Calculating Interest and Exponential Growth 1. Calculatin g 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
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Calculating Interest and Exponential Growth 1. Calculating Simple Interest A dollar today is worth more than a dollar tomorrow Because of this cost, money.
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Calculating Interest and Exponential Growth
1. 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
Calculating Interest and Exponential Growth
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)
Calculating Interest and Exponential Growth
1. Calculating Simple Interest
• Ex 1:
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
0.0455
5000 + 5000(0.045)(5)
5000 + 1125 = $6,125
Calculating Interest and Exponential Growth
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
0.067
10000 + 10000(0.06)(7)
10000 + 4200 = $14,200
Calculating Interest and Exponential Growth
2. 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 + 0.06) = (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%
Calculating Interest and Exponential Growth
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.
Calculating Interest and Exponential Growth
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 10,000(1.06)
=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 = $15,036.30Mr. Vasu has $15,036.30 after seven years.
Calculating Interest and Exponential Growth
2. Calculating Compound Interest
• Compound Interest Formula
(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
Calculating Interest and Exponential Growth
2. 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 = 6,000r = 0.05t = ?
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
Calculating Interest and Exponential Growth
2. 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 = 24,000r = 0.045t = 10
A = 24000(1.045)10
A = $37,271.27Ganiu has $37,271.27 after 10 years.
• How much did he earn in interest alone?
$37,271.27 – 24,000 =
Ganiu earned $13,271.27 in interest.
Calculating Interest and Exponential Growth
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 = $42,770.44.
Calculating Interest and Exponential Growth
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
Calculating Interest and Exponential Growth
4. 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)
Calculating Interest and Exponential Growth
• Ex 9: Henok invests $6,000 and earns 5% per year. How much will he have after six years
A(t) = 6000(1 + .05/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(1 + 0.05/2)(2●6)
A = 6000(1.025)12
A = $8,069.33
• if interest is compounded quarterly (n=4)?
A = 6000(1 + 0.05/4)(4●6)
A = 6000(1.0125)24
A = $8,084.11
4. Calculating Compound Interest
w Periodic Compounding
Semiannual
Quarterly
Monthly
Daily
Calculating Interest and Exponential Growth
• Ex 9: Henok invests $6,000 and earns 5% per year. How much will he have after six years
A(t) = 6000(1 + .05/n)6n • if interest is compounded monthly (n=12)?
A = 6000(1 + 0.05/12)(12*6)
A = $8,094.11
• if interest is compounded daily (n=365)?
A = 6000(1 + 0.05/365)(365●6)
A = $8,098.99
4. Calculating Compound Interest
w Periodic Compounding
Semiannual
Quarterly
Monthly
Daily
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
Henok’s investment gets bigger if interest compounds more frequently
Calculating Interest and Exponential Growth
Ex 10: Homer invests $1,000 at 10% for nine years
P = 1,000 r = 0.10 t = 95. Simple vs. Compound Interest
Linear vs. Exponential Functions
Simple Interest
Asimple = P + Prt
A = 1000 + 1000(0.10)(9)
Asimple = $1,900
Year A(t)
0 1,000
1 1,100
2 1,200
3 1,300
4 1,400
5 1,500
9 1,900
Compound Interest (annual)
Acompound = P(1+r)t
A = 1000(1.10)9
Acompound = $2,357.95
Year A(t)
0 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
Calculating Interest and Exponential Growth
6. 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
Calculating Interest and Exponential Growth
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?
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?
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 ?