Simple and Compound Interest

Lesson 9.11

REVIEW:Formula for exponential growth or decay

Initial amount

Rate of growth or decay

Number of times growth or decay occurs

Final amoun

t

REMINDER:Percentage increase is 1 + rate of increase.Percentage decrease is 1 – rate of decrease.

𝒚=𝒂𝒃𝒙

Interest (one type of exponential growth)

Money you earn (savings account, CD, etc.) or pay (car loan, student loan, mortgage)

Percentage of the initial deposit or loan.

Simple Interest Example #1

Calculated ONE time.You lend $100 to your little brother. He will pay you back in one year, with simple interest of 10%. How much will your brother pay you back?

𝒚=𝟏𝟎𝟎+𝟎 .𝟏𝟎 (𝟏𝟎𝟎)

𝒚=𝟏𝟎𝟎+𝟏𝟎

Original amount

Interest

¿𝟏𝟏𝟎Your little brother will pay you back $110.

Simple Interest as Exponential Growth

𝒚=𝟏𝟎𝟎¿𝒚=𝟏𝟎𝟎+𝟎 .𝟏𝟎 (𝟏𝟎𝟎)

𝒚=𝟏𝟎𝟎 (𝟏 .𝟏𝟎 )𝟏Initial amount

Rate of growth or decay

Number of times growth or decay occursFinal

amount

Factor out a 100!

𝒚=𝒂𝒃𝒙

𝟏+𝟎 .𝟏𝟎¿

Compound Interest Calculated at specific intervals (earn interest on interest) Annual interest rate is divided among these intervals.

You put $100 in the bank. The bank also pays 10% annual interest, but this interest is compounded monthly.

After 1 month

After 2 months

After 3 months

𝟏𝟎𝟎+𝟎 .𝟏𝟎𝟏𝟐

(𝟏𝟎𝟎)=$𝟏𝟎𝟎 .𝟖𝟑

𝟏𝟎𝟎 .𝟖𝟑+𝟎 .𝟏𝟎𝟏𝟐

(𝟏𝟎𝟎 .𝟖𝟑 )=$𝟏𝟎𝟏 .𝟔𝟕

𝟏𝟎𝟏 .𝟔𝟕+𝟎 .𝟏𝟎𝟏𝟐

(𝟏𝟎𝟏 .𝟔𝟕 )=$𝟏𝟎𝟐 .𝟓𝟐

Compound Interest Formula

)100(12

10.0100

After 1 month

After 2 months

After 3 months

3)12

10.01(100

𝟏𝟎𝟎 .𝟖𝟑

𝟏𝟎𝟎 .𝟖𝟑+𝟎 .𝟏𝟎𝟏𝟐

(𝟏𝟎𝟎 .𝟖𝟑)

𝟏𝟎𝟎 .𝟖𝟑(𝟏+𝟎 .𝟏𝟎𝟏𝟐

)

𝟏𝟎𝟎 (𝟏+𝟎 .𝟏𝟎𝟏𝟐

)

𝟏𝟎𝟎 (𝟏+𝟎 .𝟏𝟎𝟏𝟐

)(𝟏+𝟎 .𝟏𝟎𝟏𝟐

)

𝟏𝟎𝟎 (𝟏+𝟎 .𝟏𝟎𝟏𝟐 )

𝟐

𝟏𝟎𝟏 .𝟔𝟕

𝟏𝟎𝟏 .𝟔𝟕+𝟎 .𝟏𝟎𝟏𝟐

(𝟏𝟎𝟏 .𝟔𝟕)

𝟏𝟎𝟏 .𝟔𝟕(𝟏+𝟎 .𝟏𝟎𝟏𝟐

)

𝟏𝟎𝟎(𝟏+𝟎 .𝟏𝟎𝟏𝟐 )

𝟐

(𝟏+𝟎 .𝟏𝟎𝟏𝟐

)

Compound Interest Formula

A = Final

amount

P = Principal(initial

amount)

interest rate (r) divided by number

of times compounded in a year (n)

# of times compounded in a year (n) times the #of years (t).

𝒚=𝒂𝒃𝒙

𝑨=𝑷 (𝟏+ 𝒓𝒏 )

𝒏𝒕

𝒚=𝟏𝟐𝟎𝟎¿.𝟎𝟒𝟒¿

𝟐𝟎

Vocabulary Principal: Amount initially deposited or

borrowed.

Intervals for compounding: Annually – Monthly – Weekly – Daily – Quarterly –

1 time each year

4 times each year

12 times each year52 times each year

365 times each year

Check for Understanding Independently annotate your notes

Your notes should be able to answer: What is simple interest? What is compound interest? What are the formulas for each type of interest? Explain how to derive the formula for compound

interest.

Backup

You put $100 in the bank. The bank also pays 10% interest, but this interest is compounded monthly. How much will you earn after 3 months?

3)12

10.01(100 y

xaby

Initial amount Rate of

growth or decay

Number of times growth or decay occurs

Final amount

Compound Interest as Exponential Growth

xaby Initial Amount (amount deposited)

Rate of growth or decay

Total number of times interest calculated

Final amount

part of annual interest paid each time

𝟏+¿

Example #3: You put $1200 in a certificate of deposit account

(CD). This CD pays 4% annual interest, compounded quarterly, for 5 years. How much money will be in your account at the end of 5 years?

xaby

𝑦=1200¿.044¿20

Initial Amount (amount deposited)

Annual interest divided into four intervals

Add 1to keep original amount.

Total number of times interest calculated 4 times a year for 5 years 20 times!

Final amount

𝑦=1200 (1.01 )20¿1200 (1.22)¿ $1464.23

Example #4: Jordan plans to purchase a brand new apple

computer to bring to college. The I-Mac she wants is projected to cost $1500 at the time of her graduation in 2017. She found an account that pays 2.5% interest, compounded monthly. How much money should Jordan deposit this July, to make sure she has enough money to buy the I-Mac in June of 2017?

xaby

1500=¿¿.02512 ¿

36

Initial Amount (amount deposited)

Annual interest divided into twelve intervals

Add 1to keep original amount.

Total number of times interest calculated 12 times a year for 3 years 36 times!

Final amount

𝑃

1500=¿¿.02512 ¿

36𝑃

1500=𝑃 (1.00208333 )36

1500=𝑃 (1.077800061)1500

1.077800061=𝑃

𝑃 ≈$1391.72

Jordan must deposit about $1391.72 this July to have enough money to buy the I-Mac in June of 2015.

Process

1. Determine if you know INITIAL or FINAL amount .

2. Determine the growth rate : Divide ANNUAL rate by the

number of intervals each year. Quarterly: Divide annual by 4. Weekly: Divide annual by 52

ADD !!!

3. Determine number of times interest is compounded:

Number of times per year TIMES number of years

4. Solve for unknown!!

𝟏𝟓𝟎𝟎=𝑷 (𝟏+ .𝟎𝟐𝟓𝟏𝟐 )

𝟑𝟔𝒚=𝒂𝒃𝒙

Extension Question How much would Jordan earn in interest?

Started with:

Ended with:

Earned: ¿108.28

Interest Earned: