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MATA32H3 Midterm: Term Test 1 - Fall 2017Premium


Department
Mathematics
Course Code
MATA32H3
Professor
Raymond Grinnell
Study Guide
Midterm

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University of Toronto
Scarborough
MATA32H3
Calculus for Management I
Fall 2017
Term Test 1
Exam Guide

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Compound Interest
Compound Interest
Deals with money, time and interest
1 key equation:
𝑆=𝑃(1+π‘Ÿ)!
P ! Principle
β€’Money invested now
r ! Periodic interest rate
β€’Interest paid per compounding period
β€’Stated in percentage or annual rate
oNOTE: Must convert percentage into a decimal
n ! Number of compounding periods
β€’Counts number of times interest is paid on investment
S ! Compound Amount
β€’Amount of money we have at the end
Interest is paid exactly at the end of compounding periods
β€’Only at the end of compounds
**For one compound period, the principle (P) plus the interest on that
1st Compound ! 𝑃1+Pr =!𝑃!(1+π‘Ÿ)
2nd Compound ! 𝑃!1+π‘Ÿ+!𝑃!1+π‘Ÿβˆ™π‘Ÿ=𝑃!1+π‘Ÿ1+π‘Ÿ=𝑃!(1+π‘Ÿ)!
3rd Compound ! 𝑃!(1+π‘Ÿ)!+!𝑃!1+π‘Ÿβˆ™!π‘Ÿ=!𝑃!1+π‘Ÿ!βˆ™1+π‘Ÿ=𝑃!(1+π‘Ÿ)!
4th Compound ! 𝑃!(1+π‘Ÿ)!
NOTE: IN APPLICATIONS
β€’We often have APR β€˜a’ (Annual Percentage Rate) and frequency β€˜k’ of
compounding; and
oHow often interest is paid in a year
β€’A time β€˜t’ period
!
Khl
Compounding periods
End Start
Fixed Period of time
n!
P $
(Principle!+!Interest!on!Principle)!

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Legend:
a ! APR for period as follows
k ! Compounding frequency per year (annually)
t ! Number of years for a particular period
We have that…
!
!= r and 𝑛=π‘˜βˆ™π‘‘
Thus,
𝑆=𝑃!(1+!
π‘Ž
π‘˜)!βˆ™!
Example 1: Consider the following β€œinvestment scheme” invest $10,000 at 3.05% APR
compounding monthly for a period of 5 years.
a) Find compound amount
𝑆=10,000!1+!.!"!#
!"
!"
β‰ˆ11,645.17
b) Periodic rate (interest at the end of each month)
π‘Ÿ=
π‘Ž
π‘˜
=
0.0305
12
β‰ˆ0.002542!(0.2542%)
c) Compound Interest
Difference between beginning amount and end amount
π‘†βˆ’π‘ƒ
β‰ˆ1,645.17
Return due to interest
=
1645.17
10000
βˆ™100
β‰ˆ16.4517%
(Over 5 years accumulated)
d) How long does it take to earn $15,000
** Express terms in units of compounding (i.e. from this example, months)**
5 years
$10,000
$11,645.17 X 60
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