EXAMPLE: Use the formula for future value simple interest to find the indicated quantity. A = 22135, P = 19000, t = 39 weeks and find r =? Solution: A = P (1 + rt), Present Value (A) = 22135, Principal (P) = 19000, time (t) = 39 weeks, Rate (r) = ? 22135 = 19000(1 + r(39/52)) Convert time to decimal notation 22135 = 19000(1 + 0.75r) Divide both sides by 19000 to isolate r 22135/19000 = [19000(1 + 0.75r)]/19000 Simplify both sides 1.165 = 1 + 0.75r Subtract one from both sides 1.165 - 1 = 1 - 1 + 0.75r simplifying both sides 0.165 = 0.75r 0.165/0.75 = 0.75/0.75r 0.22 = r Divide both sides by 0.75 to solve for r Convert your decimal value to percent by Multiplying value by 100 22% = r Make the indicated conversion. Assume a 360-day year as needed. 1) 150 days to a simplified fraction of year Use I = Prt for simple interest to find the indicated quantity. 2) I = $750, r = 6%, t = 6 months. Find P. Use the formula A = P(1 + rt) to find the indicated quantity. 3) P = $7996; r = 6%; t = 10 months. Find A. Use the formula A = P(1 + rt) to find the indicated quantity. 4) Allan borrowed $6300 from his father to buy a car. He repaid him after 9 months with interest of 11% per year. Find the total amount he repaid. Find the compound interest earned. Round to the nearest cent. 5) $14,000 at 5% compounded annually for 3 years Provide an appropriate response. Round to answer to two decimal places. 6) An investment company pays 7% compounded quarterly. What is the effective rate? Solve the problem. Round to the nearest cent as needed. 7) The average cost of a 4-year college education is projected to be $130,000 in 16 years. How much money should be invested now at 6.5%, compounded quarterly, to provide $130,000 in 16 years? Solve the problem. Round to the nearest cent as needed. 8) Sandra deposits $3000 in an ordinary annuity at the end of each semiannual period at 4% interest compounded semiannually. Find the amount she will have on deposit after 25 years. Find the future value of the ordinary annuity. Interest is compounded annually unless otherwise indicated. 9) PMT = $7,500, i = 7% interest compounded semiannually for 5 years Find the monthly house payment necessary to amortize the following loan. Round the answer to the nearest cent. 10) In order to purchase a home, a family borrows $267,000 at 10.8% for 15 yr. What is their monthly payment? 1)We know that 360 days 1 year 1 1 day year 360 1 5 150 days 150 year year 360 12 2)Given that I $750, r 6%, t 6 months. Divide by 100 to write r in decimal as, r 6% 6 0.06 100 Since 1 year 12 month so divide by 12 to write t in years as t 6 month 6 1 year year 12 2 Putting the above values in the formula I Pr t we get, 1 2 2 2 2 0.06 750 P , multiply both sides by to isolate P 0.06 0.06 2 0.06 25000 P simplify both sides Thus P $25000 3)Given that P $7996, r 6%, t 10 months. 6 Divide by 100 to write r in decimal as, r 6% 0.06 100 750 P 0.06 Since 1 year 12 month so divide by 12 to write t in years as t 10 month Putting the above values in the formula A P 1 rt we get, 5 A 7996 1 0.06 7996 1 0.05 7996 1.05 8395.8 6 Thus A $8395.8 10 5 year year 12 6 4)Given that Allan borrowed $6300 so the present amount is P $6300. He repaid the amount in 9months with interest of 11% so r 11%, t 9 months. 11 Divide by 100 to write r in decimal as, r 11% 0.11 100 9 3 Since 1 year 12 month so divide by 12 to write t in years as t 9 month year year 12 4 To find the total amount repaid put the above values in the formula A P 1 rt to get, 3 A 6300 1 0.11 6300 1 0.0825 6300 1.0825 6819.75 4 Thus the total amount repaid is $6819.75 nt r 5)The compound interest formula is A P 1 n Where A the future value of investment, P the present value, r the annual interest rate, n the number of times the interest is compounded per year, t the number of years money is invested Given that P $14000, r 5% compounded annually and t 3 yrs. 5 Divide by 100 to write r in decimal as, r 5% 0.05. Since interest is compounded annually so n 1 100 Putting the above values in compound interest formula we get, 1 3 0.05 3 A 14000 1 14000 1.05 16206.75 1 So the compound interest earned is A P 16206.75 14000 $2206.75 6)Given that investment company pays 7% compounded quarterly so r 7%, n 4 as there are 4 quarters in a year. 7 Divide by 100 to write r in decimal as, r 7% 0.07 100 n r We know that effective interest rate per period i 1 1 where n is