Chapter 03 Gen Maths Page 137 Wednesday, December 15,
Chapter 03 Gen Maths Page 137 Wednesday, December 15, 1999 6:31 PMChapter 3 Financial arithmetic137Break-even analysisThe success or failure of any business enterprise can be expressed mathematically asfollows:P R CorL C Rwhere:P profit made by a businessL loss made by a businessR revenue (takings, income): money received by a company for thesale of goods and/or servicesC cost (expenses, overheads)If the revenue, R, is greater than the cost, C, a business is said to be profitable or ismaking a profit, P. However, if the revenue, R, is less than the cost, C, the business iscarrying a loss, L. In the case where there is neither a profit nor a loss, we say therevenue is equal to the cost. Such a point at which the amount made on sales equals thetotal 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 highenough to make the business viable. Hence, the aim of the break-even analysis is todetermine the number of items which must be produced and sold in order to cover theexpenses of the business.Regardless of the type of business activity a firm is involved in, there are two maintypes of costs: fixed and variable.1. Fixed costs are those which remain unchanged irrespective of changes in the quantityof production. These include rates, rent, building maintenance, interest on loans (ifany), insurance premiums, depreciation of the tools, machinery and equipment, andadministrative costs.2. Variable costs are those which change in direct proportion to production. Theseinclude purchases, transport and sales commissions.A break-even analysis performed in order to find the break-even point may involve agraphical method and an algebraic method. Break-even pointA graphical analysis involves graphing both the cost andCostrevenue equations, each of which produces a straight line.functionProfitTheir point of intersection is the break-even point. The graphLossRevenueat right represents a break-even situation.functionThe graph illustrates that initially the costs are greater thanFixed coststhe revenue. This is represented by the brown shaded region and0 Production units nis called a loss. The two lines intersect at the break-even pointindicating the costs and revenue are equal. Finally the graphillustrates that after the break-even point the revenue is greater than the costs. This isrepresented by the green shaded region and is called a profit.An algebraic analysis involves solving the cost equation and revenue equationsimultaneously to determine the number of units which must be produced and sold toreach the break-even point.
Chapter 03 Gen Maths Page 138 Wednesday, December 15, 1999 6:31 PM138General MathematicsWORKED Example 12A manufacturer of bookshelves sellsthem at an average cost of 110 each.The cost of production is made up oftwo parts: weekly fixed costs of 1000and a variable cost of 25 for eachbookshelf. Algebraically andgraphically, find the weekly productionof bookshelves required to reach thebreak-even point.Algebraic methodTHINK1Define each of the variables.2Write down the equation which definesthe cost.Note: The cost of making one bookshelfcontains the fixed amount of 1000 anda variable cost of 25 per bookshelfmanufactured.Write down the equation which definesthe revenue.Note: The revenue comes from sellingbookshelves for 110 each.Equate the cost and revenue equations.Note: In order to break even, costs mustequal the revenue.Solve for x.Subtract 25x from both sides of theequation.345Divide both sides of the equationby 85.67Evaluate.Round up the value to the nearestwhole number and answer the question.Note: x represents the number ofbookshelves and hence must be aninteger value.WRITELet x the required number of bookshelves tobe produced weeklyC the total cost of making bookshelvesweeklyR the cost of selling bookshelvesC fixed cost variable costC 1000 25xR 110xTo break even:C R1000 25x 110x1000 25x 25x 110x 25x1000 85x1000 85x------------ --------85851000------------ x85x 11.7612 bookshelves must be produced in order toreach the break-even point.
Chapter 03 Gen Maths Page 139 Wednesday, December 15, 1999 6:31 PMChapter 3 Financial arithmetic139Graphical methodTHINK12345WRITERule up a set of axes on graph paper.Label the origin. Label the horizontal axisas x, the number of bookshelves sold andscale appropriately. Label the vertical axisas and scale appropriately.Graph the cost equation. Choose twodifferent x-values, substitute them into theequation and obtain the corresponding Cvalues. Note: In order to draw any straightline graph, two points are required.Plot the points and connect them with astraight line.Graph the revenue equation.Choose two different x-values, substitutethem into the equation and obtain thecorresponding C values.Plot the points and connect them with astraight line.Label the point of intersection as thebreak-even point.Draw a dotted vertical line from thebreak-even point to the x-axis.C 1000 25xLet x 0C 1000 25 0 1000 0 1000(0, 1000)Let x 10 C 1000 25 10 1000 250 1250(10, 1250)R 110xLet x 0R 110 0 0(0, 0)Let x 10 R 110 10 1100(10, 1100) 12001000800600400200067Read the value from the x-axis.Answer the question.C 1000 25xCostsBreak-even point(11.76, 1294)RevenueR 110x2 4 6 8 10 12 14 xNumber of bookshelves soldThe dotted line intersects the x-axis atapproximately 11.8. Therefore,12 bookshelves must be produced inorder to reach the break-even point.When working with these types of problem, we may easily check that the answersobtained in either the algebraic or graphical methods are correct by entering the twoequations 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 20for X, and 0 to 2000 for Y.3. Use 2nd [CALC] and 5:Intersect to find the break-even point.
