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# Business Mathematics Jerome Chapter 05

## on Jan 24, 2014

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## Business Mathematics Jerome Chapter 05Presentation Transcript

• Linear Linear 5-1 Equations Equations Apps. Apps. Applications of Linear Equations McGraw-Hill Ryerson©
• 5-2 Linear Linear Equations Equations Apps. Apps. Learning Objectives After completing this chapter, you will be able to: LO 1. LO 1. Solve two linear equations with two variables LO 2. LO 2. Solve problems that require setting up linear equations with two variables Also Also McGraw-Hill Ryerson©
• 5-3 Linear Linear Learning Objectives Equations Equations Apps. Apps. LO 3. LO 3. Perform linear Cost-Volume-Profit and break-even analysis employing: A. A. B. B. McGraw-Hill Ryerson© - The contribution margin approach - The algebraic approach of solving the cost and revenue functions
• Linear Linear Equations Equations Apps. Apps. LO 1. LO 1. 5-4 Solving Two Equations with Solving Two Equations with Two Unknowns Two Unknowns Equations Equations 2x – 3y = – 6 x+ y = 2 (A) Solve for y (B) Solve for x (A) Solve for y 2x – 3y = – 6 Multiply by 2 x y = 2x + 2y = 4 - 5y = -10 Subtract y = 2 Divide by -5 y = 2 (B) Solve for x 2x2x 3(2) = – 6 – – 3y = – 6 Substitute y = 2 2x – 6 = – 6 2x = + 6 – 6 Check… x =0 x =0 Check… McGraw-Hill Ryerson©
• 5-5 Linear Linear Solving Two Equations with Solving Two Equations with Two Unknowns Two Unknowns Equations Equations Apps. Apps. 2x – 3y = – 6 x+ y = 2 You should always check your answer by substituting the values into each of the equations! x =0 y = 2 x =0 y = 2 Equations Equations Left Side = Right Side Equation 1 Equation 1 Left Side Right Side = 2x – 3y = 2(0) – 3(2) =–6 =–6 LS = RS McGraw-Hill Ryerson© Equation 2 Equation 2 Left Side Substituting Right Side =x+ y =0+2 =2 =2 LS = RS
• Linear Linear Equations Equations Apps. Apps. LO 2. LO 2. McGraw-Hill Ryerson© 5-6
• Linear Linear 5-7 Equations Equations Apps. Apps. York Daycare purchases the same amount of milk and orange juice each week. After price increases from \$1.10 to \$1.15 per litre for milk, and from \$0.98 to \$1.14 per can of frozen orange juice, the weekly bill rose from \$84.40 to \$91.70. How many litres of milk and cans of orange juice are purchased each week? McGraw-Hill Ryerson©
• Linear Linear 5-8 Equations Equations Apps. Apps. Purchases Purchases Let x = # litres of milk Let y = # cans of orange juice Let x = # litres of milk Let y = # cans of orange juice A. A. B. B. C. C. After price increases from \$1.10 to \$1.15 per litre of milk, and from \$0.98 to \$1.14 per can of frozen orange juice, the weekly bill rose from \$84.40 to \$91.70. McGraw-Hill Ryerson© Equations Equations Development of… (1) 1.10x + 0.98y = 84.40 (2) 1.15x + 1.14y = 91.70 Solving… Solving…
• 5-9 Linear Linear Equations Equations Apps. Apps. x litres of milk x ==##litres of milk Let Let Eliminate x Eliminate x by Dividing by Dividing by 1.10 by 1.10 Eliminate x Eliminate x by Dividing by Dividing by 1.15 by 1.15 McGraw-Hill Ryerson© Let y = # cans of orange juice Let y = # cans of orange juice Equation Equation (1) 1.10x + 0.98y = 84.40 (1.10x + 0.98y)/1.10 = 84.40/1.10 x + 0.8909y = 76.73 Equation Equation (2) 1.15x + 1.14y = 91.70 (1.15x + 1.14y)/1.15 = 91.70/1.15 x + 0.9913y = 79.74 …continue …continue
• 5 - 10 Linear Linear Equations Equations Apps. Apps. Equation Equation Equation Equation (1) (2) x + 0.8909y = 76.73 x + 0.9913y = 79.74 .1004y = 3.01 Subtract y = 29.98 i.e. 30 cans Substitute Equation (1) 1.10x + 0.98y = 84.40 Equation into 1.10x + 0.98(29.98) = 84.40 1.10x + 29.38 = 84.40 1.10x = 84.40 - 29.38 1.10x = 55.02 x = 50.02 i.e. 50 litres McGraw-Hill Ryerson©
• 5 - 11 Linear Linear Equations Equations Apps. Apps. Quantity Price \$ Litres of Milk 50 \$1.15 \$57.50 Cans of Orange Juice 30 1.14 34.20 = New Weekly Cost to Purchase McGraw-Hill Ryerson© \$91.70
• 5 - 12 Linear Linear Equations Equations Apps. Apps. LO 3. LO 3. Cost Analysis McGraw-Hill Ryerson©
• Linear Linear Equations Equations Apps. Apps. Terminology 5 - 13 Fixed Costs Fixed Costs Business Costs Business Expenses McGraw-Hill Ryerson© …do NOT change if sales increase or decrease e.g. rent, property taxes, some forms of depreciation Variable Costs Variable Costs …do change in direct proportion to sales volume e.g. material costs, direct labour costs
• Linear Linear Equations Equations Apps. Apps. Terminology Break Even Point … is the point at which neither a Profit Loss is made McGraw-Hill Ryerson© or 5 - 14
• Linear Linear Equations Equations Apps. Apps. Terminology 5 - 15 Contribution Margin …is the dollar amount that is found by …is the dollar amount that is found by deducting ALL Variable deducting ALL Variable Costs from Net Costs from Net Sales Sales and ‘contributes’ to meeting and ‘contributes’ to meeting Fixed Costs and making Fixed Costs and making aa Contribution Rate ‘Net Profit’. ‘Net Profit’. …is the dollar amount expressed as …is the dollar amount expressed as a percent (%) of Net Sales a percent (%) of Net Sales A Contribution Margin statement McGraw-Hill Ryerson©
• Linear Linear Equations Equations Apps. Apps. Terminology 5 - 16 A Contribution Margin Statement Net Sales(Price * # Units Sold) Less: Variable Costs Margin Contribution Less: Fixed Costs Net Income McGraw-Hill Ryerson© \$ x % 100 x x x x x x x x
• Linear Linear Equations Equations Apps. Apps. Scenario 1 5 - 17 Market research for a new product indicates that the product can be sold at \$50 per unit. Cost analysis provides the following information: Fixed Costs per period = \$8640 Variable Costs = \$30 per unit. Production Capacity per period = 900 units uestion: How much does the sale of an additional unit of a firm’s product contribute towards increasing its net income? McGraw-Hill Ryerson©
• 5 - 18 Linear Linear Equations Equations Apps. Apps. Formulae Formulae - To Find - Contribution Margin Contribution Rate *Break Even Point: ...in Units (x) ...in Sales \$ ...in % of Capacity CM = S - VC CR = CM/S * 100% x = FC / CM \$x = (FC / CM)* S BEPin Units/PC*100 * At Break Even, Net Profit or Loss = 0 Applying McGraw-Hill Ryerson© Formulae Formulae
