1. Suppose models the number of m&m’s in a jar after time t. How long will it take for the number of m&m’s to fall below 35? a) Determine t algebraically

b) Verify your answer using graphing calculator.

3. Solve for t :

)1log(1250)( ttf

t)6(.123,102000

Warm-up

Section 4.7 Compound Interest

1. Given a compounded interest rate,a) determine how much you have after a number of yearsb) determine how long it takes to reach a certain amount

2. Given a continuously compounded interest ratea) What rate will allow an amount to double in a certain number of years.b) How long would it take to reach a certain amount.

Suppose your parents decide to give you a gift of money when you are born. They choose to give it to you as either:

option 1: $10 per week, for the rest of your life. You do not invest the money, but keep it safely hidden in your piggy bank.

option 2: A one-time lump sum of $5000 placed in a savings account with an interest rate of 8% compounded annually.

Suppose your parents decide to give you a gift of money when you are born. They choose to give it to you as either:

option 1: $10 per week, for the rest of your life. You do not invest the money, but keep it safely hidden in your piggy bank.

option 2: A one-time lump sum of $5000 placed in a savings account with an interest rate of 8% compounded annually.

Which option would you pick if you can access the money when you turn 18?

Which option would you pick if you can access the money when you turn 18?

1. What is Compound Interest ?

Interest: Amount charged (or earned), expressed as percentage.

Compound Interest: Interest is applied to both the original and the interest that has already been made..

For example. year 1: 3% interest on $500 is (0.03)x(500) = $15.00year 2: interest is applied to previous amount and

interest: (0.03) x ($515) = $15.45

2. Determine the Future Value

Future Value is the amount in the account after a certain number of years

Suppose $5000 is compounded annually (once/year) at a rate of 8%.

# of years the

investment has been in

the bank

Balance at the start

of the current

year

Interest earned for the year (8

%)

Balance at the end of the current yearPrevious Amount +

Interest

1 5000.00 (5000)(.08) = 400

5000 + 400 = $5400

2 5400.00 (5400)(.08) = 432

5400 + 432 = $5832

3 $5832 (5832)(0.08) =466.56

5832+ 466.56 = $6298.56

4 $6298.56 $503 $6801.84

5 $6801.84 544.14 $7345.98

Let P = Principal (the amount initially invested)

1.08P(1+.08) =(1.08)2 P

1.082P(1+.08) =(1.08)3P

1.083P(1+.08) =(1.08)4 P

1.084P(1+.08) =(1.08)5 P

(1.08)2 P 0.08(1.08)2 P

(1.08)3 P

0.08(1.08)4 P

1.082 + 0.08(1.08)2 P

1.083 + 0.08(1.08)3 P

(1.08)4 P

0.08(1.08)3 P

1.084 + 0.08(1.08)4 P

P+.08P

5. Interest Compounded n times per year

represents the amount due after t years on a principal P invested at an interest rate r, compounded n times per year.

represents the amount due after t years on a principal P invested at an interest rate r, compounded n times per year.

AP 1r

n

nt

P = Principal invested (original amount)A = Amount after t years

n = # of times interest compounded per yeart = # of years

r = Interest rate (use decimal for the percentage)

What does r/n represent ?

What does the exponent nt represent?

5. Interest Compounded n times per year

Definitions:

Payments may be posted:annually: once/yearsemi-annually: twice/yearquarterly: four times/yearmonthly: 12 times/yeardaily: 365 times/year

1. Determine an amount (A)

Quarterly Compounding:

Suppose your parents invest $5000 at your birth into an account with 8% interest compounded quarterly

Quarterly Compounding:

Suppose your parents invest $5000 at your birth into an account with 8% interest compounded quarterly

How much money will you have at 18?How much money will you have at 18?

How long does it take for your parents investment of $5000 to triple?

How long does it take for your parents investment of $5000 to triple?

2. Determine how long (t)

3. Determine an effective rate (r)

Suppose you want to invest into an account that compounds interest monthly.

Suppose you want to invest into an account that compounds interest monthly.

What rate would allow your investment to double in 18 years?

What rate would allow your investment to double in 18 years?

How much should your parents invest your birth if you are to be a millionaire by the time you are 18.?

Suppose they invest at 12% compounded monthly,

How much should your parents invest your birth if you are to be a millionaire by the time you are 18.?

Suppose they invest at 12% compounded monthly,

4. Determine present value (P)

Your first credit card charges 12.49 % interest to its customers and compounds that interest monthly. Within one day of getting your first credit card, you max out the credit limit by spending $1,200.00 . If you do not buy anything else on the card and you do not make any payments, how much money would you owe the company after 6 months?

5. Determine a debt

6. Continuous Compounding Formula

represents the amount due after t years on a principal P invested

at an interest rate r, compounded continuously

represents the amount due after t years on a principal P invested

at an interest rate r, compounded continuously

APert

Gives us an upper bound on regular compound interest.

The Shampoo Formula

Suppose your parents $5000 investment is compounded continuously at 8%, what will the final value be in 18 years?

3 types of problems you will solve…

Type 1: Find time.Example 1: How long would it take for an investment to double if it invested at 8% compounded quarterly?

Type 2: Find rate.Example 2: At what annual interest rate would an investment compounded continuously double in 18 years ?

Type 3: Find Present Value. Determine how much you would need to invest to make a given amount of money after n years.

Example 3: How much would a new parent need to invest at a rate of 8% compounded quarterly if the child is to be a millionaire by the time they are 18?

3. Find an algebraic formula

How does a bank determine interest?

Is there an easier way to find out how much money you would have at 18 years old?40 years old?

Now complete table 2 and look for a formula for computing compound interest.Do you see a pattern?