Vidushi and Sanya distributed ₹ 100 each in charity. Vidushi distributes money to 5 more people than Sanya and Sanya gives each ₹ 1 more than Vidushi. How many people are recipients of the charity?

## Concept & Logic This problem links the number of recipients to the amount given per recipient via an inverse relationship, forming a quadratic equation. The total amount given by each person is fixed at ₹ 100. $$ \frac{Total\ Amount}{Number\ of\ People} = Amount\ per\ Person $$ ## Step-by-Step Solution * **Given:** Both distribute ₹ 100. Let the number of people Sanya distributes to be $S$. Then, the number of people Vidushi distributes to is $S + 5$. * **Deduction:** Amount Sanya gives to each person = $\frac{100}{S}$. Amount Vidushi gives to each person = $\frac{100}{S+5}$. * **Calculation:** The problem states Sanya gives ₹ 1 more per person than Vidushi. $\frac{100}{S} - \frac{100}{S+5} = 1$ * **Calculation:** Factor out 100 and find a common denominator: $100 \left( \frac{(S+5) - S}{S(S+5)} \right) = 1$ $100 \left( \frac{5}{S^2 + 5S} \right) = 1$ $500 = S^2 + 5S$ * **Calculation:** Solve the quadratic equation: $S^2 + 5S - 500 = 0$ Find factors of -500 that add to +5. They are +25 and -20. $(S + 25)(S - 20) = 0$. Since $S$ must be positive, $S = 20$. * **Conclusion:** Sanya distributes to 20 people. Vidushi distributes to $20 + 5 = 25$ people. Total recipients = $20 + 25 = 45$. ## Exam Strategy & Shortcut Bypass the quadratic equation by looking for two integer factors of 100 whose difference is related to the problem. We need two numbers of people that divide 100 perfectly (since usually amounts are integers in these contexts), and their difference must be 5. Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100. Look at 20 and 25. Their difference is 5. Check the amounts: $\frac{100}{20} = 5$ and $\frac{100}{25} = 4$. The difference in amounts is $5 - 4 = 1$. This perfectly matches all conditions in the question! Total people = $20 + 25 = 45$. ## Common Pitfall Solving for $S$ (20) and selecting an option if it were available, or solving for Vidushi (25). Read the final question carefully: it asks "How many people are recipients of the charity?", meaning the *total combined* number of recipients ($20 + 25 = 45$). ## Final Answer Therefore, the correct answer is **45**.