1 This assignment covers material from Units 1 to 4 in the Mathematics 244 Study Guide. It is due after Unit 4 in the Study Guide is completed. Total Marks = 150 1. Evaluate each of the following expressions. a. 3 + (4) (+5) (7) b. [2 3 (5)] [(3 + 2)] c. 56 4(7 5)2 + 17 8 + (7(23 + 5(12 3 8 + 2) 7)) 2 (2 1 x 4 y 3 ) 2 d. 2 1 2 (6 x y ) Business Mathematics (2 marks) (2 marks) (4 marks) (6 marks) 1 2. Simplify each of the following expressions into simplest terms. a. b. 2 {[ x + y + (2y)] [ x y]} [(x y)2 + (x + y)2] (x 3 + 4x 2 y + 4xy 2 + 2xy 6y 2 ) (x + 2y) (4 marks) (8 marks) Mathematics 244 / Assignment 1 3. Factor each of the following expressions completely. a. b. Business Mathematics (a + b)2 (2b)2 6b2 + ab 15a2 (4 marks) (4 marks) 3 4 c. 6a2 ab 15b2 d. 9a4 7a2b2 50b4 (6 marks) (4 marks) Mathematics 244 / Assignment 1 4. Solve for each of the variables in the questions below. a. b. Business Mathematics 8 7x + 3 0 (2 marks) 2(3 x) = 4 1.3x + 2.4y = 39.3 0.6x + 0.3y = 3.6 (8 marks) 5 c. 6 3(x + 3) y + + 2z = 1 5 2 x 5y 2 + +z= 0 3 4 2( y + 1) x 3z = 2 3 (12 marks) Mathematics 244 / Assignment 1 d. Business Mathematics 1 2 5 = (6 marks) x+3 x2 x+3 7 5. Solve each of the equations below for the variable x. a. b. 8 x(x 28) = 245 x 1 3 = x+1 x+3 (8 marks) (8 marks) Mathematics 244 / Assignment 1 6. Business Mathematics A company rents bicycles, motor scooters, and dune buggies. The total number of items in their inventory is 165. There are three times as many bicycles as dune buggies, and the number of motor scooters is four less than three times the number of bicycles. Find how many bicycles, motor scooters, and dune buggies the company has. (6 marks) 9 7. The sales of a company's products appear in the table below. Mallory's Music Stop CD Sales Last Year Category Sales (in Units) Country Rock Heavy Metal 150,000 125,000 75,000 Total 350,000 Source: Hypothetical a. Write a ratio that compares i. Country sales to Rock sales. (2 marks) ii. Rock sales to Heavy Metal sales. (2 marks) iii. Country sales to Heavy Metal sales. b. 10 (2 marks) Write a three-term ratio to compare sales in all three categories. (4 marks) Mathematics 244 / Assignment 1 8. A company's profit last year was $8000, which was 4.1% of sales. What were the company's total sales? (4 marks) 9. The table below shows the projected increase in sales for a company. Mallory's Music Stop Sales Projections Last Year's Sales (in Units) Projected Increase (%) Country Rock Heavy Metal 150,000 125,000 75,000 23 18 12 Total 350,000 Category Projected Sales this Year (in Units) Source: Hypothetical a. b. Business Mathematics Calculate the projected CD sales for this year for each music category. (3 marks) Calculate the overall projected percentage sales increase for Mallory's Music Stop. (3 marks) 11 10. The gross profit of a shoe store is 35% based on selling price. A pair of shoes costs the store $18.59. a. b. What is the selling price? What is the gross profit? (3 marks) (2 marks) 11. If the gross profit rate on the sale of a radio is 28% of the selling price, what is the gross profit rate based on the cost of the radio? (5 marks) 12 Mathematics 244 / Assignment 1 12. An office equipment store manager buys some desk lamps for $68 each. His operating expenses are 33% of the selling price and his net profit is 22% of the selling price. At what price should he sell the lamps? (5 marks) 13. A store sells an item it originally marked up 30% of the cost price, at a discount of 58% of the selling price. What percent of the cost is the resulting profit or loss on this sale? (6 marks) Business Mathematics 13 14. A clothing retailer purchased a line of fall leather coats which were priced to sell at $600 each. This price reflected a markup of 45% on the selling price. At the end of the season the retailer had three coats left, which were marked down 25% and sold. What was the retailer's actual percentage markup on the coats that were sold at 25% off? Calculate this percentage both as a percentage of the final selling price, and as a percentage of the cost price. (6 marks) 14 Mathematics 244 / Assignment 1 15. You process an order for goods with a list price of $525,000. The invoice is dated March 5, with trade discounts of 12%, 7%, and 2.5%. The invoice has terms of 10/10, 6/20, n/30, EOM. Partial payments are accepted. a. b. Business Mathematics What is the net price to the customer? (3 marks) If a payment of $150,000 is made on March 21, what is the balance left to be paid on the invoice, and what is the last day on which the company can pay the