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There are two types of costs. Fixed cost F and a variable cost. The fixed cost normally does not change for small expansion in production.Variable cost depends on the unit cost v of production. So C(x) = F + vx. Where c(x) is the cost of production. If R(x) is the revenue , then R(x) = p*x , where p is the price per unit.
So now you break even or get profit if R(x) - C(x) > 0.
px > F+vx.
So the break even point is when
px=F +vx . Or
(p-v)x = F. Or
x = F/(p-v).
If x > F/(p-v) = Fixed cost/(pirce - cost per unit) you get the profit.
If x = Fixed cost/(price -cost per unit), there is break even or no profit no loss.
x < Fixed cost /(price per unit - cost per unit) , then there is loss.
Real world example: You are making transport business with one single vehicle for simplicity . The fixed cost which includes rent per day , salary of driver per day etc = 1000 units of money. The variable cost includes deprciation of vehicle incliuding which inclides tyre and Fuel etc works out Rs 25 per kilometer. Let the revenue be R(x) = 30x, where 30 units of money is the price you charge per kilometer of running the vehicle. X is the number of kilomers of run of the vehicle.
So the cost equation C(x) = 1000+25x.
The revenue equation is R(x) = 30x
Break even is when x = F/(P-v) = 1000/(30-25) = 200 kilometer.
If you run more than 200 km(say 220km) you get a revenue of 30*220 = 6600 unit against a cost of F+25x = 1000+25*22 = 6500 unit of money. So it is a profit of 6600-6500 = 100 unit of money.
If you run less than 200 km per day you incur loss . If you run 150 km, R(x) = 300*200 = 6000 unit of moneyand C(x) =1000+25*150 = 4750 unit of money.
If you run 200 km , you are breaking even when your cost = 1000+25*200 = revenue = 30*200 unit of money.
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