123 Sectio 3.3 Exercises Part A Simplify the followig. 1. (3m 2 ) 5 2. x 7 x 11 3. f 12 4. t 8 t 5 f 5 5. 3-4 6. 3x 7 4x 7. 3z 5 12z 3 8. 17 0 9. (g 8 ) -2 10. 14d 3 21d 7 11. (2m 2 5 g 8 ) 7 12. 5x 2 4x 7 13. 25 less tha 7 times a umber is 73. What is the umber? 14. Two umbers add to 251 ad the secod is 41 bigger tha the first. What are the two umbers? Solve. 15. 5p + 12 = 39 4p 16. 5 + 48 = 7 2( 2) 17. 15x 10 = 5(x 2) 18. 2(x 5) x = 4x 7 19. 9m 3(m 3) = 15m +7 20. 8x 12x + 3 = 9x +8x 21. 45 is what percet of 39? 22. 85 is 24% of what? 23. What is 19% of 3,517? 24. What is 23% of 49? 25. Origial Price: Tax: 5% Fial Price: $239.40 26. Origial Price: $55.50 Discout: 23% Fial Price: 27. If the populatio of a tow grew 41% up to 7,191. What was the populatio last year? Assigmet 3.3a
124 28. If the price of a object dropped 35% dow to $11.44, what was the origial price? Solve. 29. 7 t + 5 = 19 30. 3 3 (x + 7) = 5 + 3x 31. 2 x 6 = 7 + 8 3 21 x 32..3(4x + 7) = 2.5x + 6 33..005x +.045 =.004x 34. x+ 7 4 5 4 = 6 x Preparatio. 35. Simplify the followig (so that there are o egative expoets). 5a 4 b 7 c 3 d 5 f 9 7g 3 h 8 j 2 k 1 Assigmet 3.3a
125 Aswers: 1. 243m 10 28. $17.60 2. x 18 29. t = 6 3. f 7 30. x = - 61 27 4. t 13 31. x = 78 1 5. 32. x = -3 81 6. 12x 8 33. x = -45 7. z 2 34. x = 27 4 13 8. 1 35. 5g 3 h 8 j 2 k 1 1 9. g 16 2 10. 3d 4 11. 128m 14 35 g 56 12. 20x 9 13. 14 14. 105,146 15. p = 3 16. No solutio 17. x = 0 18. x = -1 19. m = 2 9 20. x = 1 7 21. 115.4% 22. 354.17 23. 668.23 24. 11.27 25. $228 26. $42.74 27. 5,100 7a 4 b 7 c 3 d 5 f 9 Assigmet 3.3a
126 Compoudig Quarterly, Mothly, ad Daily So far, you have bee compoudig iterest aually, which meas the iterest is added oce per year. However, you will wat to add the iterest quarterly, mothly, or daily i some cases. Excel will allow you to make these calculatios by adjustig the iterest rate ad the umber of periods to be compouded. Remember that all iterest rates provided i the problems are aual rates. You must adjust them to fit other compoudig periods. The adjusted rate is called the periodic rate. To adjust the periodic rate i Excel, ope the FV calculatio box ad chage a 10% aual rate to quarterly, mothly, or daily as follows: Quarterly Rate:.10/4 Chagig the rate to 2.5% or.025 Mothly Rate:.10/12 Chagig the rate to.83% or.0083 Daily Rate:.10/365 Chagig the rate to.0274% or.000274 Chage te years of compoudig to quarterly, mothly, or daily as follows: Quarerly Nper: 10*4 Chagig the compoudig periods to 40 Mothly Nper: 10*12 Chagig the compoudig periods to 120 Daily Nper: 10*365 Chagig the compoudig periods to 3,650 If you assume you put $50 ito savigs ad you are comparig savigs accouts where the 10% aual iterest rate is compoudig quarterly, mothly, or daily. You ca compare the amout of iterest you will ear usig Excel as follows: Quarterly Mothly Daily Rate:.1/4 or.025 Rate:.1/12 or.00833 Rate:.1/365or.000274 Nper: 10*4 or 40 Nper: 10*12 or 120 Nper: 10*365 or 3650 Pmt: 0 Pmt: 0 Pmt: 0 Pv: -50 Pv: -50 Pv: -50 Future Value = $134.25 Future Value = $135.35 Future Value = $135.90 The more frequetly iterest is added to your savigs ad compouded, the more iterest you will ear. The above illustratio ivolves a small amout of savigs. The more the savigs ad the more ofte you add to your savigs the more differece it will make whe the iterest i added ad compouded more frequetly. The followig example illustrates savig $100 per moth for te years at 10% iterest rate compouded mothly versus aually. Aually Mothly Rate:.1 or 10% Rate:.1/12 or.00833 Nper: 10 Nper: 10*12 Pmt: -1200 Pmt: -100 Pv: 0 Pv: 0 Future Value = $19,124.91 Future Value = $20,484.50 Sectio 3.3
127 Savigs Pla Formula for a lump sum A = P 1 + r Y Savigs Pla Formula with paymet A = PMT 1 + r Y 1 r Thus we have the moster formula for a Savigs Pla that begis with a balace ad the is added to by a paymet: A = P 1 + r Y + PMT 1 + r Y 1 r A = Fial Amout PMT = mothly paymet P = Pricipal amout (begiig balace) r = aual iterest rate = umber of compoudig per year Y = umber of years So, r = periodic iterest rate (rate used i spreadsheet) Y = umber of periods (per) Spreadsheets ormally have this formula built ito their fuctios. It is kow as Future Value (FV), so you wo t eed to use this oe if you lear the spreadsheet well. Loa Paymet Formula PMT = P r 1 1 + r Y Spreadsheets also ormally have this formula built ito their fuctios. It is kow as Paymet (PMT). Fial ote usig a spreadsheet: The formulas are built so that moey goig out from you is egative ad moey comig i to you is positive. Whe you are eterig Savigs ito the spreadsheet, the paymet ad Pricipal (Preset Value) will be egative. However, for a loa, the paymet will be egative but the Pricipal (Preset Value) will be positive, because it represets moey comig to you. Sectio 3.3