invoice with no penalty? (6 marks) 15 Assignment 2 This assignment covers material from Units 5 to 7 in the Mathematics 244 Study Guide. It is due after Unit 7 in the Study Guide is completed. Total Marks = 120 1. The chart below shows the number of cheques cashed in Alberta clearing centres during the years 1992-1996. Cheques Cashed in Alberta Clearing Centres 1992-1996 (Millions of Cheques) Year Calgary Edmonton Other Total 1992 1993 1994 1995 1996 62 77 102 115 137 44 59 71 86 105 10 12 14 16 18 116 148 187 217 260 Source: Hypothetical a. Business Mathematics Construct a line chart to compare each centre's activity over the five-year period. (9 marks) 1 b. 2 Construct a pie chart for the activity that took place in 1996. (6 marks) Mathematics 244 / Assignment 2 2. Solve for x and y , graphically. (6 marks) 4x + 2y = 0 x 5y = 11 Business Mathematics 3 3. Draw a graph of the line passing through the points (6, 7) and (3, 5) . Does the point (3, 1) also lie on this line? (4 marks) 4 Mathematics 244 / Assignment 2 4. A company produces and sells 15,300 sound cards per year. They sell each of the cards for $104 wholesale; the cards cost $69.75 each to produce. The company's fixed costs per year to produce sound cards is $320,000. a. Business Mathematics What is the break-even point on sound cards in terms of dollar value and units sold? (6 marks) 5 b. 6 If demand increases and the selling price of the sound cards increases by 10%, what is the new break-even point? (6 marks) Mathematics 244 / Assignment 2 5. Assume that the number of work stoppages that occurred in the automotive industry from 1985 to 1996 were as follows: 18, 8, 14, 12, 20, 20, 10, 18, 14, 20, 16, 15. a. Given this hypothetical information, calculate: i. the arithmetic mean. ii. the median. Business Mathematics (2 marks) (2 marks) iii. the mode. (1 mark) iv. the range. (1 mark) 7 b. 8 Calculate the standard deviation. (5 marks) Mathematics 244 / Assignment 2 6. The following frequency distribution table shows the orders received by Department A of ABC Mail Order House on December 15 of last year. ABC Mail Order House Orders Department A, December 15 Value of Orders $15 to $25 $25 to $35 $35 to $45 $45 to $55 $55 to $65 $65 to $75 $75 to $85 # of Orders 5 10 20 30 20 10 5 Source: Hypothetical Business Mathematics a. Calculate the arithmetic mean. b. Calculate the standard deviation. (5 marks) (6 marks) 9 10 7. A man borrowed $460 at a simple interest rate of 9.5% for 66 days. How much must he repay? (3 marks) 8. A note for $2400 was repaid after 120 days in the amount of $2471.01. What was the simple interest rate on the note? (3 marks) Mathematics 244 / Assignment 2 9. How many years are needed for $55 to yield $8.8 in interest at the simple interest rate of 8%? (3 marks) 10. A woman received $1200 in cash as the proceeds from a demand loan from a bank. After 120 days she paid $1248. What simple interest rate did the bank charge? (3 marks) Business Mathematics 11 11. Margaret received $671.50 in cash as the proceeds from a loan. She paid the lender back $680. The rate was 10% simple interest. Find the length of the loan in months. (5 marks) 12. Two payments of $2000 each are to be received 6 and 12 months from now. If money is worth 10% simple interest, what is the total equivalent value of the payments today? (4 marks) 12 Mathematics 244 / Assignment 2 13. Payments of $2600, due 50 days ago, and $3100, due in 40 days, are to be replaced by $3000 today and another payment in 30 days. What must the second payment be if the payee is to end up in an equivalent financial position? Money now earns 8.25% simple interest. Use 30 days from now as the focal date. (10 marks) Business Mathematics 13 14. A 4-month Guaranteed Investment Certificate with a face value of $55,000 will have a maturity value of $56,100. What simple annual interest rate is it carrying? (4 marks) 15. A 7-month, $75,400 Guaranteed Investment Certificate pays simple interest of 6.85%. Calculate the maturity value. (4 marks) 14 Mathematics 244 / Assignment 2 16. On the June 12 interest payment date, the outstanding balance on Delta Nurseries' revolving loan was $65,000. The floating interest rate on the loan stood at 9.25% simple interest on June 12, but rose to 9.5% simple interest on July 3, and to 10% simple interest on July 29. If Delta made principal payments of $10,000 on June 30 and July 31, what were the interest charges to its bank account on July 12 and August 12? Present a repayment schedule supporting the calculations. (12 marks) Business Mathematics 15 17. A $5000 