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CAT 2025 Lesson : Profit & Loss - FC, VC & Break-Even

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7. Fixed Cost, Variable Cost and Break-even

Fixed Costs (FC) are constant costs which do not vary with number of units sold.
Variable Costs (VC) are those that are incurred on a specific unit of a product and vary with sales volumes.

For a factory, the fixed cost would be rent for the factory, manager's salary, etc. These costs are fixed and do not change with reasonably small changes in the output.

The variable costs would be the cost of raw materials, fuel/electricity used for production, etc. Increase or decrease in output will lead to increase or decrease in variable costs.

Total Cost = FC + VC

Contribution Margin or Variable Margin is the difference between the selling price per unit and the variable cost per unit.

Variable Margin = SP per unit - VC per unit

Profit is the excess of Variable margin of the units sold over the fixed cost.

Profit=Variable Margin×Units Sold−Fixed Cost\text{Profit} = \text{Variable Margin} \times \text{Units Sold} - \text{Fixed Cost}

Break-even point is when Total Revenue = Total Cost.

Also, as profit =0= 0, Fixed Cost =Variable Margin×Units Sold= \text{Variable Margin} \times \text{Units Sold}.

Break-even volume =Fixed CostVariable Margin= \dfrac{\text{Fixed Cost}}{\text{Variable Margin}}

Example 21

Company ABC starts an educational program in collaboration with Institute XYZ. As per the agreement, ABC and XYZ will share profit in 60:40 ratio. The initial investment of ₹100,000 on infrastructure is borne entirely by ABC whereas the running cost of Rs. ₹400 per student is borne by XYZ. If each student pays ₹2000 for the program find the minimum number of students required to make the program profitable, assuming ABC wants to recover its investment in the very first year and the program has no seat limits.
[XAT 2016]

(1) 6363            (2) 8484            (3) 105105            (4) 157157            (4) 167167           

Solution

SP = ₹ 2,0002,000 per student
VC = ₹ 400400 per student

Variable Margin =2000−400== 2000 - 400 = ₹ 16001600 per student

Fixed Costs = ₹ 100,000100,000

The minimum number of students required to recover the fixed costs is the break-even volume.

Break-even Volume =Fixed CostVariable Margin=1000001600=62.5= \dfrac{\text{Fixed Cost}}{\text{Variable Margin}} = \dfrac{100000}{1600} = \bm{62.5}

As students cannot be in decimal, 63 students are required for ABC to recover its investment.

Alternatively

To find the minimum number of students for break-even, total revenue should equal total costs. Let the number of units sold be xx.

Total Revenue =2000x= 2000x
Total Cost =100000+400x= 100000 + 400x

2000x=100000+400x2000x = 100000 + 400x

x=1000001600=62.5x = \dfrac{100000}{1600} = 62.5 (rounded to 63 students)

Answer: (1) 6363

Example 22

Some costs for a machine are fixed and do not vary with change in output, while others are variable and change at a constant rate for every additional unit produced. The average cost per unit was Rs. 6060 when 1515 units are produced and Rs. 5050 when 2020 units are produced. What is the total cost incurred when 3030 units are produced?

Solution

Let FF be the fixed cost and vv be the variable cost per unit produced.

Total Cost =Fixed Cost+(VC/Unit×Units Produced)= \text{Fixed Cost} + (\text{VC/Unit} \times \text{Units Produced})

Average cost =Total CostUnits produced= \dfrac{\text{Total Cost}}{\text{Units produced}}

When 1515 units are produced ⇒ 60=F+15v15 60 = \dfrac{F + 15v}{15}

⇒ 60=F15+v⟶(1) 60 = \dfrac{F}{15} + v \longrightarrow (1)

When 2020 units are produced ⇒ 50=F+2020 50 = \dfrac{F + 20}{20}

⇒ 50=F20+v⟶(2) 50 = \dfrac{F}{20} + v \longrightarrow (2)

Eq(11) - Eq(22) ⇒ 10=F15−F20⇒F=600 10 = \dfrac{F}{15} - \dfrac{F}{20} ⇒ F = 600

Substituting in Eq(11) ⇒ 60=60015+v⇒v=20 60 = \dfrac{600}{15} + v ⇒ v = 20

Total cost for 3030 units =F+30v=600+(30×20)=1200= F + 30v = 600 + (30 \times 20) = 1200

Answer: Rs. 12001200

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