4 2 break-even-analysis

4.2 Break-even Analysis4.2 Break-even Analysis

Break-even Analysis

4.2 Break-even Analysis4.2 Break-even Analysis

• Break-even Analysis – used to determine how much sales volume a business needs just to cover costs

• Cost – amount spent on producing x units of a product

• Revenue – amount received from selling x units of the product

• Profit – amount gained if revenue is greater than the cost

4.2 Break-even Analysis4.2 Break-even Analysis

• Cost function: yC = C(x)

• C(x) = vx + F– F is the fixed cost (overhead expenses like rent,

machineries, interest on loans)– v is the variable cost per unit (the cost of

producing each unit of the product

• Examples:

€

C(x) =10 + 2x

€

yC = 0.65x +10,000

4.2 Break-even Analysis4.2 Break-even Analysis

• Revenue function: yR = R(x)

• R(x) = px– p is the selling price per unit– If the selling price depends on the quantity

demanded, we write and p = D(x) and R(x) = xD(x).

• Examples:

€

R(x) = 3x

€

yR = 0.88x

4.2 Break-even Analysis4.2 Break-even Analysis

Break-even point – where the cost and revenue functions intersect at a point; where cost equals revenue

€

C(x) = R(x)

4.2 Break-even Analysis4.2 Break-even Analysis

• Loss – occurs when the firm makes and sells fewer units than its break-even quantity

• Profit – occurs when the firm makes and sells more units than its break-even quantity

Profit function:

€

P(x) = R(x) −C(x)

4.2 Break-even Analysis4.2 Break-even Analysis

1. Given the cost function C(x) = 10 + 2x and revenue function R(x) = 3x, find the break-even quantity and price.

€

C(x) = R(x)

€

10 + 2x = 3x

€

x =10

€

R(10) = 3(10) = 30

€

. . b e quantity: 10 units

€

. . b e price: Php30

4.2 Break-even Analysis4.2 Break-even Analysis

3. Given the cost function C(x) = 0.13x + 750 and revenue function R(x) = 0.88x, find the break-even quantity and price.

€

C(x) = R(x)

€

0.13x + 750 = 0.88x

€

x =1,000

€

R(1,000) = 0.88(1,000) = 880

€

. . b e quantity: 1,000 units

€

. . b e price: Php880

4.2 Break-even Analysis4.2 Break-even Analysis

11. An oven manufacturer claims that the total cost of producing x ovens is described by the function C(x) = 100,000 + 2,000x.

a)What is the fixed cost?b)What is the cost of producing 50 ovens?c) Graph the cost function C(x).

4.2 Break-even Analysis4.2 Break-even Analysis

11. An oven manufacturer claims that the total cost of producing x ovens is described by the function C(x) = 100,000 + 2,000x.

a)What is the fixed cost?

€

F = Php100,000

4.2 Break-even Analysis4.2 Break-even Analysis

11. An oven manufacturer claims that the total cost of producing x ovens is described by the function C(x) = 100,000 + 2,000x.

b)What is the cost of producing 50 ovens?

€

C(50) =100,000 + 2,000(50)

€

=Php200,000

4.2 Break-even Analysis4.2 Break-even Analysis

11. An oven manufacturer claims that the total cost of producing x ovens is described by the function C(x) = 100,000 + 2,000x.

c) Graph the cost function C(x).

sage: plot(100000+2000*x,(0,10),xmin=0,ymin=0)

4.2 Break-even Analysis4.2 Break-even Analysis

13. A certain car rental agency charges Php800 a day plus ten pesos per kilometer. A competitor charges 12 pesos per kilometer. Which one offers the better deal?

€

C1(x) = 800 +10x

€

C2(x) =12x

€

C1(x) =C2(x)

€

800 +10x =12x

€

x = 400

€

C1(399) = Php4,790

€

C2(399) = Php4,788

€

400 .Competitor offers better deal if distance is less than kms

4.2 Break-even Analysis4.2 Break-even Analysis

15. A food repacking company estimates sales figures of Php15.5M for their most popular item. Assuming that the item sells at Php375 each, fixed costs are Php4M, and variable cost are Php215 per unit repacked,

a)find the cost, revenue, and profit functions.b)what is the break-even sales volume?c) find the corresponding profit figures if the

actual sales will be as estimated.

