1. 1. A dollar today is worth more than a dollar tomorrow
Calculating Simple Interest
A dollar today is worth more than a dollar tomorrow
Because of this cost, money earns interest over time
If you are borrowing, you will pay interest
If you are lending/investing, you will earn interest
Simple Interest
interest on an investment that is calculated once per period, usually annually
on the amount of the capital alone
interest that is not compounded

2. 1.
Calculating Simple Interest
Principal is the initial amount invested or borrowed (the loan amount or how much you save)
Simple Interest Formula:
P = Principal
r = Annual Interest Rate
t = Number of periods (usually years) the money is being borrowed
Simple Interest = Principal times interest times years
Simple Interest = P(r)(t)
Total Owed = P + P(r)(t)

3. 1. Ex 1: Calculating Simple Interest
Mr. Vasu invests $5,000. His annual interest rate is 4.5% and he invests his money for 5 years. What is the total in his account after this time?
P =
r =
t =
Total = P + P(r)(t)
$5,000
0.045 5
(0.045)(5)
= $6,125

4. 1.
Calculating Simple Interest
Ex 2: Trayvond saves $10,000 to pay for a car. His earns 6% on his investment and invests his money for 7 years. What is the total in his account after this time?
P =
r =
t =
Total = P + P(r)(t)
$10,000
0.06 7
(0.06)(7)
= $14,200

5. 2. Constant Multiplication Factor and Interest Rate
Calculating Compound Interest
Constant Multiplication Factor and Interest Rate
The constant multiplication factor = (1 + r)
r = annual interest rate (as a decimal)
Annual interest rate and growth rate are the same thing
Ex 1: If you earn 6%, what is the constant multiplication factor: ( ) = (1.06)
Ex 2: If the CMF is 1.5, what is the growth rate?
1.5 = 1 + r; r=0.50, which is 50%

6. 2.
Calculating Compound Interest
Ex 3: Mr. Vasu invests $10,000 in an account that earns 6% annual interest that compounds annually. How much will he have in 2 years:
Year 0
Year 1
Year 2
$10,000
10,000(1.06)
= 10,600
10,600(1.06)
= $11,236
Mr. Vasu has $11,236 after two years.

7. 2.
Calculating Compound Interest
Ex 3: Mr. Vasu invests $10,000 in an account that earns 6% annual interest that compounds annually. How much will he have in 2 years:
Year 0
Year 1
Year 2
$10,000
10,000(1.06)
= 10,600
10,600 (1.06)
= $11,236
=10,000(1.06)1
=10,600
10,000(1.06)(1.06)
=10,000(1.06)2
=11,236
Ex 4: Mr. Vasu invests $10,000 in an account that earns 6% annual interest that compounds annually. How much will he have in 7 years?
10,000(1.06)7 = $15,036.30
Mr. Vasu has $15, after seven years.

8. 2. Compound Interest Formula Calculating Compound Interest
(Exponential Growth Function)
A = P(1 + r)t
A = Future Value or Final/Ending Value
P = Principal/Initial Value and Y-Intercept
r = Annual Interest Rate/Growth Rate
t = Years

9. 2. Ex 5: Aaliyah invests $6,000 and earns 5% per year.
Calculating Compound Interest
Ex 5: Aaliyah invests $6,000 and earns 5% per year.
Write an exponential growth equation for how much money Aaliyah has after t years?
A = ?
P = 6,000
r = 0.05
t = ?
A = 6000(1.05)t
How much will she have after six years if interest is compounded annually?
t = 6 years
A = 6000(1.05)6
A = $8,040.57

10. 2. Ex 6: Ganiu invests $24,000 for ten years at 4.5%.
Calculating Compound Interest
Ex 6: Ganiu invests $24,000 for ten years at 4.5%.
How much does he have in his account after the ten years?
A = ?
P = 24,000
r = 0.045
t = 10
A = 24000(1.045)10
A = $37,271.27
Ganiu has $37, after 10 years.
How much did he earn in interest alone?
$37, – 24,000 =
Ganiu earned $13, in interest.