• Linear Linear Equations Equations Apps. Apps. A 5 - 19 Formulae pplying the Formulae As in the previous scenario, the new product can be sold at \$50 per unit. Costs are as follows: Fixed Costs are \$8640 for the period , Variable Costs are \$30 per unit, and the Production Capacity is 900 units per period. CM = S - VC CR = CM/S * 100% Break Even Point: Units x = FC / CM In \$ x = (FC / CM)* S BEPin units PC*100 McGraw-Hill Ryerson© = \$50 - \$30 = \$20 = \$20/\$50 * 100 = 40% = \$8640/\$20 = 432 Units = (\$8640/\$20)* \$50 = \$21,600 = 432/ 900*100 = 48% of Capacity
• Linear Linear Equations Equations Apps. Apps. Scenario 2 5 - 20 The Lighting Division of Seneca Electric Co. plans to introduce a new street light based on the following accounting information: FC = \$3136 VC = \$157. S= \$185 Capacity = 320 units uestion: Calculate the breakeven point (BEP) …in units …in dollars …as a percent of capacity McGraw-Hill Ryerson©
• Linear Linear Equations Equations Apps. Apps. Scenario 2 5 - 21 FC = \$3136 VC = \$157. S= \$185 Capacity = 320 units …in units Break Even Point = FC / CM = \$3136/ 28 = 112 Units S – VC = CM S – VC = CM \$185 – 157 = \$28 \$185 – 157 = \$28 …in dollars = (FC / CM)* S = (\$3136/ 28) * \$185 = \$20720 …as a percent of capacity = BEPin units/PC*100 = 112/320 * 100 = 35% of Capacity McGraw-Hill Ryerson©
• 5 - 22 Linear Linear Scenario 2 -1 Equations Equations Apps. Apps. FC = \$3136 VC = \$157. S= \$185 Capacity = 320 units \$2688 Determine the BEP as a % of capacity if FC are reduced to \$2688. Formula Formula Step 1… Find CM S = \$185 VC = - 157 CM \$ 28 McGraw-Hill Ryerson© =BEPin units/PC*100 Step 2… Find Step 3… Find BEP in units = FC/CM = \$2688/ \$28 = 96 Units % of Capacity =BEPin units /PC*100 = 96/320*100 = 30% of Capacity
• 5 - 23 Scenario 2 -2 Linear Linear Equations Equations Apps. Apps. FC = \$4588 VC = \$157 S= \$185 Capacity = 320 units \$3136 \$148 VC =S*80% = \$148 Determine the BEP as a % of capacity if FC are increased to \$4588, and VC reduced to 80% of S. Formula Formula Step 1… Find CM S = \$185 VC = - 148 CM \$ 37 McGraw-Hill Ryerson© = BEPin units /PC*100 Step 2… Find Step 3… Find BEP in units = FC/CM = \$4588/ \$37 = 124 Units % of Capacity =BEPin units /PC*100 = 124/320*100 = 39% of Capacity
• 5 - 24 Scenario 2 -3 Linear Linear Equations Equations Apps. Apps. FC = \$3136 VC = \$157 S= \$185 Capacity = 320 units \$171 Determine the BEP as a % of capacity if S is reduced to \$171. Formula Formula Step 1… Find CM S = \$ 171 VC = -157 CM \$ 14 McGraw-Hill Ryerson© = BEPin units /PC*100 Step 2… Find Step 3… Find BEP in units = FC/CM = \$3136/ \$14 = 224 Units % of Capacity =BEPin units /PC*100 = 224/320*100 = 70% of Capacity
• 5 - 25 Scenario 2 -4 Linear Linear Equations Equations Apps. Apps. FC = \$3136 VC = \$157 S= \$185 Capacity = 320 units Determine the NI if 134 units are sold! Formula Formula Step 1… Find CM S = \$185 VC = - 157 CM \$ 28 CM of \$28 per unit McGraw-Hill Ryerson© NI = #Units above BEP*CM Step 2… Find BEP in units = FC/CM = \$3136/\$28 = 112 Units Units Sold 134 BEP 112 Over BEP 22 Company had a NI of 22* \$28 = \$616.