demand loan was advanced on June 3. Fixed monthly payments of $1000 were required on the first day of each month, beginning July 1. Prepare the full repayment schedule for the loan. Assume that the interest rate remained at 12.5% simple interest for the life of the loan. (10 marks) 16 Mathematics 244 / Assignment 2 Summary of Key Formulas A copy of this summary of Key Formulas will be attached to exams. On assignments and exams show all your work. This allows the marker to assess partial marks. Show the formulas you used and the substitutions made into the formulas. Applications of Algebra Order of Operations The order of operations sets the priority for calculations or defines the way in which mathematical calculations proceed, as explained below. 1. Do the work in brackets first. Start on the innermost set of brackets and work outwards, applying the following priority within each set of brackets. Brackets at the same level can be worked on simultaneously. Within the brackets: a. do multiplication and division in the order in which they come; then b. do addition and subtraction in the order in which they come. 2. Once the brackets are removed, do multiplication and division in the order in which they come. 3. Finally, do addition and subtraction in the order in which they come. Simple Average Simple average = Weighted Average Weighted average = Business Mathematics Sum of the values Total number of items Sum of (Weighting factor Value) Sum of weighting factors 1 Percentages A percent represents a fraction of a total value (the base) expressed as a portion of 100%, the total value = 100% and the percent is a portion of that value. The decimal equivalent of a percent is the percent divided by 100. 100% =1 Percentage(Rate) = Percentage Change = c = Portion 100% Base Final Value - Initial Value Initial Value V f Vi Vi 100% Vi = Initial (or beginning or original or old) value Vf = Final (or ending or new) value c = Percent change (or its decimal equivalent) Rules of Exponents Word Problems In word problems you need to go a sentence at a time. Define the variables, you need an equation for each variable you use, i.e. if you use one variable you only need one equation, if you define two variables you will need two equations etc. Business Mathematics 2 Solving Equations Always remove any fractions in equations first by multiplying both sides of the equation (and therefore each element in the equation) by an appropriate common denominator, then consolidate terms, then solve for the variables. Quadratic Equations The quadratic formula is x= b b 2 4ac 2a Arithmetic Means and Geometric Progressions The nth term of an arithmetic progression is The sum of an arithmetic progression is The nth term a geometric progression is The sum of a geometric progression is tn = a + (n 1)d Sn = n (a + t n ) 2 t n = ar ( n 1) Sn = a (1 r n ) 1 r Ratio and Proportions Ratios and Proportions are an extension of percentages. A ratio is a comparison i.e. how a value is relative to another value. A proportion represents how the total is broken into parts (proportions). Index number Index number = Business Mathematics Price or value on the selected date Basevalue Price or value on the base date 3 Mathematics of Merchandising The sequence is: List Price Base =100% Less Trade Discount(s) Net Price Net Price Base = 100% Less Cash Discount(s) Final Price (to Retailer) (Cost Price to retailer) Cost Price (to Retailer) Markup Selling Price (to Consumer) Markup can be based on SP so then the base would be SP=100% or Markup can be based on the Cost Price so then Cost = 100% Note: Markup includes profit, overhead costs, related expenses etc. Unless otherwise stated markup is assumed to be based on Selling Price Selling Price Base = 100% Less Markdown New Final Price (to Consumer) Trade Discounts N = L(1 - d) Single Trade Discount L = List price d = Rate of trade discount N = Net price N = L (1 d 1 )(1 d 2 )(1 d 3 ) Multiple Trade Discounts the d's = Various Rates of trade discounts Cash Discounts Apply any Cash Discounts that are earned. Terms of Payment: Ordinary dating (from the date of the invoice); ROG dating (from receipt of goods); EOM dating (from the end of the month) Business Mathematics 4 Markup S = Selling price (per unit) C = Cost (per unit) M = Markup E = Overhead or operating expenses (per unit) P = Operating profit (per unit) D = (Amount of) Markdown S=C+M M=E+P Markup includes any amount between Cost and Selling Price and can include profit (gross profit, overhead expenses, etc.) S=C+E+P Rate of markup on cost = M 100% C