4.2 Break-even Analysis4.2 Break-even Analysis

15. A food repacking company estimates sales figures of Php15.5M for their most popular item. Assuming that the item sells at Php375 each, fixed costs are Php4M, and variable cost are Php215 per unit repacked,

a)find the cost, revenue, and profit functions.

€

C(x) = 4,000,000 + 215x

€

R(x) = 375x

€

P(x) =160x − 4,000,000

4.2 Break-even Analysis4.2 Break-even Analysis

15. A food repacking company estimates sales figures of Php15.5M for their most popular item. Assuming that the item sells at Php375 each, fixed costs are Php4M, and variable cost are Php215 per unit repacked,

b)what is the break-even sales volume?

€

C(x) = R(x)

€

4,000,000 + 215x = 375x

€

x = 25,000 units

4.2 Break-even Analysis4.2 Break-even Analysis

15. A food repacking company estimates sales figures of Php15.5M for their most popular item. Assuming that the item sells at Php375 each, fixed costs are Php4M, and variable cost are Php215 per unit repacked,

c) find the corresponding profit figures if the actual sales will be as estimated.

€

R(x) = 375x =15.5M

€

⇒ x ≈ 41,334

€

P(41,334) =160(41,334) − 4,000,000

€

=Php2,613,440

4.2 Break-even Analysis4.2 Break-even Analysis

17. A manufacturer can sell a certain health supplement for P1,100 per unit. Its total cost consists of a fixed overhead of Php75,000 plus production costs of Php440 per unit.

a)How many units must the manufacturer sell to break even?

b)What is the manufacturer's profit or loss if 100 units are sold?

c) How many units must be sold for the manufacturer to realize a profit of Php12,500?

4.2 Break-even Analysis4.2 Break-even Analysis

17. A manufacturer can sell a certain health supplement for P1,100 per unit. Its total cost consists of a fixed overhead of Php75,000 plus production costs of Php440 per unit.

a)How many units must the manufacturer sell to break even?

€

C(x) = R(x)

€

440x + 75,000 =1,100x

€

x ≈114 units

4.2 Break-even Analysis4.2 Break-even Analysis

17. A manufacturer can sell a certain health supplement for P1,100 per unit. Its total cost consists of a fixed overhead of Php75,000 plus production costs of Php440 per unit.

b)What is the manufacturer's profit or loss if 100 units are sold?

€

P(x) = R(x) −C(x)

€

=660x − 75,000

€

P(100) = 660(100) − 75,000

€

=−9,000

€

This is a loss of Php9,000.

4.2 Break-even Analysis4.2 Break-even Analysis

17. A manufacturer can sell a certain health supplement for P1,100 per unit. Its total cost consists of a fixed overhead of Php75,000 plus production costs of Php440 per unit.

c) How many units must be sold for the manufacturer to realize a profit of Php12,500?

€

12,500 = 660x − 75,000

€

x ≈133 units

4.2 Break-even Analysis4.2 Break-even Analysis

19. As part of their course requirement, a group of students bakes pastries and sells them. The rental for the oven is Php700 for a day and the baking costs add to roughly Php25 per piece. The pastries can be sold for Php45 apiece.

a)How many pieces must the students sell to break even?

b)How many pieces must the students sell to make a profit of Php5,000?

4.2 Break-even Analysis4.2 Break-even Analysis

19. As part of their course requirement, a group of students bakes pastries and sells them. The rental for the oven is Php700 for a day and the baking costs add to roughly Php25 per piece. The pastries can be sold for Php45 apiece.

a)How many pieces must the students sell to break even?

€

C(x) = R(x)

€

25x + 700 = 45x

€

x = 35 pieces

4.2 Break-even Analysis4.2 Break-even Analysis

19. As part of their course requirement, a group of students bakes pastries and sells them. The rental for the oven is Php700 for a day and the baking costs add to roughly Php25 per piece. The pastries can be sold for Php45 apiece.

b)How many pieces must the students sell to make a profit of Php5,000?

€

P(x) = R(x) −C(x)

€

=20x − 700

€

5,000 = 20x − 700

€

⇒ x = 285 pieces