Chapter Financial arithmetic 17 Break-even analysis The success or failure of any business enterprise can be expressed mathematically as follows: P = R C or L = C R where: P = profit made by a business L = loss made by a business R = revenue (takings, income): money received by a company for the sale of goods and/or services C = cost (expenses, overheads) If the revenue, R, is greater than the cost, C, a business is said to be profitable or is making a profit, P. However, if the revenue, R, is less than the cost, C, the business is carrying a loss, L. In the case where there is neither a profit nor a loss, we say the revenue is equal to the cost. Such a point at which the amount made on sales equals the total costs is called the break-even point. For an enterprise it is very important to estimate its expected business costs (preferably, before starting the business!) to decide whether sales levels will be high enough to make the business viable. Hence, the aim of the break-even analysis is to determine the number of items which must be produced and sold in order to cover the expenses of the business. Regardless of the type of business activity a firm is involved in, there are two main types of costs: fixed and variable. 1. Fixed costs are those which remain unchanged irrespective of changes in the quantity of production. These include rates, rent, building maintenance, interest on loans (if any), insurance premiums, depreciation of the tools, machinery and equipment, and administrative costs. 2. Variable costs are those which change in direct proportion to production. These include purchases, transport and sales commissions. A break-even analysis performed in order to find the break-even point may involve a graphical method and an algebraic method. A graphical analysis involves graphing both the cost and $ Break-even point revenue equations, each of which produces a straight line. Cost function Profit Their point of intersection is the break-even point. The graph Loss at right represents a break-even situation. Revenue function The graph illustrates that initially the costs are greater than Fixed costs the revenue. This is represented by the brown shaded region and is called a loss. The two lines intersect at the break-even point 0 Production units n indicating the costs and revenue are equal. Finally the graph illustrates that after the break-even point the revenue is greater than the costs. This is represented by the green shaded region and is called a profit. An algebraic analysis involves solving the cost equation and revenue equation simultaneously to determine the number of units which must be produced and sold to reach the break-even point.
18 General Mathematics WORKED Example A manufacturer of bookshelves sells them at an average cost of $110 each. The cost of production is made up of two parts: weekly fixed costs of $1000 and a variable cost of $25 for each bookshelf. Algebraically and graphically, find the weekly production of bookshelves required to reach the break-even point. Algebraic method 12 THINK WRITE 1 2 4 5 6 7 Define each of the variables. Write down the equation which defines the cost. Note: The cost of making one bookshelf contains the fixed amount of $1000 and a variable cost of $25 per bookshelf manufactured. Write down the equation which defines the revenue. Note: The revenue comes from selling bookshelves for $110 each. Equate the cost and revenue equations. Note: In order to break even, costs must equal the revenue. Solve for x. Subtract 25x from both sides of the equation. Divide both sides of the equation by 85. Let x = the required number of bookshelves to be produced weekly C = the total cost of making bookshelves weekly R = the cost of selling bookshelves C = fixed cost + variable cost C = 1000 + 25x R = 110x To break even: C = R 1000 + 25x = 110x 1000 + 25x 25x = 110x 25x 1000 = 85x 1000 85x ----------- = -------- 85 85 1000 ----------- 85 Evaluate. x = 11.76 Round up the value to the nearest 12 bookshelves must be produced in order to whole number and answer the question. reach the break-even point. Note: x represents the number of bookshelves and hence must be an integer value. = x
Chapter Financial arithmetic 19 Graphical method THINK 1 Rule up a set of axes on graph paper. Label the origin. Label the horizontal axis as x, the number of bookshelves sold and scale appropriately. Label the vertical axis as $ and scale appropriately. 2 Graph the cost equation. Choose two different x-values, substitute them into the equation and obtain the corresponding C values. Note: In order to draw any straight line graph, two points are required. Plot the points and connect them with a straight line. Graph the revenue equation. Choose two different x-values, substitute them into the equation and obtain the corresponding C values. Plot the points and connect them with a straight line. 4 Label the point of intersection as the break-even point. Draw a dotted vertical line from the break-even point to the x-axis. Read the value from the x-axis. Answer the question. WRITE 5 $ 6 7 C = 1000 + 25x Let x = 0 C = 1000 + 25 0 = 1000 + 0 = 1000 (0, 1000) Let x = 10 C = 1000 + 25 10 = 1000 + 250 = 1250 (10, 1250) R = 110x Let x = 0 R = 110 0 = 0 (0, 0) Let x = 10 R = 110 10 = 1100 (10, 1100) 1200 1000 800 600 400 200 C = 1000 + 25x Costs Break-even point (11.76, 1294) Revenue R = 110x 0 2 4 6 8 10 12 14 x Number of bookshelves sold The dotted line intersects the x-axis at approximately 11.8. Therefore, 12 bookshelves must be produced in order to reach the break-even point. When working with these types of problem, we may easily check that the answers obtained in either the algebraic or graphical methods are correct by entering the two equations into a graphics calculator and solving. To find the break-even point graphically: 1. Enter the equations in the Y= menu. 2. Enter WINDOW settings from (for example) 0 to 20 for X, and 0 to for Y.. Use 2nd [CALC] and 5:Intersect to find the break-even point.