Rate of markup on selling price = Rate of markdown = D 100% S M 100% S Cost-Volume-Profit Analysis X= number of units sold S = selling price per unit VC = variable cost per unit FC = total fixed cost TR = Total Revenue from the sales of X units i.e. TR = (S)X TC = Total Cost of X units sold TC= (Variable costs per unit x number of units sold + Fixed costs) i.e. TC= (VC)X+FC NI = Net Income from the sales of X units NI= TR-TC = (S)X-[(VC)X +FC]= (S - VC)X - FC Business Mathematics 5 TR = (S)X TC = (VC)X + FC NI = (S - VC)X - FC NI = (CM)X - FC FC S VC FC break-even volume = CM break-even volume = unit sales at break-even point Contribution margin CM = S - VC Contribution rate CR = CM x 100% S Measures of Central Tendency & Dispersion Ungrouped Data Put the observations in order from Lowest to Highest The Mean is mean = sum.of .observations number.of .observations The sum of the observations means the total of the observations. The number of observations means how many observations have been made. The Median is the observation in the middle. If there is an odd number of observations the median is the one in the middle, if there are an even number of observations the median is the average of the middle two observations. The mode is the observation that occurs most often. There can be more than one mode The range is the highest observation - the lowest observation Business Mathematics 6 The Standard Deviation (s) is s= total square deviations total observations 1 The total square deviations is found by finding the difference between each observation and the mean (deviations from the mean), squaring each difference, and adding up the result. The total observations means how many observations have been made. Grouped Data The Mean is mean = sum.of .observations number.of .observations The sum of observations is found by calculating the midpoint of each class multiplying the midpoint by the frequency (number of observations) in each class and adding up the result. The number of observations means the total of the number of observations (the frequency) in each class. The Standard Deviation (s) is s= total square deviations total observations 1 To calculate the total square deviations; calculate the difference from the mean and class midpoint for each class, square that result for each class and multiply each class result by the frequency (number of observations) in each class and sum that result. The total observations means the total of the number of observations (the frequency) in each class. Business Mathematics 7 Simple Interest I = Prt Future Value (moving money ahead in time) S = P(1+rt) moves a lump sum of money P ahead in time Present Value (moving money back in time) P= S (1 + rt ) moves a lump sum of money S back in time Finding equivalent values of Money If the interest rate is simple interest use the simple interest formulas, if the interest rate is compound interest use the compound interest formulas. At the focal date: New debts (paying) = Old debts (owing) at the current interest rate All monies, both all the old debts and all new debts, must move to the conversion or focal date. The procedure, then, in solving such problems is as follows. Step 1. Find the original maturity value of the original debts using the original rates of interest on the original loans. Step 2. Set up a time line to help you visualize the solution of the problem. Step 3. Move all the amounts to the focal date (conversion date) at the current rate of interest. At the focal date: New debts (paying) = Old debts (owing) at the current rate Business Mathematics 8 Note that whenever the statement says simply \"due in,\" it means that the original amount borrowed plus interest is included in the value stated. On the other hand, if the statement says \"due in with interest,\" \"with interest,\" or \"plus interest,\" then the interest on the amount must be calculated to get the maturity value. Compound Interest, Annuities, Bonds, Sinking Funds, Net Present Value How much work to show for these concepts If you are not using a financial calculator, make sure on assignments and exams you show the formulas you used and the substitutions made into the formulas. If you are using a financial calculator for the compound interest, annuities, bonds, sinking funds, and net present value calculations make sure on assignments and exams you show the calculator inputs that you made to solve the questions. i.e P/Y; C/Y; N; I/Y; PV; PMT; FV and remember to mention if you used the BGN; AMORT; BOND; CF calculator functions and if you used them, make sure you show your inputs for the AMORT, BOND, CF components. This allows the marker to assess part marks. Compound Interest i = stated or nominal rate of interest number of times interest is added per year i= j m j= nominal interest rate; m= the number of compoundings per year n= the number of compoundings per year times the number of years in the financial obligation n = m (Number of years in the term) Business Mathematics 9 FV = PV (1 + i ) PV = FV (1 + i ) n n moves a lump sum of money PV ahead in time moves a lump sum of money FV back in time. maturity value compounding at a variable rate FV = PV (1 + i1 )(1 + i2 )(1 + i3 )...