Compound Interest Compound Interest ProblemsProblems

Lesson ObjectivesLesson Objectives

Use the compound interest formula to Use the compound interest formula to solve problemssolve problems

Simple interest Simple interest I = P*r*tI = P*r*tCompound interest Compound interest

Continuously compounded interestContinuously compounded interestA = PeA = Pertrt

nt

n

rPA

1

Vocabulary

simple interestprincipalrate of interestcompound interest

The formulas…The formulas…

n = # times compounded / n = # times compounded / yearyear

r = rater = rate

t = time (in years)t = time (in years)

P= principal(initial \) amt.P= principal(initial \) amt.

Continuously compounded Continuously compounded interestinterest A = PeA = Pertrt

Simple interestSimple interest I= I= (P*r*t)(P*r*t)

Compound interestCompound interest

nt

n

rPA

1

n

n ne

e

11lim

...718281827.2

The table shows some common compounding periods and how many times per year interest is paid for them.

Compounding Periods Times per year (n)

Annually 1

Semi-annually 2

Quarterly 4

Monthly 12

Example 1-2Example 1-2Suppose $10,000 is invested at 5.4%, Suppose $10,000 is invested at 5.4%,

compounded compounded monthly.monthly.

a) What is the balance after 2 yrs? 5 yrs?a) What is the balance after 2 yrs? 5 yrs?

2*12

12

054.01000,10

A

5*12

12

054.01000,10

A

240045.1000,10

78.137,11$

600045.1000,10

71.091,13$

David invested $1800 in a savings account that pays 4.5% interest compounded semi-annually. Find the value of the investment in 12 years.

Example 3: You Try!

Use the compound interest formula.

Substitute.

= 1800(1 + 0.0225)24 Simplify.

A = P(1 + )r n

nt

= 1800(1 + )0.045 t 2

2(12)

= 1800(1.0225)24 Add inside the parentheses.

#3 Continued

After 12 years, the investment will be worth about $3,070.38.

≈ 1800(1.70576) Find (1.0225)24 and round.

≈ 3,070.38 Multiply and round to the nearest cent.

Example 4-5Example 4-5

Suppose $5,000 is invested at 6%, compounded Suppose $5,000 is invested at 6%, compounded continuously.continuously.

a) What is the balance after 2 yrs? a) What is the balance after 2 yrs? You try: 5 yrsYou try: 5 yrs??

2*06.0000,5 eA 5*06.0000,5 eA

1275.1000,5

48.637,5$

34986.1000,5

29.749,6$

Example 6Example 6I have I have $2,500$2,500 to invest and need to invest and need $4,000$4,000 in in 6 6

yearsyears. I found an account that pays 8% interest . I found an account that pays 8% interest (compounded daily)(compounded daily)

A) At this rate, will I get my money?A) At this rate, will I get my money?

6*365

365

08.01500,2

A

2190000219.1500,2

97.039,4$

Kia invested $3700 in a savings account that pays 2.5% interest compounded quarterly. Find the value of the investment in 10 years.

Guided Practice. Example 7 – YOU TRY!

Use the compound interest formula.

Substitute.

= 3700(1 + 0.00625)40 Simplify.

A = P(1 + )r n

nt

= 3700(1 + )0.025 t 4

4(10)

= 3700(1.00625)40 Add inside the parentheses.

Check It Out! Example 7 Continued

After 10 years, the investment will be worth about $4,747.20.

≈ 3700(1.28303) Find (1.00625)40 and round.

≈ 4,747.20 Multiply and round to the nearest cent.

4 CORNERS : Part I

Theresa invested $800 in a savings account that pays

4% interest compounded quarterly. Find the value of the investment after 6 years.A. $1156. 79

B. $1015.79

C. $1014.39

D. $1015.85

STOP! Hw STOP! Hw

Ex.8Ex.8

Suppose $10,000 is invested at 5.4%, Suppose $10,000 is invested at 5.4%, compounded monthlycompounded monthly. . Using Log/Ln to find “t”. Using Log/Ln to find “t”.

b) What is the doubling time?b) What is the doubling time?

yearsn 86.120045.1ln12

2ln

n*12

12

054.01000,10000,20

n120045.12

0045.1ln122ln n

Ex.9 Suppose $5,000 is invested at 6%, Ex.9 Suppose $5,000 is invested at 6%, compounded continuously.compounded continuously.

b) What is the doubling time?b) What is the doubling time?

xe *06.0000,5000,10 xe 06.02

x06.2ln

yearsx 55.1106.0

2ln

Ex.10 I have Ex.10 I have $2,500$2,500 to invest and need to invest and need $4,000$4,000 in in 6 years6 years. I found an . I found an account that pays 8% interest (compounded daily)account that pays 8% interest (compounded daily)

B) What is the minimum rate I need to guarantee reaching this value?B) What is the minimum rate I need to guarantee reaching this value?

6*365

3651500,24000

x2190

36516.1

x

36516.16.1 2190

12190 x

3651000214637.1

x

%83.707834.0 x

Example 11Example 11I want to retire with $1,000,000 in thirty I want to retire with $1,000,000 in thirty

years. I can get a rate of 7%. How much years. I can get a rate of 7%. How much will I need to invest now if it is will I need to invest now if it is compounded monthly? compounded monthly?

30*12

12

07.011000000

x

360005833.11000000 x

86.205,123$x

Example 11 (contExample 11 (cont’’d)d)

You Try…What about if “Continuously”?You Try…What about if “Continuously”?

30*07.1000000 xe

1.21000000 xe

43.456,122$x