140 General Mathematics remember remember 1. The success (or failure) of any business enterprise can be expressed mathematically as follows: P = R C (or L = C R). 2. The aim of the break-even analysis is to determine the number of items which must be produced and sold in order to cover the expenses of the business.. A break-even analysis to find the break-even point may involve a graphical or an algebraic method. Graphical Method 1. Graph the costs equation. 2. Graph the revenue equation.. Obtain their point of intersection, the break-even point. Algebraic Method 1. Equate the costs equation and the revenue equation. 2. Solve for the unknown variable.. Interpret the result. G Break-even analysis Mathcad SkillSHEET Break even analysis.. WORKED Example 12 1 Darryl and Susan want to start a small pizza restaurant and plan to sell pizzas at an average price of $8.50. They estimate their fixed monthly cost as $1800 and an average variable cost of $.60 for each pizza sold. Find: a graphically, and b algebraically the monthly pizza sales required for this new business to break even. Verify your results using a graphics calculator. 2 Polina and Ric set up a florist business. They rent premises for $180 per week with other outgoings (telephone, advertising, water, electricity etc.) of $65 per week. They buy flowers from wholesalers for an average price of $1.40 and sell them for $2.10. a How many flowers do they need to sell each week to break even? b Find the profit or loss Polina and Ric will make, selling: i 200 flowers per week ii 00 flowers per week iii 500 flowers per week iv 850 flowers per week Hot Hit, a company specialising in musical video clip production, charges on average $65 000 for a 2-minute clip. The company incurs annual fixed costs of $20 000 and variable costs of $18 000 per clip. Find: a the number of musical clips to be sold before they break even b the profit or loss associated with production and sale of: i 2 musical clips ii 4 musical clips iii 6 musical clips iv 7 musical clips v 10 musical clips vi 1 musical clips.
Chapter Financial arithmetic 141 4 The Richardsons like holidaying around Australia using rented cars. They usually hire from two companies: Rent-a-bomb and Chic Lemon. The same family sedan can be hired from Rent-a-bomb for $59 per day flat rate with unlimited kilometres or from Chic Lemon for $42 per day plus a fee of 8c per kilometre over the first 100 km driven. The Richardsons are planning to rent the car for 10 days, and to travel approximately 2600 km. a b d e Which company should they choose? Which company should they choose if they are planning to rent the car for: i 8 days, travelling a distance of 1800 km ii 12 days, travelling a distance of 000 km iii 15 days, travelling a distance of km iv 7 days, travelling a distance of 1500 km v 5 days, travelling a distance of 1600 km A R = + ----------- n B R = ----------- n C R = + ----------- n D R = 4000 + ----------- n E R = ----------- n The cost equation is: 4000 4000 A R = + ----------- n B R = ----------- C R = + ----------- n D R = 4000 + ----------- n E R = ----------- n $( 000) 6 C(n) Question 5 refers to the graph at right which shows the monthly income, R, and costs, C, for a company producing widgets. 5 4 2 1 5 multiple choice a The monthly fixed costs are: 1 2 4 5 6 7 A $0 B $1000 C $1 D $ E $2 b The revenue when the company breaks even is: A $1000 B $ C $000 D $4000 E $5000 c The revenue equation is: If the company produced 4 widgets, it made a profit closest to: A $5000 B $4000 C $000 D $ E $1000 6 Peter, Mary, Melissa and Robert are in charge of finding a venue for the annual Petrou- Selo Reunion. They find that the cost of the venue depends on two factors: one is fixed, the other variable. The hiring of the hall including a band and lighting is fixed at $650, while the total cost of food and beverages depends on the number of people who attend, and is thus variable. They decide this cost to be $15 per head. Finally, the cost of the tickets will be $25 per head. a Write the above information as two equations. b Graph the two equations and obtain the break-even point. c Determine how many people must attend the reunion to break even. d Comment on the results obtained in parts b and c. e What will occur if 50 people attend the reunion? f What will occur if 70 people attend the reunion? R(n) 4000 4000 n