(1 + in ) i = n n = FV FV 1 = PV PV 1 n 1 ln (FV PV ) ln (1 + i ) Effective Interest rate f (Equivalent Interest Rate per year) f = (1 + i ) m 1 m= the number of compound periods per year Equivalent Interest Rate (Equivalent Interest Rate per interest period) i2 = (1 + i1 )m 1 /m 2 1 c = m1 Number of compoundings per year = m2 Number of payments per year i2 = (1 + i )c 1 Business Mathematics 10 Annuities: (1 + i )n 1 FV = PMT moves a group of equal payments PMT i ahead in time to immediately after the last payment is made. 1 (1 + i ) n PV = PMT moves a group of equal payments PMT i back in time to one interest period before the first payment is made. i FV ln 1 + PMT n = ln (1 + i ) i PV ln 1 PMT n= ln (1 + i ) given FV, PMT, and i given PV, PMT, and i The balance of an annuity at any point in time is the present value of the remaining payments Annuities Due (1 + i )n 1 FV (due )= PMT (1 + i ) i 1 (1 + i ) n PV (due ) = PMT (1 + i ) i i FV (due) ln 1 + PMT (1 + i ) n= ln(1 + i ) Business Mathematics i PV (due) ln 1 PMT (1 + i ) n= ln(1 + i ) 11 Deferred Annuities Deferred Annuities are annuities where payments start at a later time after the annuity begins. They require the use of annuity formulas and lump sum formulas to solve. Perpetuities Perpetuities are annuities that have no end date PV = PMT i Mortgages Mortgages are an application of annuities. Mortgages rates are usually stated as compounded semi-annually, but payments on mortgages are usually made monthly. So the semi-annual mortgage rate has to be converted to an equivalent monthly rate using the Equivalent Interest Rate formula above. before the annuity formulas can be used in mortgage calculations. Bonds: Purchase price of a bond 1 (1+ i ) n n Bond Price = b(FV ) + FV (1 + i ) i FV = Face value of the bond b = Coupon rate per interest payment interval (normally six months) i = The bond market's required rate of return per payment interval n = Number of interest payments remaining until the maturity date To find the bond price on an interest date (coupon date) use the above Bond Price formula (15-1) To find the bond price on any other date, find the bond price on the last interest date (coupon date) before the bond purchase date (using the above Bond Price formula) and move that value forward to the purchase date. Business Mathematics 12 Yield to maturity of a bond Use the above Bond Price formula to solve for the interest rate If you are using the Bond Price formula then you must use a trial and error approach to solve for the yield rate. If you are using a financial calculator, the calculator is set up to accept the inputs to do the calculation of solving for the yield rate. Sinking Funds The sinking fund method of debt repayment consists of two parts. The payment required to the sinking fund each period plus the interest that has to be paid on the debt each period, together they are the total cost each period. Book Value of the debt = The Face value of the Debt less the balance in the sinking fund Net Present Value The Net Present Value approach is used to evaluate Investment opportunities. The Present Value is used because the investment decision is being made at the beginning of the terms of the investments. Calculate the Net Present Value of each investment alternative and choose the appropriate investment. If you are using the formulas to solve for IRR (Internal Rate Of Return) then you must use a trial and error approach to solve for the internal rate of return. NPV = Net present value (of an investment) IRR = Internal rate of return (on an investment) NPV = (Present value of cash inflows) - ( Present value of cash outflows) NPV Investment Decision Criterion: Accept the investment if NPV 0. Reject the investment if NPV < 0. Business Mathematics 13 IRR Investment Decision Criterion: Accept the investment if IRR Cost of capital. Reject the investment if IRR < Cost of capital. Business Mathematics 14