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.

15. Multiple Term Ratios - Three Term Ratios to be precise

We read the triple ratio  a : b :c as a to b to c. We further write 

a : b: c  ::  A: B: C

to  say two triple ratios  a : b: c  and   A: B: C are equal or equivalent when and only when 

The foregoing hold when and only when

The latter is equivalent to saying the three fractions 

all have the same value, call it k, which in turn is equivalent to 

A = k a,  B = k b and C = k c

for the common value k.  The foregoing shows the following statement (theorem) holds.

Theorem: Two triple ratios a : b: c  and   A: B: C are equivalent, that is,

a : b: c  ::  A: B: C

when and only when (i)  there is a single constant k such that 

A = k a,  B = k b and C = k c

 (ii) the three fractions 

A a

B b

and

C c

all have the same value; the value k above; 

(iii)

a b

=

A B

and

b c

=

B C

and (iv)  a:b  = A: B and b:c = B: C  (here we could write :: in place of = sign)

---

Remark:  The three fractions 

A a

B b

and

C k

all have the same value when and only when

A a

=

B b

=

C c

 

Exercise: Generalize the foregoing explanations to cover multiple term ratios  a:b:c:d: :: A:B:C:D  

Example:  If the sides of a triangle ABC have lengths a: b: c = 3: 4: 5 then the triangle ABC is similar to the 3-4-5 right triangle. For further examples of multiple term ratios, see the explanation of similar triangles in the geometry before coordinates site area.

Further examples of multiple (term) ratios are provided by  mixtures as is or scaled-up in cooking (cake production), in construction (concrete mixing) and in chemical reactions. 

Remark - A related concepts in advanced mathematics.  In the coordinization of rays and lines in  projective geometry,. two non-zero points (x,y,z) and (X,Y, Z) belong to the same ray  when and only when (X,Y,Z) = (kx, ky, kz) for some positive real number k while  twonon-zero points (x,y,z) and (X,Y,Z) belong to the same line  when and only when (X,Y,Z) = (kx, ky, kz) for some non-zero real number k. 

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