Chapter 03 Gen Maths Page 140 Wednesday, December 15, 1999 6:31 PM140General Mathematicsrememberremember1. The success (or failure) of any business enterprise can be expressedmathematically as follows: P R C (or L C R).2. The aim of the break-even analysis is to determine the number of items whichmust be produced and sold in order to cover the expenses of the business.3. A break-even analysis to find the break-even point may involve a graphical oran algebraic method.Graphical Method1. Graph the costs equation.2. Graph the revenue equation.3. Obtain their point of intersection, the break-even point.Algebraic Method1. Equate the costs equation and the revenue equation.2. Solve for the unknown variable.3. Interpret the llSHEET3.312Break-even analysis1 Darryl and Susan want to start a small pizza restaurant and plan to sell pizzas at anaverage price of 8.50. They estimate their fixed monthly cost as 1800 and anaverage variable cost of 3.60 for each pizza sold. Find:a graphically, andb algebraicallythe 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 withother outgoings (telephone, advertising, water, electricity etc.) of 65 per week. Theybuy 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 weekii 300 flowers per weekiii 500 flowers per weekiv 850 flowers per week3 ‘Hot Hit’, a company specialising in musical video clip production, charges onaverage 65 000 for a 2-minute clip. The company incurs annual fixed costs of 230 000 and variable costs of 18 000 per clip. Find:a the number of musical clips to be sold before they break evenb the profit or loss associated with production and sale of:i 2 musical clipsii 4 musical clipsiii 6 musical clipsiv 7 musical clipsv 10 musical clipsvi 13 musical clips.
Chapter 03 Gen Maths Page 141 Wednesday, December 15, 1999 6:31 PM141Chapter 3 Financial arithmetic4 The Richardsons like holidaying around Australia using rented cars. They usually hirefrom two companies: ‘Rent-a-bomb’ and ‘Chic Lemon’. The same family sedan can behired from ‘Rent-a-bomb’ for 59 per day flat rate with unlimited kilometres or from ‘ChicLemon’ for 42 per day plus a fee of 8c per kilometre over the first 100 km driven. TheRichardsons are planning to rent the car for 10 days, and to travel approximately 2600 km.a Which company should they choose?b Which company should they choose if they are planning to rent the car for:i 8 days, travelling a distance of 1800 kmii 12 days, travelling a distance of 3000 kmiii 15 days, travelling a distance of 2000 kmiv 7 days, travelling a distance of 1500 kmv 5 days, travelling a distance of 1600 km (’000)R(n)Question 5 refers to the graph at right which shows themonthly income, R, and costs, C, for a companyproducing widgets.5 multiple choicea The monthly fixed costs are:A 0B 1000654321C(n)1 2 3 4 5 6 7 nC 1D 2000b The revenue when the company breaks even is:A 1000B 2000C 3000D 4000cE 2E 5000The revenue equation is:A R 2000 2000------------ n3B R 4000------------ n3D R 4000 2000------------ n3E R 2000------------ n3C R 2000 4000------------ n3C R 2000 4000------------ n3d The cost equation is:A R 2000 2000------------ n3B R 4000-----------3D R 4000 2000------------ n3E R 2000------------ n3e If the company produced 4 widgets, it made a profit closest to:A 5000B 4000C 3000D 2000E 10006 Peter, Mary, Melissa and Robert are in charge of finding a venue for the annual ‘PetrouSelo 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 costof 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.abcdefWrite the above information as two equations.Graph the two equations and obtain the break-even point.Determine how many people must attend the reunion to break even.Comment on the results obtained in parts b and c.What will occur if 50 people attend the reunion?What will occur if 70 people attend the reunion?
Chapter 03 Gen Maths Page 142 Wednesday, December 15, 1999 6:31 PM142General MathematicssummarySimple interestPRT Simple interest is given by I ----------- where100I 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 interestr Compound interest formula is given by A P 1 --------- 100 nwhere A amount at the end of n compounding periods, P principal, R rate of interest per periodn number of compounding periods When dealing with compound interest, the interest is calculated on the principal aswell as the interest over each time period. If solving equations where the unknown is a power, take the logarithm to base 10 ofboth sides of the equation, that is,an blog10an log10bnlog10a log10blog 10 bn --------------log 10 aStraight-line depreciation Straight-line depreciation allocates an equal amount of depreciation to each timeperiod over the asset’s useful life. The book value of an item at a specific time is defined as the difference between thepurchase 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 ------------ 100nPwhere P purchase price, R depreciation rate p.a., %T period of depreciation, yearsS scrap value, n number of time periods, years Straight-line depreciation may be illustrated graphically as a straight line with anegative 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.