128 Calculatig Paymets, Iterest Rates, ad Number of Periods Excel will help you calculate the paymet you will eed to make o a loa. It will calculate the iterest rate you would eed to ear o your savigs to realize a certai future balace. The umber of periods it will take to have your savigs grow to a certai future balace ca also be determied. Mothly Paymet Calculatio If you wated to buy a car that costs $15,000 ad you ca get a loa at 6% iterest for four years, you ca determie the mothly paymets usig the PMT Excel fuctio as follows: Rate:.06/12 or.005 (mothly iterest) Nper: 4*12 or 48 (moths) Pv: -15000 Fv: 0 Mothly Paymet = $352.28 Whe you have paid the mothly paymet for forty-eight moths you will ow the car ad the future value of the loa is zero because the loa i paid off. Beefits Versus Bodage You ca see how hard your savigs will work for you give a iterest rate ad eough time. However, iterest works agaist you whe you borrow moey. The beefits may seem great at the momet but the fiacial bodage is terrible. By calculatig the iterest you would pay o a loa to borrow a car ad the iterest you would ear by savig to be able to pay cash for the car, we ca determie the fiacial advatage of collectig iterest rather tha payig iterest. Iterest Paid o a Car Loa You calculate the amout of iterest you would pay o a four year car loa of $15,000 at 6% aual iterest usig the Excel Pmt fuctio as follows: Rate:.06/12 Nper: 4*12 Pv: -15000 Fv: 0 Mothly Paymet = $352.28 Total Paymet = $352.28*48 (Paymets) = $16,909.22 Iterest Paid =$16,909.22 (Paid) -$15,000 (Borrowed) = $1,909.22 Sectio 3.3
129 TIP: You ca have Excel calculate this for you by eterig the Pmt fuctio to calculate the mothly paymet ad the, o the formula bar at the top of the Excel sheet, multiply by 48 paymets ad subtract the $15,000 you borrowed. The formula will be as follows: =PMT(0.06/12,4*12,-15000,0)*48-15000 You ca also double click o the cell with the Pmt calculatio i it ad the formula will appear i the cell. Now you ca multiply by 48 paymets ad subtract 15000 ad eter this formula i the cell. The cell will have the aswer ad the formula will be i the formula bar. Iterest Collected o Your Savigs The iterest you will ear o your savigs of $350.00 per moth earig 6% aual iterest for 39 moths (the umber of moths we calculated above would be required to accumulate $15,000 i savigs) is calculated usig the FV fuctio i Excel as follows: Rate:.06/12 Nper: 39 Pmt: -350 Pv: 0 FV = $15,030.44 Amout Deposited i Savigs = $350*39 (deposits) = $13,650.00 Iterest Eared o Savigs = $15,030.44-$13,650.00 = $1,380.44 Agai, you ca double click o the cell cotaiig the FV calculatio ad subtract 350*39 ad eter this formula givig you the amout of iterest eared. You ca make the same adjustmet to the formula i the formula bar. The resultig formula is as follows: =FV(0.06/12,39,-350)-350*39 Total Savigs From Savig Versus Borrowig Here is how you beefited by savig ad payig cash for the car rather tha borrowig the moey to buy the car: Iterest Eared $1,380.44 Iterest Not Paid $1,909.22 Fiacial Advatage $3,289.66 You are wealthier by $3,289.66 because you collected iterest rather tha payig iterest. This practice will make a major differece i your fiacial well beig throughout your life. If you put the moey you save by payig cash for major purchases to work for you by ivestig it for your retiremet you will add greatly to your idepedet wealth. You Sectio 3.3
130 ca estimate that usig the FV fuctio i Excel as follows assumig a 6% retur o your ivestmet for 30 years: Rate:.06 Nper 30 Pv: -3289.66 FV = $18,894.13 This additio to your wealth alog with the other additios resultig from savig rather tha borrowig will make a major impact o your ultimate wealth. TIP: I all of the Excel fuctios you will be usig, you oly eed three etries or factors to calculate the fourth factor you are after. Notice that there are oly three etries i each of the above Excel fuctios. You ca leave blak ay factor ot eeded ad Excel will assume it is zero. Sectio 3.3