One event requires several connected calculations
A college tech fest has a total planning budget of ₹3,00,000. Organizers first protect 12% as an emergency reserve. The remaining amount is shared among technical events, infrastructure and publicity in the ratio 5:3:2.
A vendor quotes ₹1,50,000 and offers successive discounts of 12% and 5%. Separately, the event has ₹1,20,000 fixed cost, ₹40,000 sponsorship income, ₹350 ticket price and ₹180 variable cost per attendee. The team must find the smallest whole attendance that prevents a loss and test several attendance scenarios.
Do not substitute until every quantity has a role
| Quantity | Symbol | Value | Role |
|---|---|---|---|
| Total planning budget | B | ₹3,00,000 | Base for reserve |
| Reserve rate | r | 12% | Protected fraction |
| Fixed event cost | F | ₹1,20,000 | Independent of attendance |
| Sponsorship | S | ₹40,000 | Revenue before tickets |
| Ticket/variable cost | p,v | ₹350, ₹180 | Per attendee |
| Attendance | n | Unknown integer | Decision variable |
Every monetary expression must combine rupees with rupees. A percentage is dimensionless. Attendance is a count, so a decimal solution must be rounded upward when the goal is “at least break-even.”
Find the part, then subtract from the whole
Reserve = B × r/100
= 3,00,000 × 12/100 = ₹36,000
Usable budget = 3,00,000 − 36,000 = ₹2,64,000A quick reasonableness check helps: 10% is ₹30,000 and 2% is ₹6,000, so 12% must be ₹36,000. The usable amount must be less than the original budget and both parts must sum back to ₹3,00,000.
A ratio names shares, not rupee amounts
The parts 5:3:2 total 10. One ratio unit therefore equals ₹2,64,000 ÷ 10 = ₹26,400.
| Category | Parts | Calculation | Allocation |
|---|---|---|---|
| Technical events | 5 | 5 × 26,400 | ₹1,32,000 |
| Infrastructure | 3 | 3 × 26,400 | ₹79,200 |
| Publicity | 2 | 2 × 26,400 | ₹52,800 |
Verification: ₹1,32,000 + ₹79,200 + ₹52,800 = ₹2,64,000. Dividing any pair of allocations recovers the intended proportion.
Two percentage reductions use two different bases
After 12% discount = 1,50,000 × 0.88 = ₹1,32,000 After another 5% = 1,32,000 × 0.95 = ₹1,25,400
The discounts do not add to 17%, because the second 5% is calculated on ₹1,32,000 rather than the marked price. The remaining-price factor is 0.88 × 0.95 = 0.836. Therefore the equivalent discount is (1 − 0.836) × 100 = 16.4%.
Separate fixed amount from per-attendee contribution
Revenue R(n) = 40,000 + 350n Cost C(n) = 1,20,000 + 180n Require R(n) ≥ C(n) 40,000 + 350n ≥ 1,20,000 + 180n 170n ≥ 80,000 n ≥ 470.588...
The contribution per attendee is ₹350 − ₹180 = ₹170. Sponsorship reduces the uncovered fixed amount to ₹80,000. Because attendance must be a whole number, round upward: the minimum non-loss attendance is 471. At 471, profit is ₹70; at 470, loss is ₹100. Rounding to the nearest integer would produce the wrong operational decision.
Python program using Decimal money arithmetic
Loading source…The program separates reusable calculations into functions, validates impossible inputs and uses Decimal so base-10 money values do not inherit binary floating-point representation surprises.
Trace the complete decision
- Protect reserve.
- Find allocation base.
- Allocate budget.
- Compound discounts.
- Verify quote.
- Build break-even equation.
- Convert to whole attendance.
Press Next to begin.
Use identities and boundary tests
Allocation conservation
Discount factor
Break-even boundary
Impossible contribution
Check the reasoning
What is the equivalent discount for 12% followed by 5%?
Why is the answer rounded upward?
Extensions
- Add three ticket categories in a fixed sales ratio.
- Include a maximum venue capacity and test feasibility.
- Perform best, expected and worst attendance scenarios.
- Find the ticket price needed to break even at 400 attendees.
Explain the model, not just the arithmetic
Why do successive discounts multiply?
Each step preserves a fraction of the current price. Applying 12% then 5% preserves 88% and then 95%, giving the combined factor 0.88×0.95.
What does contribution mean?
Ticket price minus variable cost is the amount each additional attendee contributes toward uncovered fixed cost and then profit.
What assumption makes the model linear?
Ticket price and variable cost per attendee are constant, while fixed cost and sponsorship do not depend on attendance.
Always identify the base and the changing quantity
Percentages require the correct base, ratios require conservation, successive changes require multiplication, and break-even requires separating fixed and variable terms. Units and boundary checks connect the formula to a correct decision.