Chapter 03 Gen Maths Page 143 Wednesday, December 15, 1999 6:31 PMChapter 3 Financial arithmetic143nR 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 depreciationP–SU .C.D. -----------Ewhere U.C.D. unit cost depreciation, /unitP purchase price, S scrap value, E expected life of asset, number of production unitsHire-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.2400IRef ---------------------P(m 1)2Rn -----------n 1where Ref real or effective rate of interest p.a., %I total interest paid on loan, P total principal owed, m number of monthly instalmentsR 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, Ref, we seethat Ref 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 Rwhere P profit made by a businessL loss made by a businessR revenue (sales, income)C cost (expenses, overheads) A break-even analysis to find the break-even point may involve a graphical or analgebraic method.Graphical method Break-even point1. Graph the costs equation.Cost2. Graph the revenue equation.functionProfit3. Obtain their point of intersection, the break-even point.LossRevenueAlgebraic methodfunction1. Equate the costs equation and the revenue equation.Fixed costs2. Solve for the unknown variable.0 Production units n3. Interpret the result.
Chapter 03 Gen Maths Page 144 Wednesday, December 15, 1999 6:31 PM144General MathematicsCHAPTERreviewMultiple-choice3A1 Anthony earned 1016 in simple interest when he invested 19 800 for 9 months. The rateof simple interest was:A 5.13%B 6.14%C 6.84%D 7.62%E 8.21%3A2 With an interest rate of 4.11% p.a., the sum of 634 was earned in simple interest over2 1--2- years. The amount invested was close to:A 6170B 4892C 7306D 2937E 56073B3 The ‘Farmbrand’ dozen eggs (size 67) costs 2.40. With an inflation rate expected toaverage 3.5% over each of the next 4 years, the dozen eggs will then cost:A 2.50B 2.60C 2.70D 2.75E 2.803B4 An investment of 12 000 at 6.15% compounded quarterly to reach 15 000 will take close to:A 3 yearsB 4 yearsC 5 yearsD 6 yearsE 7 years3C5 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 astraight-line depreciation, is:A 790B 212C 306.67D 98.37E 131.673C6 Equipment worth 18 600 is bought for a snowboardworkshop. It depreciates at 12.5% p.a. constantdepreciation. With zero scrap value, the bookvalue of the equipment after 5 years is expectedto be:A 11 625B 6975C 8114D 5678E 73163D7 A new computer costs 3490. If depreciation is calculated at 33% p.a. (reducing balance),then the computer’s book value at the end of 5 years will be close to:A 471B 509C 432D 567E 2883D8 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 yearsB 12 yearsC 14 yearsD 15 yearsE 16 years3E9 The unit cost depreciation for a machine bought for 11 340 with scrap value of 750, anddesigned for 48 000 hours of operation is close to:A 6cB 12 cC 22 cD 35 cE 49 c
Chapter 03 Gen Maths Page 145 Wednesday, December 15, 1999 6:31 PMChapter 3 Financial arithmetic14510 A company car purchased for 28 500 and depreciating at an average rate of 47.4 c/km hastravelled 23 600 km in the first year and 20 400 km in the second year. In its third year, itsbook value will be:A 20 856B 11 186C 7644D 9670E 18 63111 A hire-purchase agreement on a loan of 3000 requires 24 monthly payments. The effectiveannual rate interest is 17%. The interest paid, in dollars, is close to:A 289B 361C 482D 679E 53112 A new cooktop worth 935 is bought under hire-purchase with a deposit of 100 and18 monthly instalments of 50. The flat rate of interest is:A 8.94%B 7.69%C 6.34%D 5.19% 105Questions 13 and 14 refer to thegraph.The graph shows the quarterly incomeand costs for a company producing gradgets.65432113 The quarterly fixed costs are:0A 0B 3.50C 3500B 5 103C 35003FE 4.88%R(n)C(n)1 2 3 4 5 6 7 n 1000 (Gradgets)D 3.5 105E 5 105D 3.5 105E 3.514 The revenue when the company breaks even is:A 5 1053F3G3GShort answer1 An investment in BT bonds of 18 000 over 3 1--2- years earned Rebecca the same amount as 21 600 invested at 7.5% in a term deposit for 5 years. Calculate the interest rate offered byBT bonds.3A2 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 2000 after 3 yearsc the number of years over which 5000 will amount to 6000 at 6.5% simple interestd the sum invested at 8% p.a. simple interest if it amounted to 4000 after 4 years.3A3 Karen borrowed 1100 from a Credit Union at 8.55% per annum. She repaid the loan in alump sum when the principal and interest amounted to 1400. How long, to the nearest year,did she keep the money?3B4 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 2000 after 3 years.c The number of years over which 5000 will amount to 6000 at 6.5% p.a. compoundedquarterly.d The sum invested at 8% p.a. compound interest if it amounted to 4000 after 4 years.3B