number of compounding n per period. Putting the given values in above formula we get, 4 4 0.07 4 i 1 1 1 0.0175 1 1.0175 1 0.07186 4 To convert in percentage multiply by 100 to get, i 0.07186 100 7.186% 7.19%, round to two decimal places Thus the effective interest rate is 7.19% 7)Given that the money is invested at 6.5% compounded quarterly to provide $130000 in 16 years so A $130000, r 6.5% compounded quarterly and t 16 yrs. 6.5 Divide by 100 to write r in decimal as, r 6.5% 0.065. Since interest is compounded quarterly so n 4 100 nt r as there are 4 quarters in a year. The compound interest formula is A P 1 n Where A the future value of investment, P the present value, r the annual interest rate, n the number of times the interest is compounded per year, t the number of years money is invested Putting the above values in compound interest formula we get, 4 16 0.065 130000 P 1 4 130000 P 1 0.01625 , simplify the right hand side 64 1 1 1 P 1.0162564 , multiply both side by 64 64 1.01625 1.01625 1.0162564 46334.83297 P, simplify both sides P $46334.83 by rounding to nearest cent 130000 So the amount to be invested is $46334.83 8)Given that Sandra deposit $3000 at end of each semiannual period at 4% interest compounded semiannually for 25 years, so PMT $3000, i 4%, t 25 years. 4 Divide by 100 to write iin decimal as i 4% 0.04 100 Since interest is compounded semiannually so n 2. The future value of an ordinary annuity is given by the formula nt i 1 1 , where n number of compounding period per year, t time in years, n i interest rate in decimal, PMT The amount of the annuity payment each period Putting the given values in above formula we get the future value of ordinary annuity as PMT FV i/n 2 25 3000 0.04 FV 1 0.04 / 2 2 3000 1 1.0250 1 253738.20436 0.02 253738.20 rounding to two decimal places Thus the future value is $253738.20 9)Given that PMT $7500, i 7% compounded semiannually, t 5 years. 7 Divide by 100 to write i in decimal as i 7% 0.07 100 Since interest is compounded semiannually so n 2. The future value of an ordinary annuity is given by the formula nt i 1 1 , where n number of compounding period per year, t time in years, n i interest rate in decimal, PMT The amount of the annuity payment each period Putting the given values in above formula we get the future value of ordinary annuity as PMT FV i/n 2 5 7500 0.07 FV 1 0.07 / 2 2 7500 1 1.03510 1 87985.4487 87985.45 rounding to two decimal places 0.035 Thus the future value is $87985.45 10)The monthly payments for a loan amount is calculated by the formula r monthly payment r principal loan amount months 1 r 1 interest rate in decimal Here r , months indicate the total number of months for which mayment is to be made. 12 Given that the family borrows $267000 at an interest rate of 10.8% for 15 years, so principal loan amount 267000,interest rate 10.8% and time is 15 years Since a year contain 12 months so the total number of months are months 12 15 180 months 10.8 To write interest rate in decimal divide by 100 to get, interest rate 10.8% 0.108 100 interest rate in decimal 0.108 Hence r 12 12 Putting the given values in monthly payment formula we get, 0.108 0.108 1 0.108 12 monthly payment 267000 267000 1 360 360 12 0.108 0.108 12 1 1 1 1 12 12 0.108 1 267000 1 3001.26642 $3001.27 by rounding to nearest cent 12 4.01661 Thus the monthly payment to be made is $3001.27 1)We know that 360 days 1 year 1 year 360 1 5 150 days 150 year year 360 12 2)Given that I $750, r 6%, t 6 months. 1 day Divide by 100 to write r in decimal as, r 6% 6 0.06 100 Since 1 year 12 month so divide by 12 to write t in years as t 6 month 6 1 year year 12 2 Putting the above values in the formula I Pr t we get, 1 2 2 2 2 0.06 750 P , multiply both sides by to isolate P 0.06 0.06 2 0.06 25000 P simplify both sides 750 P 0.06 Thus P $25000 3)Given that P $7996, r 6%, t 10 months. Divide by 100 to write r in decimal as, r 6% 6 0.06 100 Since 1 year 12 month so divide by 12 to write t in years as t 10 month Putting the above values in the formula A P 1 rt we get, 5 A 7996 1 0.06 7996 1 0.05 7996 1.05 8395.8 6 Thus A $8395.8 10 5 year year 12 6 4)Given that Allan borrowed $6300 so the present amount is P $6300. He repaid the amount in 9months with interest of 11% so r 11%, t 9 months. Divide by 100 to write r in decimal as, r 11% 11 0.11 100 9 3 year year 12 4 To find the total amount repaid