Math  /  Algebra

QuestionQuestion: Sydney invests $100\$ 100 every month into an account that pays 0.8%0.8 \% annual interest, compounded monthly. Benny invests $80\$ 80 every month into an account that pays 2.2%2.2 \% annual interest rate, compounded monthly. a) Determine the amount in Sydney's account after 10 years. b) Determine the amount in Benny's account after 10 years. c) Who had more money in the account after 10 years. d) Determine the amount in Sydney's account after 30 years. e) Determine the amount in Benny's account after 30 years. f) Who had more money in the account after 30 years. g) Write the future value function for Sydney's account where xx represents the number of months. h) Write the future value function for Benny's account where xx represents the number of months.

Studdy Solution

STEP 1

1. Sydney invests 100everymonth.<br/>2.Sydneysaccounthasanannualinterestrateof0.8100 every month.<br />2. Sydney's account has an annual interest rate of 0.8%, compounded monthly.<br />3. Benny invests 80 every month.
4. Benny's account has an annual interest rate of 2.2%, compounded monthly.
5. We need to calculate the future value of both accounts after 10 and 30 years.
6. We need to write the future value function for both accounts with respect to the number of months.

STEP 2

1. Calculate the monthly interest rate for both Sydney and Benny.
2. Determine the future value of Sydney's account after 10 years.
3. Determine the future value of Benny's account after 10 years.
4. Compare the amounts in both accounts after 10 years.
5. Determine the future value of Sydney's account after 30 years.
6. Determine the future value of Benny's account after 30 years.
7. Compare the amounts in both accounts after 30 years.
8. Write the future value function for Sydney's account.
9. Write the future value function for Benny's account.

STEP 3

Calculate the monthly interest rate for both Sydney and Benny.
For Sydney: Annual interest rate = 0.8% Monthly interest rate = 0.8%12=0.00812=0.0006667\frac{0.8\%}{12} = \frac{0.008}{12} = 0.0006667
For Benny: Annual interest rate = 2.2% Monthly interest rate = 2.2%12=0.02212=0.0018333\frac{2.2\%}{12} = \frac{0.022}{12} = 0.0018333

STEP 4

Determine the future value of Sydney's account after 10 years.
Sydney's monthly investment = $100 Number of months = 10 years \(\times\) 12 months/year = 120 months
Future value formula for monthly investments: FV=P×(1+r)n1r FV = P \times \frac{(1 + r)^n - 1}{r} where P=100 P = 100 , r=0.0006667 r = 0.0006667 , n=120 n = 120
FV=100×(1+0.0006667)12010.0006667 FV = 100 \times \frac{(1 + 0.0006667)^{120} - 1}{0.0006667}

STEP 5

Calculate the future value for Sydney's account.
FV=100×(1.0006667)12010.0006667 FV = 100 \times \frac{(1.0006667)^{120} - 1}{0.0006667} FV100×1.08310.0006667 FV \approx 100 \times \frac{1.083 - 1}{0.0006667} FV100×0.0830.0006667 FV \approx 100 \times \frac{0.083}{0.0006667} FV100×124.5 FV \approx 100 \times 124.5 FV12450 FV \approx 12450

STEP 6

Determine the future value of Benny's account after 10 years.
Benny's monthly investment = $80 Number of months = 120
Future value formula: FV=P×(1+r)n1r FV = P \times \frac{(1 + r)^n - 1}{r} where P=80 P = 80 , r=0.0018333 r = 0.0018333 , n=120 n = 120
FV=80×(1+0.0018333)12010.0018333 FV = 80 \times \frac{(1 + 0.0018333)^{120} - 1}{0.0018333}

STEP 7

Calculate the future value for Benny's account.
FV=80×(1.0018333)12010.0018333 FV = 80 \times \frac{(1.0018333)^{120} - 1}{0.0018333} FV80×1.24310.0018333 FV \approx 80 \times \frac{1.243 - 1}{0.0018333} FV80×0.2430.0018333 FV \approx 80 \times \frac{0.243}{0.0018333} FV80×132.6 FV \approx 80 \times 132.6 FV10608 FV \approx 10608

STEP 8

Compare the amounts in both accounts after 10 years.
Sydney's account: 12450Bennysaccount:12450 Benny's account: 10608
Sydney has more money after 10 years.

STEP 9

Determine the future value of Sydney's account after 30 years.
Number of months = 30 years ×\times 12 months/year = 360 months
Future value formula: FV=100×(1+0.0006667)36010.0006667 FV = 100 \times \frac{(1 + 0.0006667)^{360} - 1}{0.0006667}

STEP 10

Calculate the future value for Sydney's account after 30 years.
FV=100×(1.0006667)36010.0006667 FV = 100 \times \frac{(1.0006667)^{360} - 1}{0.0006667} FV100×1.25210.0006667 FV \approx 100 \times \frac{1.252 - 1}{0.0006667} FV100×0.2520.0006667 FV \approx 100 \times \frac{0.252}{0.0006667} FV100×378 FV \approx 100 \times 378 FV37800 FV \approx 37800

STEP 11

Determine the future value of Benny's account after 30 years.
Number of months = 360
Future value formula: FV=80×(1+0.0018333)36010.0018333 FV = 80 \times \frac{(1 + 0.0018333)^{360} - 1}{0.0018333}

STEP 12

Calculate the future value for Benny's account after 30 years.
FV=80×(1.0018333)36010.0018333 FV = 80 \times \frac{(1.0018333)^{360} - 1}{0.0018333} FV80×2.19610.0018333 FV \approx 80 \times \frac{2.196 - 1}{0.0018333} FV80×1.1960.0018333 FV \approx 80 \times \frac{1.196}{0.0018333} FV80×652.5 FV \approx 80 \times 652.5 FV52200 FV \approx 52200

STEP 13

Compare the amounts in both accounts after 30 years.
Sydney's account: 37800Bennysaccount:37800 Benny's account: 52200
Benny has more money after 30 years.

STEP 14

Write the future value function for Sydney's account where x x represents the number of months.
FV(x)=100×(1+0.0006667)x10.0006667 FV(x) = 100 \times \frac{(1 + 0.0006667)^x - 1}{0.0006667}

STEP 15

Write the future value function for Benny's account where x x represents the number of months.
FV(x)=80×(1+0.0018333)x10.0018333 FV(x) = 80 \times \frac{(1 + 0.0018333)^x - 1}{0.0018333}

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