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
 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;

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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.

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.

 17. Proportional Situations (Examples)

This lesson introduces rates and  proportionality constants with units. Rates are in fact proportionality constants. Multiplication and division of terms or quantities involving units appears in calculations below.  

Definitions

Direct Proportionality (Direct Variation):  A number or quantity Z is directly proportional to another quantity X in several circumstances when and only when the quotient    Z ÷ X = Z/X has a constant value k,.or equivalently,  there is a constant k, a number or quantity, such that Z  = k X.  That is, in each instance where we find or measure the value of X, the value of Z will be kX. 

Joint Proportionality. A number or quantity Z is directly proportional to  quantities X and Y in several circumstances when and only when there is a constant k, a number or quantity, such that Z  = k XY  That is, in each instance where we find or measure the value of X and Y, the value of Z will be kXY 

Inverse Proportionality (Inverse Variation):  A number or quantity Z is inversly proportional to another quantity X in several circumstances when and only when the product    Z*X  has a constant value k,.or equivalently,  there is a constant k, a number or quantity, such that Z  = k/ X.  That is, in each instance where we find or measure the value of X, the value of Z will be k/X. 

Joint Proportionality. A number or quantity Z is directly proportional to  quantities X and inversely proportional Y in several circumstances when and only when there is a constant k, a number or quantity, such that Z  = k X/Y  That is, in each instance where we find or measure the value of X and Y, the value of Z will be kX/Y 

Observe if Z = kXY then  X = (1/k)Z/X is directly proportionally to Z and inversely proportional to Y while Y =(1/k)Z/Y is is directly proportionally to Z and inversely proportional to X. 

Observe if Z = kX/Y then  X = (1/k)ZY is directly proportionally to Z and Y while Y =(1/k)X/Z is is directly proportionally to X and inversely proportional to Y 

Working with multiple Proportions, 
Algebraic Perspective of Proportional Situations

In situations involving multiple proportionalities, amounts are proportional to each other and to any  linear function of the amounts in questions (in which the coefficients are fixed and positive). 

Assume X, Y and Z are all nonzero.  In 3D projective geometry, thepoint (X,Y,Z) with is equivalent to another (x,y,z) when and only when X:Y:Z = x:y:z when and only when the three (cross) ratios  x/X  y/Y and z/Z have the same value k when and only when there is a constant k such that x =kX, y=kY and z =kZ when and only when corresponding internal ratios are equal: that is when and only when  X:Y = x;y, Y:Z = y:z and Z:X = z:x  See the earlier discussion of multiple ratios. 

Now the equations  x =kX, y=kY and z =kZ  can be used backwards and forwards. If k is unknown, it can be found given from one of the equations  Then it can be use in the other two. 

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