Symbiosis runs a Corporate Training Programme. At the end of running the first programme, its total takings were ₹ $38950$. There were more than $45$ but less than $100$ participants. What was the participant fee for the programme?

### Concept & Strategy The total revenue (takings) is the product of the number of participants and the fee per participant. Because the number of participants must be a whole number, the correct participant fee must be a perfect divisor of the total takings. ### Step-by-Step Solution * **Given:** * Total takings = ₹ $38950$ * Number of participants ($n$) is an integer where $45 < n < 100$. * Options for fee ($f$): $410$, $450$, $500$, $510$. * Formula: $n \times f = 38950 \implies n = \frac{38950}{f}$ * **Calculation / Deduction:** 1. We need to find which option perfectly divides $38950$ to yield an integer between $46$ and $99$. 2. **Test Option (c) 500:** $38950 / 500 = 389.5 / 5$. This will not result in a whole number. Reject. 3. **Test Option (b) 450:** $38950 / 450 = 3895 / 45$. Since $3895$ does not end in $0$, it is not perfectly divisible by $45$ (which requires divisibility by both $5$ and $9$). Sum of digits of $3895 = 25$, not divisible by $9$. Reject. 4. **Test Option (d) 510:** $38950 / 510 = 3895 / 51$. $51$ is $17 \times 3$. Since the sum of digits of $3895$ is $25$, it's not divisible by $3$, so it can't be divisible by $51$. Reject. 5. **Test Option (a) 410:** $38950 / 410 = 3895 / 41$. Let's divide: $41 \times 100 = 4100$. Since $3895$ is close to $4100$, let's try $41 \times 95$. $41 \times (100 - 5) = 4100 - 205 = 3895$. Thus, the number of participants is exactly $95$. 6. Check the constraint: Is $95$ between $45$ and $100$? Yes. ### Exam Strategy & Shortcut Use divisibility rules to eliminate options rapidly. You know $n = 38950 / f$. Ignore the trailing zeros: you are essentially dividing $3895$ by $41$, $45$, $50$, or $51$. Since $3895$ doesn't end in $0$, it cannot be divided evenly by $50$. Since the sum of digits ($3+8+9+5 = 25$) is not a multiple of $3$, it cannot be divided by $45$ (needs $9$) or $51$ (needs $3$). By process of elimination, only $410$ remains. You don't even need to calculate $3895 / 41 = 95$ to know it's the right answer. ### Common Pitfall Attempting to formulate complex algebraic equations or guess the number of participants first is a massive time sink. The fastest route is recognizing the integer constraint and testing the given options as divisors. ### Final Answer Therefore, the correct answer is ₹ 410.