142 General Mathematics summary Simple interest Simple interest is given by I = PRT ----------- where 100 I = interest, $ P = principal, $ R = rate of interest p.a., % T = term of interest, years The total amount is given by A = P + I. When calculating simple interest, the interest earned is the same for each time period. Compound interest r Compound interest formula is given by A = P 1 + -------- n 100 where A = amount at the end of n compounding periods, $ P = principal, $ R = rate of interest per period n = number of compounding periods When dealing with compound interest, the interest is calculated on the principal as well as the interest over each time period. If solving equations where the unknown is a power, take the logarithm to base 10 of both sides of the equation, that is, a n = b log 10 a n = log 10 b nlog 10 a = log 10 b Straight-line depreciation Straight-line depreciation allocates an equal amount of depreciation to each time period over the asset s useful life. The book value of an item at a specific time is defined as the difference between the purchase price and the accumulated depreciation at that time. RT The book value is given by B.V. = P 1 --------. 100 P S The depreciation rate is given by R = ----------- 100 np where P = purchase price, $ R = depreciation rate p.a., % T = period of depreciation, years S = scrap value, $ n = number of time periods, years Straight-line depreciation may be illustrated graphically as a straight line with a negative gradient. Reducing-balance depreciation Reducing-balance depreciation involves a fixed percentage applied to an amount (book value) changing at the beginning of each time period. n = log 10 b -------------- log 10 a
Chapter Financial arithmetic 14 R The book value is given by B.V. = P 1 -------- 100 where P = purchase price, $ R = allowed annual rate of depreciation, % n = number of depreciation periods, years Reducing-line depreciation may be illustrated graphically by an exponential curve. Unit cost depreciation P S U.C.D. = ----------- E where U.C.D. = unit cost depreciation, $/unit P = purchase price, $ S = scrap value, $ E = expected life of asset, number of production units Hire-purchase, flat rate and real rate of interest Flat rate is the simple rate of interest charged on the original sum borrowed. Real or effective rate is the rate of interest being paid on the average principal outstanding. 2400I R ef = --------------------- Pm ( + 1) 2Rn = ----------- n + 1 where R ef = real or effective rate of interest p.a., % I = total interest paid on loan, $ P = total principal owed, $ m = number of monthly instalments R = flat rate of interest p.a., % n = total number of instalments to be made Comparing the flat rate of interest, R, with the effective interest rate, R ef, we see that R ef 2R. Break-even analysis The break-even point is a point at which the level of sales equals the total costs, R = C. P = R C or L = C R where P = profit made by a business L = loss made by a business R = revenue (sales, income) C = cost (expenses, overheads) A break-even analysis to find the break-even point may involve a graphical or an algebraic method. Graphical method 1. Graph the costs equation. 2. Graph the revenue equation.. Obtain their point of intersection, the break-even point. Algebraic method 1. Equate the costs equation and the revenue equation. 2. Solve for the unknown variable.. Interpret the result. n $ Break-even point Cost function Profit Loss Revenue function Fixed costs 0 Production units n
144 General Mathematics CHAPTER review Multiple-choice A 1 Anthony earned $1016 in simple interest when he invested $19 800 for 9 months. The rate of simple interest was: A 5.1% B 6.14% C 6.84% D 7.62% E 8.21% A 1 B B C C D D E 2 With an interest rate of 4.11% p.a., the sum of $64 was earned in simple interest over 2 -- years. The amount invested was close to: 2 A $6170 B $4892 C $706 D $297 E $5607 The Farmbrand dozen eggs (size 67) costs $2.40. With an inflation rate expected to average.5% over each of the next 4 years, the dozen eggs will then cost: A $2.50 B $2.60 C $2.70 D $2.75 E $2.80 4 An investment of $12 000 at 6.15% compounded quarterly to reach $15 000 will take close to: A years B 4 years C 5 years D 6 years E 7 years 5 A laser printer is purchased for $790. It has an expected lifetime of 6 years and zero residual (scrap) value. The amount of depreciation to be allowed per year, in dollars, assuming a straight-line depreciation, is: A $790 B $212 C $06.67 D $98.7 E $11.67 6 Equipment worth $18 600 is bought for a snowboard workshop. It depreciates at 12.5% p.a. constant depreciation. With zero scrap value, the book value of the equipment after 5 years is expected to be: A $11 625 B $6975 C $8114 D $5678 E $716 7 A new computer costs $490. If depreciation is calculated at % p.a. (reducing balance), then the computer s book value at the end of 5 years will be close to: A $471 B $509 C $42 D $567 E $288 8 A piece of equipment, originally worth $49 600, diminishes at a rate of 14.5% p.a. The owner decides to replace the equipment when its book value falls below $5000. The time passing before the next replacement is required is close to: A 10 years B 12 years C 14 years D 15 years E 16 years 9 The unit cost depreciation for a machine bought for $11 40 with scrap value of $750, and designed for 48 000 hours of operation is close to: A 6 c B 12 c C 22 c D 5 c E 49 c
Chapter Financial arithmetic 145 10 A company car purchased for $28 500 and depreciating at an average rate of 47.4 c/km has travelled 2 600 km in the first year and 20 400 km in the second year. In its third year, its book value will be: A $20 856 B $11 186 C $7644 D $9670 E $18 61 11 A hire-purchase agreement on a loan of $000 requires 24 monthly payments. The effective annual rate interest is 17%. The interest paid, in dollars, is close to: A $289 B $61 C $482 D $679 E $51 12 A new cooktop worth $95 is bought under hire-purchase with a deposit of $100 and 18 monthly instalments of $50. The flat rate of interest is: A 8.94% B 7.69% C 6.4% D 5.19% E 4.88% Questions 1 and 14 refer to the graph. The graph shows the quarterly income and costs for a company producing gradgets. 1 The quarterly fixed costs are: A $0 B $.50 C $500 D $.5 10 5 E $5 10 5 14 The revenue when the company breaks even is: A $5 10 5 B $5 10 C $500 D $.5 10 5 E $.5 Short answer $ 10 5 6 5 4 2 1 n 1000 (Gradgets) 1 An investment in BT bonds of $18 000 over -- years earned Rebecca the same amount as 2 $21 600 invested at 7.5% in a term deposit for 5 years. Calculate the interest rate offered by BT bonds. 2 Find: a the simple interest on $2900 for 8 months at 5.85% p.a. b the annual rate of simple interest so that $1600 will amount to $ after years c the number of years over which $5000 will amount to $6000 at 6.5% simple interest d the sum invested at 8% p.a. simple interest if it amounted to $4000 after 4 years. Karen borrowed $1100 from a Credit Union at 8.55% per annum. She repaid the loan in a lump sum when the principal and interest amounted to $1400. How long, to the nearest year, did she keep the money? 4 Find each of the following and compare with the results from question 2. a The amount owing on $2900 for 8 months at 5.85% p.a. compounded daily. b The annual rate of compound interest so that $1600 will amount to $ after years. c The number of years over which $5000 will amount to $6000 at 6.5% p.a. compounded quarterly. d The sum invested at 8% p.a. compound interest if it amounted to $4000 after 4 years. 1 0 R(n) 1 2 4 5 6 7 C(n) F F G G A A B B
146 General Mathematics C D E F G test yourself CHAPTER 5 Machinery bought for $29 600 is expected to have a scrap value of $1200 after 10 years. a Assuming straight-line depreciation, find the yearly depreciation charge. b Find the amount by which depreciation at the end of the second year exceeds the depreciation at the end of the sixth year. 6 What was the original cost of an item which has a book value of $644 after 7 years, if the depreciation rate is 18% p.a.? 7 A photocopier is purchased for $11 400 and depreciates at an average rate of 2.1 10 c per copy. a Calculate the book value of the photocopier after 4 10 8 copies were made. b After how many copies will the photocopier be written off (that is, have a zero scrap value)? 8 A new refrigerator is bought under a hire-purchase agreement for $00 with $00 deposit and 24 monthly instalments of $15 each to be made. Calculate: a the amount of interest charged b the flat rate of interest c the effective rate of interest. 9 A manufacturer sells wholesale its filing cabinets at $85 each. The variable cost of each filing cabinet is $8 and the annual fixed costs are $68 400. Find the break-even point. Analysis 1 Michelle is planning to travel as a backpacker around Australia on her summer holiday next year. She estimates that she needs to save $1400 over the next 12 months. She has already saved $1095 and is considering two options. Option A a Buying government bonds paying 9.85% p.a. i How much interest will Michelle earn on this investment? ii How long will it take for Michelle to save the remaining amount of $05 required for the holiday? Option B b c Opening a Super-Saver investment account which pays 9.55% p.a. compounded daily. i How much interest will Michelle earn on this investment? ii How long will it take for Michelle to save the remaining amount of $05 required for the holiday? Michelle chose Option B. She also realised that she would not have enough money for the trip. Therefore, 65 days later, she saved some more money and was willing to invest it again. Advise Michelle how much she has to add to make $1400 at the end of 12 months. 2 a Ken bought for his business, Ken s Lawn Mowing, a new mower for $4850. It depreciates at the rate of 18% of its value at the start of each year. i Find the value of the equipment at the end of 6 years, correct to the nearest dollar. ii Approximately how much time will pass before the equipment will be worth less than $500? b Ken decided to buy a new utility truck and trailer which totalled $19 890. He traded in his old car and small trailer for $2600 as a deposit. The balance was to be paid on hirepurchase over 4 years, in weekly payments of $115. i How much will Ken pay over the period of 4 years? ii How much interest will Ken pay over that period? iii Calculate the flat rate of interest charged. iv Calculate the effective rate of interest.