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Class 8 — Linear Equations

Break-even: the number that decides your day

Rent is due whether you sell one glass or a hundred. So every day starts in a hole, and the only question that matters before you open is: how many do I have to sell just to climb out? That number is break-even, and it is a linear equation doing an honest day's work.

Two kinds of cost

Fixed costs do not change with how much you sell — the rent for your spot. Variable costs change with every unit — the lemon, sugar and cup in each glass.

Break-even is where what you keep from each sale has finally paid off the fixed cost.

The equation

Break-even quantity = fixed cost ÷ profit per unit.

If rent is ₹20 and you keep ₹5 on each glass, you need 20 ÷ 5 = 4 glasses before the day is worth anything. Glass five is the first rupee you actually earn.

Written as an equation: 5n − 20 = 0, so n = 4. This is precisely the Class 8 linear equation chapter, with a reason to exist.

A worked example from a real day

Rent for the spot (fixed)₹20
Price per glass₹12
Cost to make one glass (variable)₹7
Profit kept per glass₹5
Break-even20 ÷ 5 = 4 glasses

Now raise the rent to ₹60 for a busier street. Break-even jumps to 12 glasses. The better location is only the better decision if it brings at least 8 more customers — which is a calculation you can do before you move, not after.

Try it: how many glasses before you stop losing money?

Rent is due whether you sell one glass or a hundred.
4 glasses
every glass after the 4th is pure profit

Practice

Fixed cost ₹90, profit per unit ₹6. What is the break-even quantity?
Show the answer
15 units
90 ÷ 6 = 15.
A stall pays ₹150 rent, sells at ₹25 and spends ₹10 per unit. How many to break even?
Show the answer
10 units
Profit per unit = 25 − 10 = ₹15. 150 ÷ 15 = 10.
A cart breaks even at 20 glasses with ₹100 of fixed costs. What does it keep per glass?
Show the answer
₹5
100 ÷ n = 5 when n = 20. Rearranged: 100 ÷ 20 = ₹5.

Learn it by running one

In Lemonade Empire your child runs a lemonade cart and uses this exact concept to decide what to buy and what to charge. Free, no signup, plays in a browser.

▶ Play Lemonade Empire free

Common questions

What is the break-even point in simple terms?
The number of sales at which you have exactly covered your costs — no profit, no loss. Everything after it is profit.
Which NCERT chapter does break-even belong to?
It is not named as a chapter, but it is exactly what Class 8 linear equations are for. Writing 5n − 20 = 0 and solving for n is the whole idea.
Why do children find this harder than it looks?
Because it needs two ideas at once — that some costs do not move, and that each sale contributes a fixed amount toward them. Seeing the number change as the rent changes is what makes it click.

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