# Algebraic Representation of BEA

Basically, there are two approaches to the break even analysis:

1) Equation technique 2) Contribution margin technique

1) Equation technique: A simple equation that represents that relationship of income statement is considered here. For any break even or profit estimate Situation, the equation reads:

2) Contribution Margin Technique: Here, with the help of the concept of contribution margin, the break even point is calculated. The unit contribution is divided into total fixed expenses to arrive at the number of units to be sold to break even. So the equation reads.

Unit Sales Price – Unit Variable Overheads = Unit Contribution

1) BEP (units) = Fixed Overheads / Per Unit Contribution

2) BEP (Rs) = Fixed Overheads / Per Unit Contribution x Per Unit sales Price

or = BEP (Units) x Sales Price (p.u).

3) BEP (Rs) = Fixed Overheads x Total Revenue / Gross Contribution Margin
4) BEP (Rs) = Fixed Overheads / P / V ratio (%).

Illustration

XYZ Ltd plans to sell a toy motor at the Gujarat Diwali fair. The motors are purchased at Rs 5/- each on the express condition that all unsold motors shall be returned. The booth rent at the fair is Rs 2,000/- payable in advance. The motors will be sold at Rs 9/- each. Determine the number of motors which must be sold.

a) to break even b) to earn Rs 400/- as profits
We shall solve it with both aforesaid methods.

1) equation Technique : Here

In the case (a):

Profits = 0 (since it s BEP)

Variable overheads = Rs 5/- p.a

Sales price = Rs 9/- p.u.

Suppose the number of motors sold is A, then,

9A = 5A + 2,000 + 0

4A = Rs 2,000

A = 500(units)

In the case (b)

Apart from other information profit is expected to the tune of Rs 400/- . Then,

9A = 5A + 2,000 + 400

9A – 5A = Rs 2,400

4A = Rs 2,400

A = 600 (units).

2) Contribution Marginal Technique : Here, in the case (a):

BEP (units) = Fixed Overhead/Unit Contribution Margin

= 2,000 / 9 – 5 = 2,400 /4

BEP (units) = 500 units

and BEP (Rs) = BEP (units) x Sales Price (p.u)

= 500 x Rs 5 = Rs 2,500

In the case (b); to earn Rs 400/- as yield we must obtain Rs 400/- over and the fixed overheads. So, now,

BEP (units) = Fixed Overheads + Expected Profit / Per Unit Contribution

= 2,000 + 400 / 9 – 5 = 2,400 / 4

BEP (units) = 600 units.

And BEP(Rs) = BEP (Units) x Sales Price (p.u)

= 600 x Rs 5

= Rs 3,000

In case XYZ Ltd sells 750 motors in the fair find (I) Margin of Safety (2) Profits at 750 units.

1) Margin of safety = Sales – sales at BEP / Sales x 100

= 750 – 500 / 750 x 100

= 250 –100 / 750 = 33.33%

OR

Margin of Safety = Operating Profits x100 / Gross Contribution Margin

= (Gross Contribution Margin – FC) x 100 / Gross Contribution margin

Now, here at the level of 750 units:

Gross Contribution Margin:

= (750 x 9) – (750 x 5)

= Rs (6,750 – 3,750) = Rs 3,000

Now, margin of safety

= (3,000 – 2,000) x 1000 / 3,000

= 1,000 x 100 / 3,000 = 33.33%

And (2) to calculate profits at a sale of 750 motors

Profit = Total Revenue – Total Cost

= Total Revenue – Total Variable Cost – Total Fixed Cost

= Gross Contribution Margin – Total fixed cost

Rs 3,000 — Rs 2,000 = Rs 1,000.