Interest Rates and Units
First Example (Interest Rates Without Units). The amount
A in a simple interest bank account at t years after a
deposit of an amount (principal) P is given by
where r is the annual interest rate. The
number t of years may be a whole number or a whole number
plus a fraction. The interest rate r is given here as a
percentage, e.g. 3.5%. The rule 100% = 1 implies that
1% = 0.01 = [1/100].
Now the slope of the above graph is
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|
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DA
Dt |
= |
A2-A1
t2-t1 |
= |
(P+Prt2)-(P+Prt1)
t2-t1 |
= |
| |
|
|
|
The units of this slope m = Pr is units A over the
those of the quantity, more precisely the number t.
Recall t is the number of years. Now if the amount
of money A is measured in dollars
then the slope will have units [dollars /1].
To devise a second approach, let
be a measure of time (with units) and
let i = r/
yr = [(r)/yr ]. Then
|
A = P+Prt = P+P |
r
year
|
·(t year) = P+PiT |
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Second Example (Interest Rate With Units).
The amount A in a simple interest bank account at
time T since the deposit of the amount (principal) P is
also given by
where
i is the interest rate per year or per annum. The units of i are
a percentage per year, that is i = [1%/\yr ].
This yields a graph similar, very similar, to the previous one.
Now the slope
|
|
|
|
|
DA
DT |
= |
A2-A1
T2-T1 |
= |
(P+PiT2)-(P+PiT1)
T2-T1 |
= |
| |
|
|
|
The units of this slope m = Pi is units of money over units of
time. If time is measured in terms in years and money in dollars,
then this slope will have units [\dollars /\year ] or
dollars per year.
Third Example (Interest Rate Without Units). With interest compounded annually, an
initial deposit P grows to amount A = P(1+i)n after n
years where i is the annual or yearly interest rate. The
interest rate i is usually given as a percentage. Compute the
final amount A in the case where an initial deposit of $100.00
compounds at 4% per year, for 3 years.
In the requested computation, i = 4% = 0.04, P = $100.00 and n = 3.
Therefore
A = P(1+i)n = $100.00(1+4%)3 = ¼.
REMARK. Daily,
weekly, monthly and annual interest rates are given by percentages or pure
numbers (the unit free approach). For example, a yearly or annual interest rate
of 5% is given by the number 5% = 5 ×[1/100] = [5/100]. Second, interest rates
per day, week, month or year refer to a percentage over a period of time. With
the latter, for example [5%/year] represents a 5% per year interest.
FOOTNOTE: Some conventions like these are needed for the consistent use of
units in computations. Without any such conventions, the use of units in
financial computations will depart from the practice in technology and
science.
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Why Slopes
and
More Math
understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
|
Content Guide
Foreword
Chapter Descriptions
1. Introduction
Preview -why slopes in 1983
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review (optional)
9 On Calculus Studies
11 Slope of Slope
13 Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17 What is Area
18 Integration
18 Area Calculation
18 Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc
Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity Motions
10 Slopes without Units.
10 Units & Slopes
10 Units in Cost vs. Quantity
10 How Units Appear
10 Unit Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units
Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area.
Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
Theorem
Integration
& Lipschitz
Continuity
These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
Range Theorem is a postscript,
not in printed version.
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