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YOU are better than YOU think. Show yourself how:
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-/[]\- Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite. Logic
Mastery 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;
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-/[]\- 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. |
Chapter 14
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Now multiplying a quantity P by a nonzero number4
(or quantity) and then dividing by the same number yields the quantity P,
no matter what P equals. In our situation,
On your first reading of the shorthand solution you may assume the letter
have the values given above. On your second reading, pretend A, i
and n have values not known to you - for instance they might be hidden in
an sealed enveloped. This second viewpoint is a key to algebra.
The second equality follows from the replacement of P×1.251272
by its equal $1350. The initial amount invested was $1078.90 to the nearest
penny or cent. This yields the solution of one problem. In the next solution
method, we will see how to algebraically describe the arithmetic solution of all
similar problems.
P = (P×1.251272) ¸1.251272
= ($1350) ¸1.251272 = $1078.90
ALGEBRAIC SHORTHAND SOLUTION. There is a pattern in the first solution. This
pattern can be followed if we had the same problem again, but with different
numbers. For instance, how would we solve the above problem if the 9% was
replaced by 8%? We will try to capture the pattern using shorthand notation.
This approach is given next. It requires a little more work. But it will give a
formula for solving many similar problems.
First, note the following simple idea. The compound interest formula A = P(1+i)n
says the symbol A and the expression P(1+i)n
both stand for, represent or give the final amount in the account.5
5More precisely we can say:
Here the symbol A and the more complicated expression or symbol P(1+i)n both represent the same quantity. So we take the liberty of using one in place of the other, as convenient.
- The symbol A is shorthand for a quantity.
- The expression P(1+i)n when computed gives the same quantity.
Second, we can assume and use the rule: when B is a nonzero number or
quantity, then
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We now use the above ideas. We will apply this rule with B = (1+i)n. That is, the value of B is given by (1+i)n whenever the latter is computed. So
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using the formula - number substitution. The substitutions A = $1305 , i = [.09/12] = 0.0075, and n = 30 leads to
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Remark (Algebraic Viewpoint). The formula we have found, namely
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More Chapter Sections:
www.whyslopes.com
2. Three Skills for AlgebraForeword, Chapters
& Appendices
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
Real Player Videos
Perfect arithmetic skills with whole numbers & fractions after or besides chapters 1 to 14.
Arithmetic Videos Summary Addition with Decimals Subtraction with Decimals Multiplication with Decimals Fraction Arithmetic Recognizing Primes Long Division for Decimals Square Root Simplification Greatest Common Divisors Least Common Multiples 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 lessonSolving 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
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