• 5 - 26 Scenario 2 -5 Linear Linear Equations Equations Apps. Apps. FC = \$3136 VC = \$157 S= \$185 Capacity = 320 units What unit sales will generate NI of \$2000? Formula Formula Step 1… Find CM S = \$185 VC = - 157 CM \$ 28 CM of \$28 per unit McGraw-Hill Ryerson© #Units above BEP = NI/CM Step 2… Find BEP in units = FC/CM = \$3136/\$28 = 112 Units NI/CM NI/CM = \$2000/\$28 per Unit = 72 Units above Break Even 72 Units + 112 BEP Units 72 Units + 112 BEP Units = Total Sales Units = 184 = Total Sales Units = 184
• 5 - 27 Scenario 2 -6 Linear Linear Equations Equations Apps. Apps. FC = \$3136 VC = \$157 S= \$185 Capacity = 320 units What are the unit sales if there is a Net Loss of \$336? Formula Formula Step 1… Find CM S = \$185 VC = - 157 CM \$ 28 CM of \$28 per unit McGraw-Hill Ryerson© # Units below BEP = (NI)/CM Step 2… Find BEP in units = FC/CM = \$3136/\$28 = 112 Units (NI)/CM (NI)/CM = (\$336)/\$28 per Unit = 12 Units below Break Even 112BEP --12 Units Below BEP 12 Units Below 112 = Total Sales Units = 100 = Total Sales Units = 100
• 5 - 28 Scenario 2 -7 Linear Linear Equations Equations Apps. Apps. FC = \$3136 VC = \$157 S= \$185 Capacity = 320 units 272 The company operates at 85% capacity. Find the Profit or Loss. Step 1… Find CM S = \$185 VC = - 157 CM \$ 28 Step 2… Find BEP in units = FC/CM = \$3136/\$28 = 112 Units CM of \$28 per unit Formula Formula 320*.85 320*.85 = 272 = 272 Units Units Production 272 Production 272 BEP 112 BEP 112 Over BEP 160 Over BEP 160 # units above BEP *CM = NI 160 Units * \$28 = Profit \$4480 160 Units * \$28 = Profit \$4480 McGraw-Hill Ryerson©
• Linear Linear Equations Equations Apps. Apps. Case 5 - 29 The Marconi Co. year end operating results were as follows: Total Sales of \$375000 Operated at 75% of capacity Total Variable Costs were \$150000 Total Fixed Costs were \$180000 What was Marconi’s BEP expressed in dollars of sales? McGraw-Hill Ryerson©
• Linear Linear Equations Equations Apps. Apps. 5 - 30 Case The Marconi Co. year end operating results were as follows: Total Sales of \$375000 Operated at 75% of capacity Total Variable Costs were \$150000 Total Fixed Costs were \$180000 What was Marconi’s BEP expressed in dollars of sales? What information is needed to calculate the \$BEP? What information is needed to calculate the \$BEP? 1. Number 1. Number of of Units sold Units sold McGraw-Hill Ryerson© 2. VC 2. VC per per Unit Unit 3. CM 3. CM 4. Total 4. Total Costs Costs 5. BEP 5. BEP in \$ in \$
• Linear Linear Equations Equations Apps. Apps. Case 5 - 31 The Marconi Co. year end operating results were as follows: Total Sales of \$375000 Operated at 75% of capacity Total Variable Costs were \$150000 Total Fixed Costs were \$180000 What was Marchoni’s BEP expressed in dollars of sales? 1. Number 1. Number of of Units sold Units sold 2. VC 2. VC per per Unit Unit 3. CM 3. CM McGraw-Hill Ryerson© Let S = \$1 and X be the Number of \$1 Units sold Sales of \$375 000 = 375000 Total Units sold Total VC = Total Unit Sales S \$1.00 VC .40 CM \$ .60 \$150000 375000 = \$0.40pu
• Linear Linear Equations Equations Apps. Apps. Case 5 - 32 The Marconi Co. year end operating results were as follows: Total Sales of \$375000 Operated at 75% of capacity Total Variable Costs were \$150000 Total Fixed Costs were \$180000 What was Marchoni’s BEP expressed in dollars of sales? 4. Total 4. Total Costs Costs 5. BEP 5. BEP in \$ in \$ McGraw-Hill Ryerson© TC = FC + VC = \$180 000 + 0.40X \$BEP = (FC/CM)*S = (\$180000/0.60)*\$1.00 # Of Units = (300000)*\$1.00 # Of Units = \$300000 \$BEP
• Linear Linear 5 - 33 Equations Equations Apps. Apps. This completes Chapter 5 McGraw-Hill Ryerson©