|
YOU are better than YOU think. Show
yourself how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
-/[]\-
||
_ / \
||||||||||||||||||||||||||||
What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
| |
www.whyslopes.com
Volume 2, Three Skills
for Algebra
Even if you have seen a site topic, look through site
coverage. It will often, if not always, give a
unique viewpoint not seen your earlier education as a student or instructor.
site entrance Road
Safety Message Proper
Use of Equal Sign
24 Your Money
[ Back ] [ Up ] [ Next ]
Chapter 24
Personal Money, Investment
and Pension Computations
Previous: 23 Summation Notation
Geometric sums appear in calculations involving (i) loan and
mortgage repayments, and (ii) the purchase cost for a pension plan, here an
annuity with a constant monthly payments This chapter indicates how and why
geometric sums appear in money matters.
1. Two Calculations
A Multiple Investment Calculation
Initial Example: John's investments earn 6 percent
compounded monthly. His investment began with the initial amount of 5000
dollars. Then
- 10 months after his initial investment, he invests another 3000 dollars,
- 2 years after his initial investment, he invests another 2000 dollars,
- 30 months after his initial investment, he invests another 6000 dollars,
- 40 months after his initial investment, he invests another 3500 dollars,
Find the total value of his investments after five years.
Solution: Five years equal sixty months. The monthly
interest rate is i = [6%/12] = 0.5% = 0.005.
- His first 5000 dollars stays invested for five years = sixty months. It
grows to 5000(1+.005)60 dollars.
- His second investment of 3000 dollars is invested for 60-10=50 months.
Thus at the end of five years, it has grown to 3000(1+.005)50
dollars.
- His third investment of 2000 dollars is invested for sixty months minus
two years, that is, 60-24 months = 36 months. Thus at the end of five years,
it has grown to 2000(1+.005)36 dollars.
- His fourth investment of 6000 dollars is invested for 60-30=30 months.
Thus at the end of five years, it has grown to 6000(1+.005)30
dollars.
- His fifth investment of 3500 dollars is invested for 60-40=20 months. Thus
at the end of five years, it has grown to 3500(1+.005)20 dollars
At the end of five years his investment is worth
S = $5000(1.005)60 + $3000(1.005)50
+ $2000(1.005)36 +
$6000(1.005)30
+ $3500(1.0005)20
This gives
S = $5000·(1.34885) + $3000·(1.283226)
+ $2000·(1.19668) + $6000·(1.1614)
+ $3500·(1.1049)
Therefore his investments grows to
S = $6744.25+$3849.68+$2393.36+$6968.40+$3867.13
Addition now implies
S = 23822.82
dollars. This gives the value of his investments after five years.
1.1 A Debt Calculation - Many Amounts
Borrowed
Second Example: Fred can borrow money at 6 percent
compounded monthly. He initially borrows 5000 dollars. Then
- 10 months after his initial loan, he borrows another 3000 dollars,
- 2 years (24 months) after his initial loan, he borrows another 2000
dollars,
- 30 months after his initial loan, he borrows another 6000 dollars,
- 40 months after his initial loan, he borrows another 3500 dollars,
Find the total amount of the debt after five years.
The solution of this second example involves exactly the same numbers and
computations as the solution first example. For instance, the initial amount of
5000 dollars grows to a debt of
5000(1.005)60 = 6744.25
dollars after five years. Likewise, the second debt of 3000 dollars grows to
a debt of
3000(1.005)60-10 = 3000(1.005)50
= 3849.68
dollars. This second example shows that the same arithmetic or algebra may
give the solution of two different problems - recycled mathematics.
Chapter Sections: [Multiple Investments and Loans (Debts] [ 24 Periodic Deposits ] [ 24 Account Tracking ] [ 24 Pension Plans ]
| |
www.whyslopes.com
Volume 2, Three Skills for Algebra -
Preview, starter & further lessons for logic and algebra
to (i) improve work & study skills; (ii) to to ease or avoid
algebra (math) fears & difficulties; and (iii) to fill gaps in the
exposition of mathematics.
Foreword, Chapters and Appendices follow.
Foreword
1. Introduction
2. Implication Rules
3. Chains of Reason
4. Romeo and Juliet
4. Induction Mathematical
5 Knowledge Islands
6 Old Language
7 Arith Skill Check
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable
9. Algebra Talk
10 Two More Skills
11 Why Shorthand
12 Shorthand Usage
13 What's Next
14 Compound Interest
15 Linear Equations
PS I. Distributive Law
PS II. Polynomials
16 Painless Proofs
17 Pythagoras
18 Rules of Algebra
19 Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums
23 Summation Notation
24 Your Money
25 Induction & Recursion
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
A. Advice For Learning
Words Before Symbols:
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
Complex number: starter lesson
Solving Linear Equations:
A. Letters and Lengths
B. & C. Solving Linear Eq'ns
with stick diagrams.
(i) x + 20 = 29
(ii) 2x + 5 = 20
(iii) 3x + 10 = 32
(iv) 5a + 16 = 3a+ 24
(v) (½)x + 8 = 24½
(vI) (¾)a + 16 = (¼)a+ 24
(vii) (¾)q + 17 = 32
(viii) 13 =[2/3]x +7 twice
(x) Animated Examples
(i) Integral Coefficients (A)
(ii) Integral Coefficients (B)
(iii) Fractional Coefficients
(iv) With
Parameters
Problem Solving with Linear
Equations in one or many
unknowns, and in essentially
one unknown - Symbols before
words.
C. Solving Linear Eq'ns
without
Stick Diagrams
D.
Problems in
essentially one unknown
E: 2D Systems - Sub Methods.
F. Larger Systems
|