11. 3.
Analyzing
Compound Interest Formula
Ex 7: The following function represents how much money Lashawn has in her account after t years: A(t) = 6,500(1.17)t
What is the y-intercept?
The coefficient is 6,500, so the y-intercept is 6,500.
What is the constant multiplication factor?
The base is 1.17, so the CMF is 1.17.
How much money does Lashawn invest at the beginning into her account?
The y-intercept is where t=0, the initial value. So, she started with $6,500.
What is the annual interest rate?
CMF = (1+r) = 1.17, so r = 0.17 or 17%
How much Lashawn have after twelve years?
A(t) = 6,500(1.17)12 = $42,

12. 3.
Analyzing
Compound Interest Formula
Ex 8: The following function represents the number people living the Chinese city of Kunming: C(t) = 50,000(2)t
What is the y-intercept?
Coefficient is 50,000, so the y-intercept is 50,000.
What is the constant multiplication factor?
The base is 2, so the CMF is 2.
How many people were initially in Kunming?
The y-intercept is where t=0, the initial value. So, the initial population was 50,000 people.
What is the annual growth rate in population?
CMF = (1+r) = 2, so r = 1 or 100% growth
How many people in Kunming after 10 years?
C(t) = 50,000(2)10 = 51,200,000 people

13. 4. A = P(1 + r/n)nt Compound Interest Formula
Calculating Compound Interest
w Periodic Compounding
Semiannual
Quarterly
Monthly
Daily
Compound Interest Formula
with Periodic Compounding
A = P(1 + r/n)nt
A = Future Value or Final/Ending Value :
P = Principal/Initial Value and Y-Intercept
r = Annual Interest Rate/Growth Rate
t = Years
n = Periods per Year (1, 2, 4, 12, 365)

14. 4.
Calculating Compound Interest
w Periodic Compounding
Semiannual
Quarterly
Monthly
Daily
Ex 9: Henok invests $6,000 and earns 5% per year. How much will he have after six years
A(t) = 6000( /n)6n
if interest is compounded annually (n=1)?
A = 6000(1.05)6
A = $8,040.57
if interest is compounded semi-annually (n=2)?
A = 6000( /2)(2●6)
A = 6000(1.025)12
A = $8,069.33
if interest is compounded quarterly (n=4)?
A = 6000( /4)(4●6)
A = 6000(1.0125)24
A = $8,084.11

15. 4.
Calculating Compound Interest
w Periodic Compounding
Semiannual
Quarterly
Monthly
Daily
Ex 9: Henok invests $6,000 and earns 5% per year. How much will he have after six years
A(t) = 6000( /n)6n
if interest is compounded monthly (n=12)?
A = 6000( /12)(12*6)
A = $8,094.11
if interest is compounded daily (n=365)?
A = 6000( /365)(365●6)
A = $8,098.99
Henok’s investment gets bigger if interest compounds more frequently
Annually
Semi-Annually
Quarterly
Monthly
Daily
n = 1
n = 2
n = 4
n = 12
n = 365
$8,040.57
$8,069.33
$8,084.11
$8,094.11
$8,098.99

16. 5. Simple vs. Compound Interest Linear vs. Exponential Functions
Ex 10: Homer invests $1,000 at 10% for nine years
P = 1,000 r = 0.10 t = 9
Simple Interest
Compound Interest (annual)
Asimple = P + Prt
A = (0.10)(9)
Asimple = $1,900
Acompound = P(1+r)t
A = 1000(1.10)9
Acompound = $2,357.95
Year
A(t)
1,000 1
1,100 2
1,200 3
1,300 4
1,400 5
1,500 9
1,900
Year
A(t)
1,000
1000(1.1)0 1
1,100
1000(1.1)1 2
1,210
1000(1.1)2 3
1,331
1000(1.1)3 4
1,464
1000(1.1)4 5
1,611
1000(1.1)5 9
2,358
1000(1.1)6

