Maths shelf · Class 10
The first profitable ticket
Every ticket brings in ₹50, but the show has costs. Which ticket is the first one that actually makes money?
What you will learn: Model takings and costs as two linear equations, find where they meet by graph and by algebra, and explain what the meeting point means.
What should I know first?
If this feels new, try these first. It is fine to go back.
- Compare mobile data plans with equations Class 9
- Two colours per kit Class 8
Small test at the end: scroll down to the end of this page for five short questions that check what you have learnt. Tap an answer, then press Check answer. Go to the questions ↓
Jump to: Check yourself · For grown-ups · For teachers
Safety first
Use made-up prices. Never collect real money without a grown-up.
Keep it safe:
- Use made-up prices for practice. Real ticket money is handled only by grown-ups.
- If you use coins or counters to act it out, keep them away from babies: nothing goes in the mouth.
You will need
- Graph paperFor the two lines.
- A ruler and pencilTo draw straight lines.
- A calculator (optional)To check the table.
Time: About 45 minutes.
What do you think?
Tanmay says 25 tickets are enough, because 25 × ₹50 = ₹1,250 is more than ₹1,200. Is he right?
Make a prediction and write one reason for it. Then tap the guess closest to yours to see how to test it. It is fine to change your mind after checking.
Yes
Let's check. What do 25 snack kits cost? Add that to ₹1,200 and compare with ₹1,250.
No
How many tickets are enough, then? Make a table for 10, 20, 30, 40 and 50 tickets.
It depends
On what? Write the takings and the costs as rules with x for tickets, and see where they meet.
Not sure
That is fine. Work out the takings and the costs for 25 tickets, then for 50.
Need a hint?
Each ticket brings ₹50 but also costs ₹20 for a snack kit. How much does each ticket really add?
The story
The class is putting on a show at the fair to raise money for the library. Tickets are ₹50. The hall, lights and sound cost ₹1,200 whatever happens, and every person who comes gets a snack kit costing ₹20. "Sell 25 tickets and we have ₹1,250, that's more than ₹1,200!" says Tanmay. Dhruv shakes his head. "You forgot the snack kits." Tara draws two lines on graph paper.
The idea
Fixed costs are paid whatever happens (the hall: ₹1,200). Variable costs grow with each person (₹20 per snack kit).
Takings y = 50x and costs y = 1200 + 20x are a pair of linear equations. Each has a straight-line graph.
The break-even point is where the lines cross: solve 50x = 1200 + 20x to get x = 40, y = 2000.
Below 40 tickets the cost line is higher (a loss). Above 40 the takings line is higher (a profit of ₹30 for each extra ticket).
Worked example
Solving for the break-even point
Find where takings equal costs.
- Set them equal: 50x = 1200 + 20x.
- Take 20x from both sides: 30x = 1200.
- Divide by 30: x = 40.
- Takings at 40: 50 × 40 = ₹2,000. Costs: 1200 + 20 × 40 = ₹2,000. They match.
The show breaks even at 40 tickets. Ticket 41 is the first profitable ticket.
Try it
Write the rules.
Write takings and costs for x tickets.
Make a table.
Work out both for 0, 10, 20, 30, 40, 50 and 60 tickets.
Draw the lines.
Tickets along the bottom, rupees up the side. Plot both rules and join each with a ruler.
Find the crossing.
Read where the lines cross. Then solve the equations to check.
Explain the next ticket.
Work out profit at 40, 41 and 50 tickets. How much does each ticket after 40 add?
Change one thing.
The snack kit now costs ₹25. Find the new break-even point. Then try raising the ticket price instead.
What you make (paper or offline)
Your break-even card
Make a break-even card for a stall or show of your own: price, fixed cost and cost per customer.
Show the table, the graph and the equation, and mark the first profitable sale.
Add one "what if" line: what happens if a cost changes?
Make it on paper. Keep it with you. Nothing to upload.
What did we find out?
Read this after you have tried it. Look back at your guess and your reason first.
Takings: y = 50x. Costs: y = 1200 + 20x. Both are straight lines.
They meet where 50x = 1200 + 20x, so 30x = 1200 and x = 40: ₹2,000 in, ₹2,000 out.
At 40 tickets the show breaks even: no profit and no loss. Profit starts with ticket 41, and each extra ticket adds ₹30.
Check yourself
Five questions about what this page teaches. Choose an answer and press Check answer, or open "See the answer and why". Nothing is saved or sent: your choices stay on this screen and disappear when you leave or reload the page.