put the above values in the formula A P 1 rt to get, Since 1 year 12 month so divide by 12 to write t in years as t 9 month 3 A 6300 1 0.11 6300 1 0.0825 6300 1.0825 6819.75 4 Thus the total amount repaid is $6819.75 nt r 5)The compound interest formula is A P 1 n Where A the future value of investment, P the present value, r the annual interest rate, n the number of times the interest is compounded per year, t the number of years money is invested Given that P $14000, r 5% compounded annually and t 3 yrs. 5 0.05. Since interest is compounded annually so n 1 100 Putting the above values in compound interest formula we get, Divide by 100 to write r in decimal as, r 5% 1 3 0.05 3 A 14000 1 14000 1.05 16206.75 1 So the compound interest earned is A P 16206.75 14000 $2206.75 6)Given that investment company pays 7% compounded quarterly so r 7%, n 4 as there are 4 quarters in a year. Divide by 100 to write r in decimal as, r 7% 7 0.07 100 n r We know that effective interest rate per period i 1 1 where n is number of compounding n per period. Putting the given values in above formula we get, 4 4 0.07 4 i 1 1 1 0.0175 1 1.0175 1 0.07186 4 To convert in percentage multiply by 100 to get, i 0.07186 100 7.186% 7.19%, round to two decimal places Thus the effective interest rate is 7.19% 7)Given that the money is invested at 6.5% compounded quarterly to provide $130000 in 16 years so A $130000, r 6.5% compounded quarterly and t 16 yrs. Divide by 100 to write r in decimal as, r 6.5% 6.5 0.065. Since interest is compounded quarterly so n 4 100 nt r as there are 4 quarters in a year. The compound interest formula is A P 1 n Where A the future value of investment, P the present value, r the annual interest rate, n the number of times the interest is compounded per year, t the number of years money is invested Putting the above values in compound interest formula we get, 0.065 130000 P 1 4 4 16 130000 P 1 0.01625 , simplify the right hand side 64 1 1 1 P 1.0162564 , multiply both side by 64 64 1.01625 1.01625 1.0162564 46334.83297 P, simplify both sides 130000 P $46334.83 by rounding to nearest cent So the amount to be invested is $46334.83 8)Given that Sandra deposit $3000 at end of each semiannual period at 4% interest compounded semiannually for 25 years, so PMT $3000, i 4%, t 25 years. 4 0.04 100 Since interest is compounded semiannually so n 2. Divide by 100 to write iin decimal as i 4% The future value of an ordinary annuity is given by the formula nt i 1 1 , where n number of compounding period per year, t time in years, n i interest rate in decimal, PMT The amount of the annuity payment each period PMT FV i/n Putting the given values in above formula we get the future value of ordinary annuity as 3000 0.04 FV 1 0.04 / 2 2 2 25 3000 1 1.0250 1 253738.20436 0.02 253738.20 rounding to two decimal places Thus the future value is $253738.20 9)Given that PMT $7500, i 7% compounded semiannually, t 5 years. 7 0.07 100 Since interest is compounded semiannually so n 2. Divide by 100 to write i in decimal as i 7% The future value of an ordinary annuity is given by the formula nt i 1 1 , where n number of compounding period per year, t time in years, n i interest rate in decimal, PMT The amount of the annuity payment each period PMT FV i/n Putting the given values in above formula we get the future value of ordinary annuity as 7500 0.07 FV 1 0.07 / 2 2 2 5 7500 1 1.03510 1 87985.4487 87985.45 rounding to two decimal places 0.035 Thus the future value is $87985.45 10)The monthly payments for a loan amount is calculated by the formula r monthly payment r principal loan amount months 1 1 r interest rate in decimal Here r , months indicate the total number of months for which mayment is to be made. 12 Given that the family borrows $267000 at an interest rate of 10.8% for 15 years, so principal loan amount 267000,interest rate 10.8% and time is 15 years Since a year contain 12 months so the total number of months are months 12 15 180 months To write interest rate in decimal divide by 100 to get, interest rate 10.8% 10.8 0.108 100 interest rate in decimal 0.108 12 12 Putting the given values in monthly payment formula we get, Hence r 0.108 0.108 1 0.108 12 monthly payment 267000 267000 1 360 360 12 0.108 0.108 12 1 1 1 1 12 12 0.108 1 1 3001.26642 $3001.27 by rounding to nearest cent 12 4.01661 Thus the monthly payment to be made is $3001.27 267000