his wife receives a raise of ₹ $840$, what is the difference between their annual incomes after their raises?

### Concept & Logic When two distinct values are increased by the exact same percentage, the difference between the absolute increases is directly proportional to the difference between the original base values. ### Step-by-Step Solution **Given:** * David's raise = ₹ $800$ (which is $8\%$ of his income) * Wife's raise = ₹ $840$ (which is $8\%$ of her income) **Calculation:** * Find David's original annual income ($D$): $$8\% \text{ of } D = 800$$ $$D = \frac{800}{0.08} = 10000$$ * Find his wife's original annual income ($W$): $$8\% \text{ of } W = 840$$ $$W = \frac{840}{0.08} = 10500$$ * Calculate their incomes **after** the respective raises: $$\text{David's New Income} = 10000 + 800 = 10800$$ $$\text{Wife's New Income} = 10500 + 840 = 11340$$ * Find the absolute difference: $$\text{Difference} = 11340 - 10800 = 540$$ ### Exam Strategy & Shortcut Work strictly with the differences to bypass full income calculations entirely. The difference in raise amounts = $840 - 800 = 40$. This ₹ $40$ represents $8\%$ of the difference in their base incomes. $$8\% \text{ of Base Difference} = 40$$ $$\text{Base Difference} = \frac{40}{0.08} = 500$$ The final difference after the raise will simply be the Base Difference plus the Difference in Raises: $$\text{Final Difference} = 500 + 40 = 540$$ ### Common Pitfall Students frequently calculate the difference in the base incomes (₹ $500$) and hastily select option (c). However, the question specifically requests the difference **after** their raises, requiring the addition of the ₹ $40$ raise discrepancy to the base discrepancy. ### Final Answer **Therefore, the correct answer is ₹ 540.**