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it 's all about compound interest
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R e v i e w !• Interest: I= Prt • Simple interest: A = P + Prt = P(1 + rt)• Compound Interest: A = P(1 + r)t
• Other compounding periods: semiannually(2), quarterly(4), monthly(12), weekly(52), daily(365)…
mt
m
rPA
1
You deposit $10000 in an account that pays 12%
annual interest. Find the balance after I year if the interest is compounded with the given frequency.a.Annuallyb.Quarterlyc.Monthlyd.Weeklye.Daily
a) annually b) quarterlyc)monthly
d) weekly e.) daily
A=10000(1+ .12/1)1x1
= 10000(1.12)1
≈ $11200
A=10,000(1+.12/4)4x1
=10000(1.03)4
≈ $11225.09
A=10,000(1+.12/365)365x1
≈10,000(1.000329)36
5
≈ $11,274.75
A=10,000(1+.12/12)12x1
=10000(1.01)12
≈ $11268.25
A=10,000(1+.12/52)52x1
=10000(1.00231)52
≈ $11273.41
A=P(1+r/m)mt
How Frequent?
Compounded annually, quarterly, monthly, weekly or daily… ?
C O N T I N U O U S
C O M P O U N D INTEREST
A=P(1+r/m)mt
How many periods?
Construct a new formula
mt
m
rPA
1
A Little Math Trick
11
rtk
Pk
As m gets large...
Call it “e”
Continuous Compound Interest
Note that here the exponent is “ rt ”, NOT “ mt ” as in the earlier formula.
Compare
How oftencompounded Computation
yearly
semi-annually
quarterly
monthly
weekly
daily
hourly
every minute
every second
e = 2.718 281 828 459 …
• Just like π, e is an irrational number which can not be represented exactly by any finite decimal fraction. • However, it can be approximated by
for a sufficiently large x
e
e
ex
x
11
A = P e rt
Example
Another Example
1. If $ 8000 is invested in an account that pays 4% interest compounded continuously, how much is in the account at the end of 10 years.
2. How long will it take an investment of $10000 to grow to $15000 if it is invested at 9% compounded continuously?
1. If interest is compounded continuously at 4.5% for 7 years, how much will a $2,000 investment be worth at the end of 7 years.
2. How long will it take money to triple if it is invested at 5.5% compounded continuously?
If $ 8000 is invested in an account that pays 4% interest
compounded continuously, how much is in the account at the end
of 10 years.
Formula: A =P ert A= $ 8000 e .04(10)
A= $ 11,934.60
How long will it take an investment of $10000
to grow to $15000 if it is invested at 9% compounded continuously?
Formula: A =P ert 15000 = 10000 e .09t
1.5 = e .09t
Ln (1.5) = ln (e .09t) Ln (1.5) = .09 t So t = ln(1.5) / .09 t = 4.51
It will take about 4.51 years
If interest is compounded continuously at 4.5% for 7
years, how much will a $2,000 investment be worth at the end
of 7 years.
Formula: A =P ert A= $2,000 e .045(7)
A= $ 2,740.52
How long will it take money to triple if it is
invested at 5.5% compounded continuously?
Formula: A =P ert 3P = P e .055t
3 = e .055t
Ln 3 = ln (e .055t) Ln 3 = .055t So t = ln3 / .055 t = 19.97
It will take about 19.97 years
Which function eventually
exceeds the other as x
approaches infinity?
y= 100x30
y= 3.5x
C O M P A R I S O N OF EXPONENTIAL
GROWTH PHENOMENA
y=x3
y=2x
X x3 2x
1 1 2
2 8 4
3 27 8
4 64 16
5 125 32
6 216 64
7 343 128
8 512 256
9 729 512
10 1000 1024
In the long run, exponential growth will always end up
ahead of polynomial growth.
Which function eventually
exceeds the other as x
approaches infinity?
y= 100x30
y= 3.5x
₱50 and increases by ₱50 each week
₱5 and doubles each week
Or
W 0 1 2 3 4 5 6 7 8
1 5 10 20 40 80 160 340 680 1,360
2 50 100 150 200 250 300 350 400 450
₱5 and doubles each week
Or₱50 and increases by ₱50 each week
y= 5(2)w
y= 50 + 50w
Option A: ₱ 1000 would be deposited on Dec. 31st in a bank account bearing your name and each day an additional ₱1,000 would be deposited ( until January 31st)Option B: One penny (.01 ) would be deposited on Dec. 31st in a bank account bearing your name. Each day, the amount would be doubled ( until January 31st )
B(t)= 0.01(2)t
t= time in # of days since
Dec. 31
A(t) = ₱ in
account after t days
t= time in # of days since
Dec. 31
A(t) = ₱ in
account after t days
0 1000 0 .01
1 2000 1 .02
2 3000 2 .04
10 11000 10 10.24
21 22000 21 20,971.52
31 32000 31 21,474,836.48
A(t)=1000t + 1000
Linear function grows by addition and exponential
function grows by multiplication
I. Solve the ff.1. An amount of $2,340.00 is
deposited in a bank paying an annual interest rate of 3.1%, compounded continuously. Find the balance after 3 years.
2. How long will it take $4000 to triple if it is invested at 5% compounded continuously?
II. Compare the ff.a. polynomial and exponential
growth.b. Linear and exponential growth.
God bless!
T H A N K S F O R
L I S T E N I N G ! ! !
Ma’am DianN:)