How do you want to answer? You choose, and you can change at any time.
See the answer and why
- A. It exactly breaks even Not this one. It breaks even at 40 tickets, not 30.
- B. Yes: ₹1,500 is more than the ₹1,200 hall cost Not this one. The ₹20 snack kits must be counted too.
- C. Yes: it takes ₹1,800 and costs ₹1,500 Not this one. The two amounts are swapped.
- D. No: it takes ₹1,500 but costs ₹1,800 Correct. Takings 50 × 30 = 1,500. Costs 1,200 + 20 × 30 = 1,800. A loss of ₹300.
See the answer and why
- A. 40 Correct. Each ticket adds 50 − 20 = 30 towards the ₹1,200: 1,200 ÷ 30 = 40.
- B. 17 Not this one. That divides by 50 + 20. The kit cost is taken away, not added.
- C. 60 Not this one. That is 1,200 ÷ 20. It uses the kit cost instead of what each ticket adds.
- D. 24 Not this one. That is 1,200 ÷ 50. It forgets the ₹20 snack kit for each person.
See the answer and why
- A. 50x + 20x = 1200 Not this one. The snack kits are a cost, so they belong with the ₹1,200.
- B. 50x = 1200 + 20x Correct. Takings on the left, all the costs on the right.
- C. 1200 + 50x = 20x Not this one. The ₹1,200 is a cost, not takings.
- D. 50x = 1200 Not this one. This leaves out the ₹20 for each person.
See the answer and why
- A. 24 tickets Not this one. That is 1,200 ÷ 50, which ignores the kits.
- B. 48 tickets Correct. Each ticket now adds 50 − 25 = 25: 1,200 ÷ 25 = 48.
- C. Fewer than 40, because each kit costs more Not this one. Costs went up, so more tickets are needed, not fewer.
- D. 40 tickets Not this one. That was with ₹20 kits. A higher cost needs more tickets.
See the answer and why
- A. ₹500 Not this one. That is 10 extra tickets × ₹50. Each extra ticket adds only ₹30 after its kit.
- B. ₹300 Correct. Takings ₹2,500 minus costs 1,200 + 1,000 = ₹2,200 gives ₹300.
- C. ₹2,500 Not this one. That is the takings, not the profit.
- D. ₹1,300 Not this one. That is 2,500 − 1,200. It forgets the 50 snack kits.
Think back
Why does profit start with ticket 41 and not ticket 40?
What would you change first to make the show break even sooner?
Investigate further
- If tickets cost ₹60 instead, where is the new break-even point?
- Compare with the Class 9 "mobile data plans" activity: what is the same about the two crossing lines?
Can you use it here?
A stall sells lemonade at ₹15 a glass. It costs ₹300 to set up, and ₹5 for each glass. How many glasses to break even?
Ask someone at home
Ask someone at home about a business they know (a shop, a tuition class, a taxi). What costs are fixed, and what changes with each customer?
No one free to join in? That's fine. You can do it by yourself with drawings, and ask a grown-up to help with the materials.
For grown-upsTap a heading to open it. The safety notes are at the top of the page.
Before you start
- Graph paper, a ruler and a pencil. Keep coins and counters away from babies; nothing in the mouth.
Questions that help
- "Which costs are fixed? Which grow?"
- "What does each extra ticket really add?"
- "What happens just after the lines cross?"
What's really going on
Break-even is the point where takings equal costs. Every small business plans with it.
Children often forget the cost that grows with each customer, as Tanmay did.
The graph and the algebra give the same answer; checking one with the other builds confidence.
Other ways to do it
- Use a real local example: a chai stall, a tuition batch, a school trip.
- No graph paper? A table alone still finds the point.
For teachersLearning aims, classroom ideas and what to look for.
Learning aims
- Form a pair of linear equations from a real situation.
- Solve them graphically and algebraically.
- Interpret the solution as a break-even point, with profit and loss on either side.
Subject links
- CBSE Class X Mathematics: pair of linear equations in two variables.
- Class 9 experience "Linear plans" (mobile data plans).
- Commerce and economics: fixed and variable costs.
These links are general. They have not been checked against a specific board syllabus.
In the classroom
- Groups plan a real fundraiser with their own prices and present the break-even point.
- Change one number at a time and record how the crossing point moves.
What to look for
- Includes both fixed and per-person costs.
- Explains the crossing point in words.
- Uses "each ticket adds ₹30" as a check.