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 11. Jungle Jim owes three debts: $500 due in one year plus interest at 6% compounded semi-annually, $2000 due in two years, $1000 due in three years plus interest at 5% compounded monthly. He wishes to discharge these debts by paying $500 now and two equal but unknown payments in one and two years respectively. Find the size of the equal payments if money is, at present, worth 12% compounded quarterly. Use a focal date of two years. (16 marks) 12. What are the future and present value of an annuity of $100 payable at the beginning of each quarter for 15 years if the interest rate is 12% compounded quarterly? (5 marks) 13. If the present value of an annuity due of $400 payable semi-annually is $5600 and interest is computed at 6% compounded semi-annually, what is the number of payments? (6 marks 14. Josie spends $60 at the end of each month on cigarettes. If she stops smoking and invests the same amount in an investment plan paying 6% compounded monthly, how much will she have after five years? (4 marks) 15. A 20-year loan requires semi-annual payments of $1333.28 including interest at 10.75% compounded semi-annually. What is the original amount of the loan and what will be the balance of the loan 8 years later (just after the scheduled payment)? (6 marks) 16. What price will a finance company pay to a merchant for a conditional sale contract that requires 15 monthly payments of $231 beginning in six months? The finance company requires a rate of return of 18% compounded monthly. (7 marks) 17. What amount can be paid at the end of every month in perpetuity from an endowment of $350,000 which is earning 5.4% compounded monthly? (3 marks) 18. Pascal has just agreed with his financial planner to begin a voluntary accumulation plan. He will invest $500 at the end of every three months in a balanced mutual fund. a. How much will the plan be worth after 20 years if the mutual fund earns 8% compounded quarterly? (4 marks) b. If Pascal does not make any more quarterly payments but leaves the money in the fund for another 30 years, how much will he have accumulated at the end of 50 years, assuming the interest rate does not change