17. 6. Compound Interest Formula with Periodic Compounding
Finding the Initial Value
Of Exponential Growth/Interest
Compound Interest Formula with Periodic Compounding
A = P(1 + r/n)nt
To find the Initial Value, we need to solve for P
We will be given: A, r, n, t

18. 6.
Finding the Initial Value
Of Exponential Growth/Interest
Ex 1: The future value of an investment at the end of five years is $25,000. What is the initial investment if you earned 10% interest, compounded annually?
A = 25,000
P = ?
r = 0.10
n = 1 (annually)
t = 5 (years)
25000 = P( /1)(1*5)
25000 = P( )
= P
$15, = P
This initial value was $15,523.03
Check:
(1.1)5
= 25,000
Go to five decimal places

19. 6.
Finding the Initial Value
Of Exponential Growth/Interest
Ex 2: The future value of an investment at the end of seven years is $35,000. What is the initial investment if you earned 5% interest, compounded quarterly?
A = 35,000
P = ?
r = 0.05
n = 4 (quarterly)
t = 7 (years)
35000 = P( /4)(4*7)
35000 = P( )
= P
$24, = P
This initial value was $24,717.69
Check:
(1.0125)28
= 35,000
Go to five decimal places

20. 6. Finding the Initial Value Of Exponential Growth/Interest Check:
Ex 3: You decide you need $50,000 to go to graduate school in five years. You find an investment that pays 12% interest, compounded monthly. How much money will you need to invest today, to go to graduate school in five years ?
A = 50,000
P = ?
r = 0.12
n = 12 (monthly)
t = 5 (years)
50000 = P( /12)(12*5)
50000 = P( )
= P
$27, = P
This initial value was $27,522.48
Check:
(1.01)60
= 50,000
Go to five decimal places

21. Both are rates per one year7.
What is Annual Percentage Yield?
Comparing
APY vs. APR
The Annual Percentage Rate (APR) is the rate of interest earned per year.
This is the rate that we’ve used in all of our problems so far
The Annual Percentage Yield (APY) is the actual annual percent earned when you calculate for compounding
APY ≥ APR
Both are rates per one year

22. 7.
What is Annual Percentage Yield?
Comparing
APY vs. APR
Ex 1: Annual Percentage Rate (APR) is 10% and interest is compounded annually. What is APY?
r = APR = 0.10
n = 1 (annually)
t = 1 (years)
1 + APY = (1 + r/n)(n*t)
1 + APY = ( /1)(1*1)
1 + APY =
APY = 0.10 = 10%
If annual compounding
APY = APR
Go to five decimal places

23. 7.
What is Annual Percentage Yield?
Comparing
APY vs. APR
Ex 2: Annual Percentage Rate (APR) is 10% and interest is compounded quarterly. What is APY?
r = APR = 0.10
n = 4 (annually)
t = 1 (years)
1 + APY = (1 + r/n)(n*t)
1 + APY = ( /4)(4*1)
1 + APY =
APY = = %
10.38% > 10%
APY > APR bcs of compounding
Go to five decimal places

24. 7.
What is Annual Percentage Yield?
Comparing
APY vs. APR
Ex 3: Annual Percentage Rate (APR) is 10% and interest is compounded daily. What is APY?
r = APR = 0.10
n = 365 (annually)
t = 1 (years)
1 + APY = (1 + r/n)(n*t)
1 + APY = ( /365)(365*1)
1 + APY =
APY = = %
10.52% > 10%
APY > APR bcs of compounding
Round to five decimal places

25. 7.
What is Annual Percentage Yield?
Comparing
APY vs. APR
Why does it matter?
Loans will advertise APR (even though you pay higher APY because of compounding)
Investments will advertise APY (since it is higher than APR)