Chapter 03 Gen Maths Page 146 Wednesday, December 15, 1999 6:31 PM146General Mathematics3C5 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 thedepreciation at the end of the sixth year.3D6 What was the original cost of an item which has a book value of 6344 after 7 years, if thedepreciation rate is 18% p.a.?3E7 A photocopier is purchased for 11 400 and depreciates at an average rate of 2.1 10 3 cper copy.a Calculate the book value of the photocopier after 4 108 copies were made.b After how many copies will the photocopier be written off (that is, have a zero scrap value)?3F8 A new refrigerator is bought under a hire-purchase agreement for 3300 with 300 depositand 24 monthly instalments of 153 each to be made. Calculate:a the amount of interest chargedb the flat rate of interestc the effective rate of interest.3G9 A manufacturer sells wholesale its filing cabinets at 85 each. The variable cost of each filingcabinet is 38 and the annual fixed costs are 68 400. Find the break-even point.Analysis1 Michelle is planning to travel as a backpacker around Australia on her summer holiday nextyear. She estimates that she needs to save 1400 over the next 12 months. She has alreadysaved 1095 and is considering two options.Option Aa 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 305 required forthe holiday?Option Bb 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 305 required forthe holiday?c Michelle chose Option B. She also realised that she would not have enough money for thetrip. Therefore, 65 days later, she saved some more money and was willing to invest itagain. Advise Michelle how much she has to add to make 1400 at the end of 12 months.CHAPTERtestyourself32 a Ken bought for his business, ‘Ken’s Lawn Mowing’, a new mower for 4850. Itdepreciates 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 hisold 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.
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 revenue equations, each of which produces a straight line. Their point of intersection is the break-even point. The graph at right represents a break-even situation.
Part One: Heir of Ash Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 .
SAU PhD Maths Questions Papers Contents: SAU PhD Maths Que. Paper-2014 SAU PhD Maths Que. Paper-2015 SAU PhD Maths Que. Paper-2016 SAU PhD Maths Que. Paper-2017 SAU PhD Maths Que. Paper-2018 SAU PhD Maths
TO KILL A MOCKINGBIRD. Contents Dedication Epigraph Part One Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Part Two Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18. Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26
DEDICATION PART ONE Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 PART TWO Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 .
Gen 3, 6.5 kV Gen 3, 900 V Gen 2, C2M Family 1.2 kV Gen 1, 1.2 kV Gen 3, 1.2 kV Scaling of State-of-Art Gen-3 SiC Power MOSFETs in R&D RCh/RON becomes larger for lower-V MOSFETs. For Gen-3 1200V MOSFET, RCh 40% of total RON. Future Prospective Reduce RCh/RON by: o Improving MOS INV o Higher packing density
gen-26 davar nanabhoy s practical book keeping & accountancy 12 07/07/1966 gen-27 ghatalia s v practical auditing 15 07/07/1966 gen-28 gupta rup ram advanced accounting 22.5 07/07/1966 gen-29 gupta rup ram auditing 12.5 07/07/1966 gen-30 gupta rup ram text book of auditing 7.5 07/07/1966 gen-31 gupta rup ram cost accounting 9 07/0
SPARC @ Oracle 16 x 2nd Gen cores 6MB L2 Cache 1.7 GHz 8 x 3 rd Gen Cores 4MB L3 Cache 3.0 GHz 16 x 3rd Gen Cores 8MB L3 Cache 3.6 GHz 12 x 3rd Gen 48MB L3 Cache 3.6 GHz 6 x 3 Gen Cores 48MB L3 Cache 3.6 GHz T3 T4 T5 M5 M6 S7 32 x 4th Gen Cores 64MB L3 Cache 4.1 GHz DAX1 M7 8 x 4th Gen Co
Year 7 & 8 Numeracy Workbook. Week Topic AFL 1 Addition 2 Subtraction 3 Mental Maths 4 Multiplication 5 Division 6 Mental Maths 7 BIDMAS 8 Percentages 9 Mental Maths 10 Simplifying Fractions 11 Adding Fractions 12 Mental Maths 13 Fractions-Decimals-Percentages 14 Ratio 15 Mental Maths 16 Collecting Like terms 17 Substitution 18 Vocabulary and Directed Numbers 19 Word Based Puzzle. Week 1 Maths .
